(£<0^)0,,-

u: tr v=p

£

ev(K-i®Ei),

v\ tr u=p

® Fi + Fi 0 k-i)e

=

£

0 Fi)

( F i a - j e , -

0 Fi + Fi 0 ifi) £

Bi/(Fi 0 k i ) .

CHAPTER

5

The Symmetries TU, T& of an Integrable U - M o d u l e 5 . 1 . T H E CATEGORY C[

5.1.1. In this chapter we fix i E I. Let C[ be the category whose objects are Z-graded Q(i;)-vector spaces M = 0 n G Z M n provided with two locally nilpotent Q(i>)-linear maps Ei,Fi : M —> M such that (a) Ei{Mn)

C M n + 2 and Fi{Mn) C Mn~2 for all n;

(b) EiFi - FiEi : Mn —• Mn is multiplication by [n]i for all n; the morphisms in the category are Q(?;)-linear maps preserving the Zgrading and commuting with Ei,Fi. This is the same as the category C of integrable U-modules in the case where I = {i} and X = Y = Z with i = 1 E F, i' = 2 E X. On the other hand, in the general case, any object M in C' may be regarded, for any z, as an object of C[ with the Z-grading Mn = ®A6X;(i,A>=n^A- (We forget the action of E j , F j for j ^ i.) : M —> M are given by For M E C[ and p E Z, the operators E?/\p]l*T/\p\i

i f f > 0, and by 0 if p < 0.

5.1.2. For M E C[, let c : M —> M be the Q(t;)-linear map given by vn~l + ?;~ n+1 — 2g = c(x) = ^ ( z ) + * . -1\2 (wi-w< r

+

vn+1 4- ?;~ n-1 — 2 ' , * (wi-^ r

for x € M n . It is easy to check that c is a morphism in C[. For n E Z, we set s n =

,

~ and < =

Proposition 5.1.3. Let n E Z and me n

, ^-i

~ •

N. n

(a) The subspace M (m) = {x E M | £ ^ m ) x = 0} is c-stable and c satisfies the identity (c — s n )(c — s n +2) • • • (c — s n + 2 m ) = 0 on M n ( m ) . (b) TTie subspace Mn[m] = {x E M n | i ^ ( m ) x = 0} zs c-stable and c satisfies the identity (c — — • • • (c — s / n _ 2 m ) = 0 on Mn[m).

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_2, © © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010

40

5.1. The Category C[

41

(c) c : Mn —> Mn is a locally finite, semisimple linear map. (d) If n > 0, the eigenvalues of c : Mn —• Mn are contained in {s n , s n + 2 , s n + 4 , • • • }; if n < 0, the eigenvalues of c : Mn —> Mn are contained in {s-n, S-n-f2j S-n+4, • • • }• Since c commutes with Ei, it leaves Mn(m) stable. We prove (a) by induction on ra. Assume first that m = 0. If x G Mn(0), then by definition, (c—s n )x = 0. Assume now that m > 0 and that the result is already known for m - 1. If x G M n ( m ) , then EiX G Mn+2(m - 1). By the induction hypothesis, we have (c — s n + 2 ) ( c — s n +4) • • • (c — sn+2Tn)EiX = 0. Since = 0. cEi = EiC, it follows that Ei(c - s n +2)(c - s n +4) • • • (c - sn+2m)x Applying Fit we get F i F i ( c - s n + 2 ) ( c - s n + 4 ) • • • ( c - s n + 2 r n ) x = 0. Hence, by the definition of c, we have (c —s n )(c —s n + 2 )(c —5 n+ 4) * • • ( c — s n + 2 m ) x = 0. This proves (a). The proof of (b) is entirely similar. We prove (c). Assume first that n > 0. Since So,Si, s 2 , . . . are distinct elements of Q(t>), we see from (a) that c : Mn(m) —> Mn(m) is locally finite, semisimple (note that Mn = U m > o M n ( m ) ) . Assume next that n < 0. Since Sq, s'_1, S'_2, . . . are distinct elements of Q(v), we see from (b) that c : Mn[m] —> Mn[m] is locally finite, semisimple. It follows that c : Mn —• Mn is locally finite, semisimple (note that Mn = U m >oM n [ra]). This proves (c). Now (d) follows from the earlier points and the identity s'n = S-N. The proposition is proved. 5.1.4. By taking the eigenspaces of c : M —• M we obtain a canonical direct sum decomposition of M (as an object in C'I) into subobjects with the property that c acts on each subobject as scalar multiplication by sm for some m G N. Proposition 5.1.5. Let M G C[ be such that c = sm on M for some me N . (a) If Mn + 0, then n G { - m , -m + 2, -m + 4 , . . . , m). (b) If n and n + 2 belong to {—m, —m + 2, —m + 4 , . . . , m}, then both FiEi : Mn —> Mn and EiFi : Mn+2 —> Mn+2 are given by multiplication by [1 + m / 2 + n/2]i[m/2 - n/2fc ^ 0. Hence Ei : Mn Mn+2 and Fi : n+2 n M —> M are isomorphisms. (c) Let x G Mn where n G {—m, —m + 2, — m + 4 , . . . , m}. There are unique elements z G M~m and z' G M m such that x — j?(m/2~n/2)z' —

E{m/2+n/2)z

(d) In the setup of (c), we have ^(1+m/2"n/2)Z'

F(-l+m/2-n/2)z,

_

^(1+m/2+n/2)z

= E(~l+m/2+n/2)z

and

42

5. The Symmetries T,i e, Ti'e of an Integrable U-Module (d) In the setup of (c), we have ^ l + m / 3 " n / 2 V = E(r1+rn/2+n/2)z

F(-H-m/2-n/2)

, _

p(l+m/2+n/2)

and

(a) follows immediately from 5.1.3(d). If x G M n , we have FiEi(x) = (c - sn)x = (sm - sn)x; if ?/ e M n + 2 , we have EiFi(y) = (c - s'n+2)(y) = (Sm ~ s'n+2)y = (sm - s-n-2)y = («m - sn)y. We have sm - sn = [l+m/2+n/2 [m/2 - n/2]< and (b) follows. Now (c),(d) clearly follow from (b). L e m m a 5.1.6. Let M G C[ and let y G Mn - {0} be such that Eiy = 0. Then (a) n > 0 and F?+ly = 0 and (b) y i E?+1M. (a) is already contained in the proof of 3.5.8. We prove (b). By 5.1.4, we may assume that M is as in 5.1.5. Assume that y = E™+ly' for some y' G M. We may assume that y' G M~n~2 — {0}. By 5.1.5(b), we then have is a non-zero multiple of y'\ in particular, F?+1y ± 0, a that F?+1E?+1y' contradiction. 5 . 2 . FIRST PROPERTIES OF T / E , 7 ^ " E

5.2.1. We fix e = ±1. Let M G C[. We define two Q(v)-linear maps T/ e , T"e : M M (called symmetries) by

7j,e(*) =

T

"AZ)

=

Y ,

E

(-1

a,b,c;—a+b—c=n

)bvl{-ac+b)Fla)E\b)Flc)z,

(-l)bvf~ac+b)Ela)F^Elc)z

for ^ G M n ; the integers a, 6, c are restricted as shown; although the sums are infinite, for any given all but finitely many terms of either sum are zero.

Proposition 5.2.2. Let m > 0 and let j, h G [0, m] be such that j + h = m. (a) If r) G Mm is such that E^ = 0, then T(e(Ft^j)r)) = ( - 1 Y v f ^ ^ F ^ r j . (b) / / £ G M~m

is such that F£ = 0, thenT^E^()

= (-1 )'v;ah+s)Ejh)(.

5.2. First Properties ojTi

We prove (a). Assume that a-b F^E^F^F^

=

e,

T"e

+ c = m-2j.

43

Using 3.4.2, we have

c+ j F^E^F^ri c C + j"

=

Now E^T]

b - c + h' F ^ F ^ ^ E ^ r j . t .i

£

t> o . j

= 0 if 6 ^

thus F(«)F(c+j~

c

|J

b)^

b

_ r c + i i \b -[ J M

c + h'

b

7i" i a

It remains to show that £ ( - W a,byc>0;a — b+c=h—j

•ac+b) c + j' c

a + j' 'K b i a i

= (-i yv?i+l)h

for any j, h > 0. The term corresponding to a, b, c is zero unless a < h, b < a + j and . and [ c + j ]. by c < h, hence the sum is finite. We replace [ a + J ]. = (—l) c [~ J c ~ 1 ] i - The sum becomes e(—ac+c—hj — h) -3 ~ 1'

h-j-ave(a+hj+j)

a> 0

c

i c=0

We replace the sum over c by

a>0

a + j' i h—c

and we obtain

LaJ t L

n

J

If a > h, we have [£]. = 0; if 1 < a < h, we have p " 1 ] . = 0; thus, if a > 0, we have [£]. . = 0. Hence only a = 0 contributes to the sum, which becomes (_1 )h-jve(hj+j)

= (-1

)jvfhj+j\

as required. This proves (a). The proof of (b) is entirely similar (it can also be reduced to (a)).

5. The Symmetries T,i e, Ti'e of an Integrable U-Module

44

Proposition 5.2.3. We have (a) Vi eT'l_e = 7y;_e2Ve = 1 : M (b)

M.

= ( - l j ^ f l j ^ x for all x G M*.

Let m,j,h,r} 5.2.2(a),(b)

be as in 5.2.2(a).

Let £ = F ^ r j .

=

We have, using

= if

v

Since the vector space M is generated by vectors of the form F^rj as above, we have T'l_eT'i e = 1. The identity T'i eT[[_e = 1 is proved in a similar way (or it can be deduced from the previous identity). This proves (a). To prove (b), we may assume that x = F^rj = We have T"ex = h hj+h) j ( - 1 ) vf E\ \ and T'iex = ( - 1 y v f ^ F ^ r j = ( - 1 It remains to note that h = j -f t. Proposition 5.2.4. For any z G Ml we have (a) - v f - 2 ) T l [ _ e ( F i Z ) = E i T ' ^ i z ) ; (b) -v-*T>l-e(Eiz) = FiTZ_ e (z); (c) -v~etT'i

e(Fiz)

=

EiT'^z);

(d) - v f ^ T I J E i z ) = FiTiJz); (e) T('_ez e

M-';

(f) T'i ez € M-K (e),(f) are clear from the definition. To prove (a),(b), we may assume that 2 = F^j)r) = E\h\ where m, j,/i, 77 are as in 5.2.2(a) and £ = F^rj; then h = j + t. We have T " - e ( z ) =

( - 1 )hv~e{jh+h)E\j)t

= (-1

)hv;e(jh+h)F^h)V.

If j = m, then both sides of (a) are zero. So it suffices to prove (a) assuming j < m and h > 0. We have T!:-e(Fiz) = \j +1 }at_e(F^n) = \j + i ] ( ( _ i

= [7 + l i . i y . r f - " ? )

5.3. The Operators

L"

45

and EiT"_e(z) = ( - 1 )hv7e{jh+h)\j + l]iE?+1)b (a) follows. If h = m, then both sides of (b) are zero. So it suffices to prove (b) assuming h < m and j > 0. We have 1X-e(Eiz)

= [h + 1 ]iT>:_e(E?+1)0

= (-l)h+1[h

+

+ l]iF^+1)r); (b) follows. The proof of and Fi(T"_ez) = (-l)hv;eijh+h)[h (c),(d) is entirely similar; alternatively, we can deduce (c),(d) from (a),(b) using 5.2.3. 5.2.5. If M is an object of Cf (an integrable U-module), then we can regard it as an object of C[ as in 5.1.1; hence the symmetry operators T / E , T " E : M —• M are well-defined. All the previous results will hold for these operators. Lemma 5.2.6. Let M be an integrable U-module and let z G M. fi eY and let p! — fi — (fi, i')i G Y. Then (a) T'[_e{K„z)

(b) ru(K„z)

=

Let

K»V[_e{z);

= K„rite(z).

We may assume that z G M A . Then, from the definition, T"_ez and T'iez belong to M ^ W . Hence acts on either of these two vectors as multiplication by (fi, A) — (i,X)(fi,i') = (fi — (fi,i')i,X) = (fi', A). The lemma follows. Proposition 5.2.7. Let M be as in the previous lemma. For any A G X, the operator T"_e : M —> M defines an isomorphism of the X-weight space of M onto the Si(X)-weight space of M. The inverse of this isomorphism is the restriction ofT'i e. This follows immediately from the previous lemma. 5 . 3 . T H E OPERATORS

Let M,N be two objects of C[. By 3.4.3 and 3.5.2(a), the tensor product M N is again an object of if x G Ml,y G Ns, the degree of x ® y is t -f s and 5.3.1.

Ei(x <8>y) = EiX®y

+ v\x ® E{y,

F{(x ®y)=x®Fiy

We define linear maps L", L\ : M N -> M N by

+ v^8FiX y.

46

5. The Symmetries T,i e, Ti'e of an Integrable U-Module

L[{x ® j,) = £„ v?n-1)/2{nhFln)x ® £$">„, where {n}* is as in 4.1.4. Although the sums are infinite, any vector in M N is annihilated by all but finitely many terms in the sum; hence the operators are well defined. These maps are a special case of the operators defined by 0 , 6 of 4.1, in the case I = {«}. (See 4.1.4.) Lemma 5.3.2. We have LJL? = L'(L[ = 1. This follows immediately from the identity 1.3.4(a); alternatively, it can be deduced from 4.1.3. G Ns, we have

Lemma 5.3.3. For x G

FiL'({x ®y) = L'((x ® Fiy + v'FiX y). This can be deduced from the defining identity for 9 (see 4.1.2) or it can be checked directly. Proposition 5.3.4. Let M,N be objects of C\. For any z G M N, we have (*)T!>A(L>!(z)) = (Tl[1®n[l)(z). (In the left hand side of (a) we have the action of T"x on M N; in the right hand side we have the action of T(\ on M and on N.) Assume that (a) holds for z = x y where x G Mm,y G Nn; we claim that (a) must also hold for z' — x y. Using 5.3.3, 5.2.4(a), and our hypothesis, we have Tl[1(L'l(z')) =

Tl[1(FiLKz))

= - v ^ E J ^ m i z ) ) =

®T''a){Z).

On the other hand, again using 5.2.4(a), we have m:i ®

= =

®

T

2

"AF*y)+vtn'AFi*)

-„r 2j> ®

= - v r ' - ' E , ^

E T

® nhv

' "Ay - vfvt'EJZrX

®

QT^Xz).

Our claim is proved. For q > 0, let Zq be the subspace of M N spanned by vectors of the form x (g> F^y where x, y are homogeneous and Eiy = 0. It is clear that

5.3. The Operators L • , L"

47

M N = Zq- Hence to prove (a) it suffices to prove the statements (b),(c) below. (b) The identity (a) holds for any z £ Zo. (c) If (a) holds for all z € Zq, then it holds for all z G Zq+1. We prove (c). Let x G Ml,y G Ms be such that Eiy = 0. By the assumption of (c), we have that (a) holds for z = x® F^y. By the earlier part of the proof, it follows that (a) holds for z\ = [q + \}%x F^q+1^y + v*~2qFiXF^y. Again by the assumption of (c), we have that (a) holds for It follows that (a) holds for z3 = [q + l]iX®F^+1)y; = v*~2qFiX®Flq)y. since [q + l]i ^ 0, it also holds for Thus (c) is proved. It remains to prove (b). Let x G Mm,y G Nn be such that Eiy = 0. From the definition, we have L"(x y) = x y. We must prove that T['x(x y) = T['x(x) T^iy). We may assume that n > 0. We have

Ti:

® y) =

£

(

-

1

1

)

®

y)

a,b,c;—a+b—c=m+n _

, v-a'c-a"c+b'+b"+b%"-b'n+a a"+a"(Tn+2c-2b')

^ a' ,a" ,b' ,b" ,c; —

a'—a"+b'+b"—c=m+n

x E ^ F P E ^ x V E ^ F ^ y .

Here we substitute E^F^y a" + g. We have F^y

a',a" ,b' ,g,c\ —

= [ a ' \ b " + n ] . i ^ 6 ' ' - 0 ? / and we set b" =

— 0 unless g < n. We obtain

a'+b'—c=m+n—g;g
n-g

a' ,b' ,g,c;—a'+b'—c=m+n—g;g
a"

xE^FpElc)x®F^y J2 ( - l f v r ' ^ ' E ^ F ^ E ^ x ( - l ) X n ^ ( n > 2 / a',b',c;-a'+b'-c=Tn ?(n)ot

The proposition is proved.

— T"

r,"

Complete Reducibility Theorems

6.1.1. In this chapter we assume that the root datum is both F-regular Let J5, BV be as in 4.1.2. Applying M(S ® 1) to the identities at the end of 4.2.5, where m : U 0:

equivalently, setting ft< p =

tr u
£6gBi/(-1)

tr

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_2, © Springer Science+Business Science+Business Media, LLC 2010 2011

we have

48

6.1. The Quantum Casimir Operator

49

6.1.2. If M G Chi, then for any ra G M , we have that S7(ra) = f2< p (m) G M is independent of p for large enough p. We can write Q(m) = £ 6 ( - l ) t r ' 6 'v|6|5(6 _ )6* + m and we have k^iEin

(a)

= KiQEi,

nFi = Fikinki,

q.k„ = k11q.

as operators on M. It follows that for m G M A , we have Q.Ei(m) = v~2{iA+i,)EiQim) and ftF<(m) = ^ 2 < i , A > F ^ ( m ) . 6.1.3. Remark. Let us define an isomorphism of Q(i?)-algebras S : U —• U°p p by S(u) = S(u) (S is the antipode.) For any ue U, we have S(u)f2 = QS(u) : M —> M. Indeed, it suffices to check this for the generators Ei,Fi, K^ where it follows from the formulas above. 6.1.4. Let C be a fixed coset of X with respect to the subgroup Z[7] C X. Let G : C —> Z be a function such that A) (a) G(\) - G(\ - i') = i • for all A G C and all t G / . Clearly, such a function exists and is unique up to addition of an arbitrary constant function C —» Z. L e m m a 6.1.5. Let A, A' G CnX+. TTien A = V.

Assume that A > A' and G(A) = G{A').

We can write A' = A — — i2 — • • - — i'n for some sequence ii, i2,..., in I. Using 6.1.4(a) repeatedly, we see that n 4 ) = £ i p • z p (i p , A) £ zp • iq. p—\ 1
G(A) - G{A - i[ - i2

Using our assumption, we have therefore that n (a)

£ p=l

Since A G

W

1
we have («, A) G N for all i, hence

(b) Similarly, since A' G

n ^ 2 i p ' i p ( i p , A) > 0. p= l , we have n • ip(ip,\') P =i

> 0,

in

6. Complete Reducibility Theorems

50

or equivalently, n

(c)

^ ip • ip{ip, X-i[-if2 P= l

i'n)> 0.

Adding (b),(c) term by term gives n

n

• P

l

A) - 2

^

ip ' ig - 2 y ^ ip • ip > 0.

l
=l

Introducing here the equality (a), we obtain —2 ip • ip > 0. Since •fcp> 0 for all p, it follows that n = 0; hence A = A' as required. 6.1.6. Let M eC. For each Z[/]-coset C in X we define Mc = ©AecMA. It is clear that M = © c - ^ c as a vector space and that each M c is a U-submodule. Hence M = ®cMc as an object in C. Proposition 6.1.7. Let M € Chi. (a) Assume that there exists C as above such that M = Mc - Let G : C —• Z 6e as tn We define a linear map E : M —* M by E(m) = vG^m A for all A € C and all m G M . Then the operator ftS : M —• M is in the commutant of the U-module M. Moreover, the Q {v)-linear map fiS : M —• M is locally finite. (b) Assume that M is a quotient of the Verma module M\>. Then QE : M —• M is equal to vG(x ) times identity. (c) Let M be as in (a). Then the eigenvalues of CIS : M —> M are of the form vc for various integers c. We prove (a). We have for A,m as above: QE Ei(m) = vGtx+i,)SlEi(m) =

=

t,G(A+t')-G(A)-<.<«,A+i'>J5;.nS(m)

E^m) =

E ^ m )

and SIZ Fi(m) = vG(x~i^Q,Fi(m) = vG(\-i')+i-W,\) = vG(x-i')-G(X)+i.i(i,x)F.nE(rn) = FinE(m). Moreover, f2E maps each weight space of M into itself. This proves the first assertion of (a). To prove the second assertion, it suffices to show that

6.2. Complete Reducibility in Chi n C

51

the restriction of f£E to any weight space is locally finite. Let ra G Mx. Let M' be the U+-submodule of M generated by ra. Let M" be the U-submodule of M generated by ra. We have M" = U ~ M ' . We have d i m M ' < oo since M G Chx. It follows easily that all weight spaces of M" are finite dimensional. In particular, the A-weight space of M" is finite dimensional. This weight space is stable under f2E and it contains ra. Thus, QH : M —• M is locally finite. We prove (b). From the definition, 0,3 acts on the A'-weight space of M as multiplication by vG(x ) times identity. Since this weight space generates M as a U-module, we see that (b) follows from (a). (Note that (a) is applicable to M.) We prove (c). Let M be the sum of the generalized eigenspaces of : c M —> M corresponding to eigenvalues of form v for various integers c. We must show that M = M. By the argument in the proof of (a), we may assume that, for any A G C, we have CIMW = £ V > A d i m M A < oo. We will prove that, for any A G C, we have Mx C M, by induction on d = ^M(A). If d = 0, there is nothing to prove. Assume now that d > 1. Let Ai G C be maximal such that Ai > A and MXl ^ 0. Let m\ be a non-zero vector in MXl. Let Mi be the U-submodule of M generated by m \ . Clearly, dM/Ml(X) < dM(X). Hence, by the induction hypothesis, we have (M/M\)x C {M/M{f. On the other hand, by (b), we have M\ c M. It follows that Mx C M. The proposition is proved. The operator

: M —> M is called the quantum Casimir operator.

6 . 2 . C O M P L E T E REDUCIBILITY IN Chi N C

L e m m a 6.2.1. Let M G C. Assume that M is a non-zero quotient of the Verma module M\ and that M is integrable. Then (a) A G X+ and (b) M is simple. (a) follows from 3.5.8 applied to a non-zero vector ra G Mx. We prove (b). Assume that M' is a subobject of M distinct from M and 0. Then clearly, M'x = 0. We can find X' G X maximal with the property that M'x ^ 0. Then X' < A. Let m' be a non-zero vector in M'x>. By the maximality of A', we have Eim! = 0 for all i. By 3.4.6, there exists a morphism of U-modules from the Verma module My into M' whose image contains ra'. Let M" be the image of this homomorphism. Clearly M" is integrable (since M is integrable). Applying (a) to M" we see that A' G X+.

52

6. Complete Reducibility Theorems

Applying 6.1.7(b) to M and to M" and to the Z[/]-coset of A (or A') in X, we see that Q,E(m) = vG^m for all m G M and QE(m) = vG^x'^m for all m G M". (G as in 6.1.4.) It follows that G{A) = G(X'). This contradicts 6.1.5 since < A. The lemma is proved. Theorem 6.2.2. Let M be an integrable U-module in Chi. Then M is a sum of simple U-submodules. We may assume that M ^ 0. By 6.1.6, we may also assume that M = Mc for some Z[/]-coset C in X. We choose a function G : C —• Z as in 6.1.4 and we define f2S : M —> M (in the commutant of M) as in 6.1.7. By writing M as a direct sum of the generalized eigenspaces of $IE : M —> M (see 6.1.7), we may further assume that there exists c G Z such that (f£E — vc) : M —• M is locally nilpotent. Let P = {me M\Eim = 0 Vi}. We have P = ©A <=cPX where Px = P n Mx. For any non-zero element m G P A , the U-submodule of M generated by m is of the type considered in 6.2.1; hence it is a simple subobject of M. Thus the U-submodule M' of M generated by P is a sum of simple U-submodules. Let M" — M/M'. Assume that M" ± 0. Then we can find Ai G C maximal such that M"Xl ^ 0. Let mi be a non-zero vector of M"Xl. We have ^ m j = 0 for all i. Applying 6.2.1 and 6.1.7 to the U-submodule of M" generated by mi, we see that Ai G X+ and f2E(mi) = v G ( Al >mi. Since (ftE - vc) : M -> M is locally nilpotent we see that (QE - vc) : M" —> M" is locally nilpotent. Hence we must have c = G(Ai). Let mi G M A l be a representative for mi. As in the proof of 6.1.7, the U + -submodule M\ of M generated by mi is a finite dimensional Q{v)~ vector space which is the sum of its intersections with the weight spaces of M. Hence we can find A2 G C maximal such that M\ fl MX2 ± 0. Let be a non-zero vector in M\ fl Mx*. We have Eim2 = 0 for all i. Applying 6.2.1 and 6.1.7 to the U-submodule of M generated by m2, we see that A2 G X+ and QE(m2) = vG^x^m2. From the definition of c, we have that A2 G X+; from the G{A2) = c. Hence G(Ai) = G(A2). Note that Ai G definitions, we see that X2 > Ai. Using 6.1.5, we deduce that Ai = \ 2 . It follows that Mi is the one dimensional subspace spanned by mi; hence we must have Ei{m\) = 0 for all i, or equivalently, mi G P . This implies that mi = 0 , a contradiction. We have proved that M" — 0. Hence M = M' and therefore M is a sum of simple U-submodules. The theorem is proved. Corollary 6.2.3. (a) For any A G X+, the U-module A\ is a simple object

6.3. Affine or Finite Cartan Data

53

of C'. (b) If A, A' € X+, the U-modules A\, Av are isomorphic if and only if A = A'. (c) Any integrable module in Chx is a direct sum of simple modules of the form A A for various A G X+. (a) follows from 6.2.1 since A A is integrable. To prove (b), it suffices to note the following property which follows from the definitions: given the U-module M = A\ where A G X+, there is a unique element Ai G X such that MXl ^ 0 and Ai is maximal with this property; we have Ai = A. We prove (c). From Theorem 6.2.2, it follows that any integrable module in Chl is a direct sum of simple objects of Chl which are necessarily integrable. Let M' be one of these simple summands. Let A G X be maximal such that M' A / 0. Let ra be a non-zero vector in M'x. Then Eim = 0 for all i. Using 3.5.8, we can find a non-zero morphism A\ —> M' (in C'). Since both AA and M ' are simple, this must be an isomorphism. 6 . 3 . A F F I N E OR F I N I T E CARTAN DATA

6.3.1. In this section we assume that (/, •) has the following positivity property: E i . i • jxixj > 0 for all (XJ) G N 7 . This certainly holds if (/, •) is of affine or finite type. We first prove an irreducibility result for certain Verma modules. Proposition 6.3.2. Let A £ X be such that (i, A) < - 1 for all i. M\ is simple.

Then

Assume that M has a non-zero U-submodule M' distinct from M. Let A' G X be maximal with the property that M'x ^ 0. Let ra' be a non-zero vector in M'x . Then Eim' = 0 for all i. Hence there is a homomorphism of U-modules My —• M' whose image contains ra'. Using 6.1.7 for M\ and My, we see that nS(ra') = vG^m' and ftE(m') = v G ( A ' W . It follows that G(A) = G(A'). We have A' < A hence we can write A' = A — i[ — i'2 i'n for some sequence ii,i2, • • ->in in / with n > 1. As in 6.1.5, from G(A) = G(A'), we deduce ( a ) £ p = i H ' ipfypi A) = ]C p < g e [i ) n ] V ' V Hence ( E ^ i V) • ( E ^ i *,) = E ^ i V • + 1). By our assumption, the left hand side is > 0 and the right hand side is < 0. This contradiction proves the proposition. 6.3.3. In the remainder of this section, we assume that ( / , •) is of finite type. In this case the root datum is automatically ^-regular and X-regular.

54

6. Complete Reducibility Theorems

Proposition 6.3.4. (a) For any A E X+, we have dimA* < oo. (b) Let M EC be such that dim M < oo. Then M is integrable and M E Ch%, hence (by 6.2.2), it is a direct sum of simple U-modules isomorphic to A\ for various A € X+. Let A' E X be such that A*' ^ 0. Using 5.2.7 several times, we see that we also have A ^ A ^ ^ 0 for any w E W. In particular, we have A™o(a ) ^ 0. It follows that A' < A and wo(X') < A. The last inequality implies, in view of 2.2.8, that w0(A) < A7. Thus, we have w0{\) < X' < A. These inequalities are satisfied by only finitely many A'. Since each weight space of AA is finite dimensional, we see that (a) holds. Now (b) is immediate since the root datum is X-regular. The proposition follows. 6.3.5. The following result is a variant of the complete reducibility theorem 6.2.2: we assume (see 6.3.3) that the Cartan matrix is of finite type but we do not need the condition that our module is in Ch%. Proposition 6.3.6. Let M be an integrable U-module. of simple U-modules of form A\ for various X E X+.

Then M is a sum

Let m E M** and let M' be the U + -submodule of M generated by m. Since M is integrable, there exist ai E N such that m = 0 for i E I. Hence there exists A' E X+ such that E^t,X = 0 for all i. It follows that u —> um gives a surjective linear map U + / ( ] T \ Using 3.5.6, we see that U + / ( E i U + ^ < i , A > + 1 ) ) is isomorphic as a vector space to Ay, hence it is finite dimensional, by 6.3.4. Thus, dimM 7 < oo. Let M" be the U-submodule generated by M'. Since M' is stable under U + and U°, M" is equal to the U~-submodule generated by M'. By the argument above, the U~-submodule generated by a vector in M is finite dimensional. Since M' is finite dimensional, the U~-submodule generated by M' is finite dimensional. Thus, dim M" < oo. We have shown that m is contained in a finite dimensional U-submodule of M. Thus, M is a sum of finite dimensional U-submodules. By 6.3.4(b), each of these is a sum of simple U-modules of form AA for various A E X+. The proposition follows.

CHAPTER

7

Higher Order Quantum Serre Relations

7.1.1. In this chapter we assume that we are given i ^ j in I and e = ±1. Given n, m £ Z, we set fi,r,n,m;e =

£

(-1)

£ f

r-\-s=m

To simplify notation we shall write / n , m ; e instead of / i j ; n , m ; e when conE N. venient, and we shall set a = —(i,jf) G N, a' = L e m m a 7.1.2. We have (in Uy) (a) - « r ( a B - 2 m ) ^ / + m ; e + /+ m : e £?i = [m + l ] i / + m + 1 ; e ; (b) - F i / + m . e + / + m ; e F i = [an - m + We prove (a). The left hand side of (a) is

r+s=m er(an—ra-1-1)

V,

=

E

}iElr)E^n)Els+1))

>+ 1

er(an-m+l)-m-l)r j

r+s=m-fl

(-i)r(«:

^ e r ( a n - m + l ) M ^ ^(r) ^(n) ^(s)

It remains to observe that er(an — m-f 1) — e(m-f-l))

r]i + « f ( a n " m + 1 ) [ « ] i = < r ( Q n - m ) [ m + l]i

Vi

if r + 5 = m + 1. We prove (b). Using the identity

f , e [

n )

e

{ n )

f ,

Vi

Ki~ v, Vi - v~

K-j

E(N-I)

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_13, © Springer Science+Business Media, LLC 2010 2011

55

7.1. Higher Order Quantum Serre Relation 56

we see that the left hand side of (b) is J 2 ( - 1 )r+1vr(on"m+1)(F<(r)F<£7jn)^') r-f s=m Vi — v-

+ =

Y, (-1 Y v ^ - ^ E ^ E ^ E ^ F i r+s=m ( 1 )^+l- u er ( QW - m+1 )(

Y

r V S lK V lK E( )E^) i ^ i- i~ -i Vi

~ i

r+s=m _

V ~

r + l

K j -

V j ~

1

l

K - i

E

{ r - l ) E(n)

%

E(s)^

3

Vi - v r er(an-m+l)Vj

~ Vj v

r+s=m

+

^

(

^ ^ ( a n - m + l )

i ~

v

i

i

~~

V

„.s+2?—an £> Vi4

K

V i1

i ~ ViK~i 1

r+s=m— 1

i

i

* r+s=ra — 1 1)r-lT;e(r+l)(an-m+l)

J

^ ( r - l ) ^ (n)^ j»

„.—s — 2 r + a n

(

K-j ^ r ^ n ) ^ - ! )

-

r+s=m

+

E[s-\) V

£(r)^(n)^(s)

1

and (b) follows. From Lemma 7.1.2 we deduce by induction on p > 0 the following result. L e m m a 7.1.3. We have (b) F^ f+ Vu/ L I Jn,m;e — z^p'=0\

1

[ocn-m+p'l JXj> r+ p(p-p') [ p' J i -ep'*Jn,m-p';eri

J ui

L e m m a 7.1.4. We have e(n-l)

V

F j t t ,m:e

f+ _ fn,m;eFj — K-ej —e Jn—l,m;—e V J ~ VJ

-e(n-l)

K

V•

K ' + f m;e' —e eJn—l, v i ~ j

7.1. Higher Order Quantum Serre Relation

57

We have p.JJn,m\e f+ =

_ Jf+ p. n,m;ex 3

__

( r+s=m

:=

„

1

V

'

(_iyver(<*n-m+l)Yj

^

ti+1 j?. _

iyver(<*n-m+1)E(r)Vj

3 ~

V

-n+l+a'rf>

V

j

J

l

V

j^

n

is

~3

E{n-1)gjs)

' n-\-a'rj> A

~3

'

^r^n-l)^*)

V

r+s=m

~~ j

We now use the identity v f = v f ; the lemma follows. Proposition 7.1.5. (a) If n < 0 or m < 0, then fn,m,e = 0. (b) If ra > an, then fn!m;e = 0. (a) is obvious. In particular (b) holds for n < 0. Hence, to prove (b), we may assume that n > 0 and that (b) holds with n replaced by n — 1. For such fixed n, we see from 7.1.2(b) that /,J" + 1 commutes with Fi and from 7.1.4 and the induction hypothesis, that f^an+i-e commutes with Fj. (We have an+ 1 > a(n — 1) hence the induction hypothesis is applicable to /n-i,on+i;±i*) It is trivial that fn,Qn+i;e commutes with Fh for any h^ i,j. Thus, fn,Qn+i;e commutes with Fh for any h G I. Using 3.2.7(a), it follows that / n ,an+i;e is a scalar multiple of 1. On the other hand, it belongs to f ( a n + i ) i + n j and (an + l)i + nj ^ 0. It follows that / „ i Q n + i ;e = 0. We now show, for our fixed n, that / n ,m;e = 0 whenever m > an. We argue by induction on m. If ra = an + 1, this has been just proved. Hence we may assume that m > cm+1. Using the induction hypothesis we see that the left hand side of the identity - v - ( a n ~ 2 m + 2 ) £ t / £ m _ i ; e + f£,m-i;eEi = [m}ifn,m;e ( s e e 7.1.2) is zero. Hence we have [ra]i/+ m ; e = 0. We have ra 0, hence f£m.e = 0. It follows that / n ,m;e = 0 and the induction is completed. The proposition is proved. 7.1.6. The identities / n ,m;e = 0 (ra > an;n > 1) in f are called the higher order quantum Serre relations. For n = 1 and ra = a + 1, they reduce to the usual quantum Serre relations. Corollary 7.1.7. For any n,m> e(m)0(

n)

=

0 such that m > an + 1, we have £

r+s'=rn;m—an<s'

where

7

, =

7s<0<

r)0<">0<S'>

<m

(identity

in

7.1. Higher Order Quantum Serre Relation 58

From 7.1.5 we see that

/n,m-g;i

m—an—l

9=

Y

= 0 for 0 < q < m — an — 1; hence

(-l)qvrq+qfn,m-q-Aq)

is zero. On the other hand, using the definitions, we have m—an—1 q=0

=

£

r+s=m—q

C r ^ W '

r+s'=m

where Cr,y =

Sm-an-l(_l)r+^r(an-m+9+l)-mg+g

If 0 < s' < m — an — 1, we may replace the range of summation above to 0 < q < s' and the sum will not change, since for 0 < s' < q, the binomial coefficient [* ]. is zero. Hence for such s' we have c r y = ( - l ) ^ , r ( a n - m + 1 ) X ^ - l ) ^ 1 - * 0 [',']<• By 1.3.4, the last sum is zero unless sf = 0. Thus, for 0 < s' < m — an — 1, we have
N o t e s on Part I 1. The Hopf algebra U has been defined in the simplest case (quantum analogue of SL2) by Kulish and Reshetikhin [10] and Sklyanin [14] and, in the general case, by Drinfeld [2] and Jimbo [5], [6]. The definition given here is different from the original one; the two definitions will be reconciled in Part V. 2. The bilinear form ( , ) in 1.2.3 turns out eventually to be the same as that of Drinfeld [3]. 3. The idea of defining the A-form ^f and ^ U of f and U (see 1.4.7, 3.1.13) in terms of ^analogues of divided powers appeared in [12]. (In the classical case, the Z-forms of enveloping algebras were defined in terms of divided powers with ordinary factorials by Chevalley and Kostant [9], for finite types, and by Tits, for infinite types.) 4. The theorem in 2.1.2 is due to Iwahori, for finite types, and to Matsumoto and Tits [1], in the general case. The statement in 2.2.7 can be deduced from a theorem of Tits on Coxeter groups, see [1], ch. 4, p.93, statement Pn. 5. The notion of Cartan datum (resp. root datum), see 1.1.1 (resp. 2.2.1), is closely related to (but not the same as) that of a generalized Cartan matrix (resp. a realization of it) in [7]. In fact, an irreducible generalized Cartan matrix is the same as an irreducible Cartan datum up to proportionality (see 1.1.1). 6. The commutation formulas in 3.1.7, 3.1.8, are closely connected with Drinfeld's description [3] of U (in the formal setting) as a quantum double. Their consequence, Corollary 3.1.9, is the quantum analogue of an identity of Kostant [9] (it was shown to me by V. Kac). 7. The definition 3.5.1 of integrable U-modules is the quantum analogue of Kac's definition [7] of integrable modules of a Kac-Moody Lie algebra. 8. The definition of universal 7£-matrices is due to Drinfeld [3]. The characterization of a modified form of the 7^-matrix given in 4.1.2, as well as in 4.1.3, appeared in [13]. Propositions 4.2.2 and 4.2.4 are due to Drinfeld [3]. 9. The formulas for the operators e ,T" e (in 5.2.1) are new (they are classical for v =1). An identity like 5.3.4(a) (with a different definition of T ^ ) is stated in [8] and [11]. 10. The definition of the quantum Casimir operator (see 6.1) is due to Drinfeld [4]. The proof of the complete reducibility theorem 6.2.2 is inspired by the proof of the analogous result in the non-quantum case (Kac [7]). 11. A number of statements of Drinfeld in [3] were given without proof; some of the proofs were supplied by Tanisaki [15].

60

References REFERENCES

1. N. Bourbaki, Groupes et algebres de Lie, Ch. Hermann, 1968. 2. V. G. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254-258. 3. , Quantum groups, Proc. Int. Congr. Math. Berkeley 1986, vol. 1, Amer. Math. Soc., 1988, pp. 798-820. 4. , On almost cocommutative Hopf algebras, Algebra and analysis 1 (1989), 30-46. 5. M. Jimbo, A q-difference analogue of U(Q) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69. A q-analog of U(gl(N + 1)), Heche algebras and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252. 7. V. G. Kac, Infinite dimensional Lie algebras, Birkhauser, Boston, 1983. 8. A. N. Kirillov and N. Yu. Reshetikhin, q-Weyl group and a multiplicative formula for universal R-matrices, Commun. Math. Phys. 134 (1990), 421-431. 9. B. Kostant, Groups over Z, Proc. Symp. Pure Math. 9 (1966), 90-98, Amer. Math. Soc., Providence, R. I.. 10. P. P. Kulish and N. Yu. Reshetikhin, The quantum linear problem for the sineGordon equation and higher representations, (Russian), Zap. Nauchn. Sem. LOMI 1 0 1 (1981), 101-110. 11. S. Z. Levendorskii and I. S. Soibelman, Some applications of quantum Weyl groups, J, Geom. and Phys. 7 (1990), 241-254. 12. G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237-249. 13. Canonical bases in tensor products, Proc. Nat. Acad. Sci. 89 (1992), 81778179. 14. E. K. Sklyanin, On an algebra generated by quadratic relations, Uspekhi Mat. Nauk 40 (1985), 214. 15. T. Tanisaki, Killing forms, Harish-Chandra isomorphisms and universal R-matrices for quantum algebras, Infinite Analysis, World Scientific, 1992, pp. 941-961.

Part II G E O M E T R I C REALIZATION OF f

The algebra f has a canonical basis B with very remarkable properties. This gives an extremely rigid structure for f and also (in the F-regular case) for each A,\. Part II will introduce the canonical basis of f. At the same time, f will be constructed in a purely geometric way, in terms of perverse sheaves on the moduli space of representations of a quiver. Chapter 8 contains a review of the theory of perverse sheaves over an algebraic variety in positive characteristic. As far as definitions are concerned, it would have been possible to stay in characteristic zero and use 2>-modules instead of perverse sheaves. This would certainly have been more elementary, but would have deprived us of the possibility of using the Weil conjecture and its consequences which are available in the framework of perverse sheaves on varieties in positive characteristic. In Chapter 9 we introduce a class of perverse sheaves attached to a quiver and the operations of induction and restriction for perverse sheaves in this class. In Chapter 10, we study the Fourier-Deligne transform of perverse sheaves in our class. This is necessary for understanding the effect of changing the orientation of the quiver. In Chapter 11 we study linear categories with a given periodic functor (a functor which has some power equal to identity). These are needed to handle the case where the Cartan datum is not symmetric. (The geometry associated to a non-symmetric Cartan datum is very closely related to that associated to a symmetric Cartan datum, together with an action of a finite cyclic group.) In Chapter 12 we study quivers with a cyclic group action. The geometric construction of f and of its canonical basis (up to signs) is given in Chapter 13. In Chapter 14, we discuss various properties of the canonical basis. For example, the property expressed in Theorem 14.3.2 is responsible for the existence of a canonical basis in the simple integrable U-modules (see Theorem 14.4.11). Perhaps the deepest property of the canonical basis is

62

Part II. Geometric Realization of f

expressed by the positivity theorem 14.4.13, which states (for symmetric Cartan data) that the structure constants of f are given by polynomials with positive integer coefficients. Theorem 14.4.9 gives a natural bijection between the canonical basis of f for a non-symmetric Cartan datum and the fixed point set of a cyclic group action on the canonical basis of the analogous algebra corresponding to a symmetric Cartan datum.

CHAPTER

8

Review of the Theory of Perverse Sheaves

8.1.1. Let p be a fixed prime number. All algebraic varieties will be over an algebraic closure k of the finite field Fp with p elements. Let X be an algebraic variety. We denote by V(X) = Vbc(X) the bounded derived category of Q/-(constructive) sheaves on X (see [1, 2.2.18]); here, I denotes a fixed prime number distinct from p and Q/ is an algebraic closure of the field of adic numbers. Objects of T>(X) are referred to as complexes. For a complex K £ T>(X), we denote by Ti^K the n-th cohomology sheaf of K (a Q/-sheaf on X). We denote by D{K) E *D{X) the Verdier dual of K. The constant sheaf Q/ on X will be denoted by 1. For any integer j , let K »—> K[j] be the shift functor T>{X) —> V(X); it

8.1.2. Let M(X) be the full subcategory of T>(X) whose objects are those K in T>(X) such that, for any integer n, the supports of both HnK and 7 i n D { K ) have dimension < —n. In particular, 7inK and 7 i n D ( K ) are zero for n > 0. The objects of M(X) are called perverse sheaves on X. M(X) is an abelian category [1, 2.14, 1.3.6] in which all objects have finite length. The simple objects of A4(X) are given by the Deligne-GoreskyMacPherson intersection cohomology complexes corresponding to various smooth irreducible subvarieties of X and to irreducible local systems on

functors of truncation

and perverse cohomology, which in [1] are denoted

for any K E T>(X) and any n. For fixed K, we have r
G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_13, © Springer Science+Business Media, LLC 2010 2011

63

8. Review of the Theory of Perverse Sheaves

64

For any n G Z, let M(X)[n] be the full subcategory of T>(X) whose objects are of the form K[n] for some K e M(X). 8.1.3. A complex K G T>(X) is said to be semisimple if for each n, (a) there exists : (H n K)[—n] —• r
is a semisimple object of

M(X). n

It follows that K is isomorphic to ®n(H K)[-n]

in

V(X).

8.1.4. Let f : X Y be a smooth morphism with connected fibres of dimension d. We have D(f*K) = f*{D{K))[2d\ for K G V{Y). (We will ignore the Tate twist.) If K G A4(r), then f*K G M{X)[-d) (see [1, 4.2.4]) and K i-> f*K defines a fully faithful functor from M{Y) to M(X)[-d\ (see [1, 4.2.5]). 8.1.5. Let / : X —> Y be a proper morphism with Y smooth. f\ 1 G T>(Y) is a semisimple complex. (See [1, 5.4.5, 5.3.8].)

Then

8.1.6. More generally, let / : X —• Y be a morphism. Assume that we are given a partition X = X 0 UXiU- • - U X m such that X<j = XoUXiU- • - U X j is closed for j = 0 , 1 , . . . , m. (We define X<j = 0 for j < 0.) Assume that, for /• f" each j, we are given morphisms Xj —U Zj - A Y such that Zj is smooth, f j is a vector bundle, f j is proper and f j f j = f j where f j : Xj —> Y is the restriction of f". Then / j l G T>(Y) is a semisimple complex. Moreover, for any n and j, there is a canonical exact sequence (in M.(Y))\ (a)

0-

H"(f,),i

^ Hn(f<,)a

^ mu
- 0

where f<j : X<j Y is the restriction of / . The proof is essentially the same as that in [5, 3.7]; it is based on the theory of weights in [1]. 8.1.7. G-equivariant complexes. Let m : G x X —> X be the action of a connected algebraic group G on X; let ir : G x X —» X be the second projection. A perverse sheaf K on X is said to be G-equivariant if the perverse sheaves 7r*i^[dim G] and m* if [dim G] are isomorphic. More generally, a complex K G M(X)[n] is said to be G-equivariant if the perverse sheaf K[—n] is G-equivariant. We denote by A 4 G ( X ) the full subcategory of M . ( X ) whose objects are the G-equivariant perverse sheaves on X. More generally, we denote by

8.1. Theory of Perverse Sheaves

65

the full subcategory of M(X)[n\ whose objects are of the form K[n) where K G M G ( X ) . Here are some properties of G-equivariant complexes. (a) If A G M G ( X ) , and B € M(X) is a subquotient of A, then B G

M.G{X)[TI)

M

G

( X ) .

(b) Assume that G acts on two varieties X',X and that / : X' —• X is a morphism compatible with the G-actions. If K G Aic(X), then G M G ( X ' ) for all n. If K' G M G ( X ' ) , then Hn{f>Kf) G M G { X ) Hn(f*K) for all n. (c) Assume that / : X —> Y is a locally trivial principal G-bundle (in particular G acts freely on X and trivially on Y). Let d — dimG. If K G M(X)[n], then we have K G MG(X)[TI} if and only if K is isomorphic to f*K' for some K' G M(Y)[n + d]. The functor M(Y)[n + d] -> MG(X)[n] (Kf -> f*K') and the functor MG{X)[n\ -»• M(Y)[n + d] (K f\,K := (H~n~df*K)[n + d]) define an equivalence of the categories MG(X)[n],M{Y)[n + d\. 8.1.8. A semisimple complex K on a variety X with a G-action is said to be G-equivariant if for any n G Z, HnK is a G-equivariant perverse sheaf on X . Let / : X —> Y be as in 8.1.7(c). If K' is a semisimple complex on Y, then f*K' is a G-equivariant semisimple complex on X. Conversely, if K is a semisimple G-equivariant complex on X, then K is isomorphic to f*K' for some semisimple complex K' on Y, which is unique up to isomorphism. In fact, we have K' ^ foK where, by definition, fbK = ®nfo((HuK)[-n]) n and h((H K)[-n]) G M(Y)[-n + d] is as in 8.1.7(c). 8 . 1 . 9 . Let A, B be two G-equivariant semisimple complexes on a variety X with G-action; let j be an integer. We choose a smooth irreducible algebraic variety T with a free action of G such that the Q/-cohomology of T is zero in degrees 1 , 2 , . . . , m where m is a large integer (compared to \j\). Let us consider the diagram

X 4-TxX

G\(r x X)

where the maps s,t are the obvious ones. Then s*A,s*B are semisimple G-equivariant complexes; since t is a principal G-bundle, the semisimple complexes t\,s*A, t\,s*B on G \ ( T x X ) are well-defined. Let u : G\(TxX) —> {point} be the obvious map. Consider the Q/-vector space

8. Review of the Theory of Perverse Sheaves

66

By a standard argument (see [6, 1.1, 1.2]), we can show that this vector space is canonically attached to A,B,j\ it is independent of the choice of m and T provided that m is sufficiently large. We denote this vector space by G; A, B). 8.1.10. We give some properties of T>j(X, G, A, B). (a)

G; A, B) =

G; B, A).

(b) T>j(X, G; A[n], B[m}) = T>j+n+rn(X, (c) D j (X,G;A®AUB)

G; A, B) for all n, m G Z.

= D j (X, G; A, B) © D,- (X, G; AUB).

(d) If A,B are perverse sheaves, then D j ( X , G ; A , B ) = 0 for j > 0; if, in addition, A,B are simple, then Do(X, G;A, B) = Qj, if B = DA and D 0 (X, G; A, B) = 0, otherwise. (e) There exists j0 G Z such that T>j(X, G; At B) = 0 for j > j0. (f) If A!,B' (resp. A",B") are G'-equivariant (resp. G"-equivariant) semisimple complexes on a variety X' (resp. X") with a G'-action (resp. G"action) where G' ,G" are connected algebraic groups, then A' A" and B' B" are G' x G"-equivariant semisimple complexes on X' x X" and we have a canonical isomorphism D j ( X ' x X", G' x G"; A' ® A", B' 0 B") A

=

'>B') ®

G"; A", B").

j'+j"=j

The sum is finite by (e). Properties (a),(b),(c) are obvious; (d) follows from [5, 7.4]; (e) follows from (d); (f) follows from the Kunneth formula. 8.1.11. Fourier-Deligne transform. We fix a non-trivial character Fp —• Q*. The Artin-Schreier covering k —> k given by x —> xp — x has Fp as a group of covering transformations. Hence our character Fp —> gives rise to a Q^-local system of rank 1 on k; its inverse image under any morphism T : X' —• k of algebraic varieties is a local system CT of rank 1 on X'. Let E —• X and E' —> X be two vector bundles of constant fibre dimension d over the variety X. Assume that we are given a bilinear map T : E Xx E' —• k which defines a duality between the two vector bundles. We have a diagram E E Xx E' A E' where s, t are the obvious projections.

8.1. Theory of Perverse Sheaves

67

The Fourier-Deligne transform is the functor $ : T>(E) —> V(E') defined by T>(E); it is known that the Fourier inversion formula = j*K holds for K e T>(E), where j : E ^ E is multiplication by —1 on each fibre of E. 3> restricts to an equivalence of categories M(E) —> M(E'); hence it defines a bijection between the set of isomorphism classes of simple objects in M(E) and the analogous set for M(E'). It also commutes with the n functors K ^ H K. 8.1.12. Let A (resp. A') be an object of V(E) (resp. V(E')). Let be the obvious maps of E, E', E Xx E' to the point. We have m(A 0

=

u,u',ii

0 A').

Indeed, from the definitions, we see that both sides may be identified with ii\(s*A®t*A'®Cr[d\). 8.1.13. Let T : kn —> k be a non-constant affine-linear function. Let u : kn —> {point} be the obvious map. We have U\(CT) = 0. The proof is left to the reader.

CHAPTER

9

Quivers and Perverse Sheaves

9.1.1. By definition, a (finite) graph is a pair consisting of two finite sets I (vertices) and H {edges) and a map which to each h E H associates a We say that h is an edge joining the two vertices in [h]. We assume given a finite graph (I, H,h [/i]). An orientation of our graph consists of two maps H —> I denoted h »-> h! and h > h" such that for any h E H, the two elements of [h] are precisely /i',/i". We assume given an orientation of our graph. Thus we have an oriented graph (=quiver). Note that

9.1.2. Let V be the category of finite dimensional I-graded vector spaces V = 0ieiVi; the morphisms in V are isomorphisms of vector spaces comFor each v — i>\\ E N[I] we denote by Vu the full subcategory of V whose objects are those V such that dim Vi — v\ for all i E I. Then each object of V belongs to Vu for a unique v E N[I] and any two objects of V„ are isomorphic to each other. Moreover, Vv is non-empty for any v E N[I]. Given V E V, we define G v = {g E GL(V)|g(Vi) = Vi for all i E 1}

Then GV is an algebraic group (isomorphic to Yl\ GL(V{)) acting naturally

9.1.3. Flags. A subset I' of I is said to be discrete if there is no h E H If v E N[I], we define the support of v as {i E I\u\

0}. We say that v

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_13, © Springer Science+Business Media, LLC 2010 2011

68

9.1. The Complexes Lu

69

Let X be the set of all sequences v = (i/ 1 , i / 2 , . . . , vm) in N[I] such that vl is discrete for all I. Now let V 6 V and let i/ G X be such that dim V j = Yli for all i G I. A flag of type is in V is by definition a sequence (a)

/ = (V = V ° D V 1 D • - • D V m = 0)

of I-graded subspaces of V such that, for I = 1 , 2 , . . . , m, the graded vector space V * - 1 / V ' belongs to Vui. If x G E v , we say that / is x-stable if x h(VLH>) C for all I = 0 , 1 , . . . , r a and all h. Let Tv be the variety of all flags of type v in V. Let Tv be the variety of all pairs ( x , f ) such that x G E v and / G Tv is x-stable. Note that G y acts (transitively) on Tu by g : / —• gf where / is as in (a) and = (V = gV° 3 gV1 D - • • D gVm = 0). Hence Gv acts on f v by gf 9 -(xj) (gx,gf)Let ir u : T v —> E v be the first projection. We note the following properties which are easily checked. (b) T v is a smooth, irreducible, projective variety of dimension

E

i\KV the second projection T v

Fu is a vector bundle of dimension Y^

h-,i'
v

h'vh"-

(c) T v is a smooth, irreducible variety of dimension

/(-) =h,l'
+E

i\1
(d) 7TU is a proper Gv-equivariant morphism. Let Lu = (TT^))! G £>(EV). By (c),(d) and by 8.1.5, Lu is a semisimple Since D(l[f(u)]) = 1 [f(u)] on f v complex on E V . Let Lu = Lu[f(v)]. (see (c)) we have D(LU) = Lu. We denote by Vv the full subcategory of A4(Ev) consisting of perverse sheaves which are direct sums of simple perverse sheaves L that have the following property: L[d] appears as a direct summand of Lv for some d G Z and some v G X such that dim V j = Y^i v\ for all i G I. We denote by Qv the full subcategory of P(EV) whose objects are the complexes that are isomorphic to finite direct sums of complexes of the form L[d'\ for various simple perverse sheaves L G TV and various d' G Z. Any complex in Qv is semisimple and GV-equivariant. Prom 8.1.4, we see that Vv and Qv are stable under Verdier duality.

9. Quivers and Perverse Sheaves

70

9.1.4. Let v — (i/1, i / 2 , . . . , vm) G X. Assume that for some j we write v3 = is[ + v^ where € N[I] have disjoint support. Let 1 2 j l - - -jVm) € X. It is clear that Lv = Lu> u' = (I/ , V ,..., v ~ , v{, and / ( u ) = f(v'). Hence Lu = L„'. Thus, in the definition of Vw, we may restrict ourselves to sequences u = (v l , v 2 , . . . , is™) G such that each u3 is of the form ni for some i G I and some n > 0. Since there are only finitely many such u (subject to dim Vi = Yli for all i G I) we see that Vv has only finitely many simple objects, up to isomorphism. 9.1.5. In the special case where V is such that J \ d i m V | i is discrete, we have E v = 0 and Vv has exactly one simple object up to isomorphism, namely 1. 9.1.6. Let K,K'

E Qv- The following two conditions are equivalent:

(a) K 2 K'\ f o r a11 simple (b) dim D j (Ev, Gv; K, DB) = dimT>j(Ev,Gv;K,,DB) objects B G Vw and all j G Z. It is clear that (a) implies (b). Assume now that K,K' are not isomorphic. Now if is a direct sum of complexes L[n] where L runs over the isomorphism classes of simple objects L of Vw and n G Z; let m(L, n) G N be the number of times that L[n] appears in this direct sum. We define similarly m'(L,n) by replacing K by K'. Since K,K' are not isomorphic, we can find Lo, n o such that m(L 0 ,n 0 ) ^ m'(Lo,no) and such that m(L, n) = m'(L, n) for all L and all n < no. By (b), we have

Y^ m{L, n) dim D J + n ( E v , G v ; L, DB) L,n

=

m'(L,n) d i m T > j + n ( E v , G v ' , L, DB) L,n

for all simple objects B G TV and all j G Z. Using 8.1.10(d), we rewrite this as follows: m(B,-j)

+ Y^ L

(c)

=m'(B,-j)

J2

n) dim D J + n ( E v , Gv; L, DB)

n;n<—j

+ J2 L

X)

rn'(L,n)dimT>j+n(Ev,Gv;L,DB).

n\n<—j

We apply this to B = LQ and j = —no. Since by our assumption, m(L,n) = m'(L,n) for n < no, we see that (c) implies m(Lo, n o) = m'(Lo,no). This is a contradiction. Thus the equivalence of (a),(b) is proved.

9.2. The functors Ind and Res

71

9 . 2 . T H E FUNCTORS IND AND R E S

9.2.1. Let T , W be two objects of V. We can form E T , E W and their product ET X Ew- This has an action of GT X GW (product of actions as in 9.1.2). We define a full subcategory VT,W of M{ET X EW) and a full subcategory QT,W of P ( E T x EW), as a special case of the definitions of 7V, QV in 9.1.3; indeed, T x W and ET X EW are special cases of V and E v where the oriented graph in 9.1.1 has been replaced by the disjoint union of two copies of that oriented graph. Prom the definitions it is clear that any simple object B G Vt,w is the external tensor product B' ® B" of two simple objects B' G Vt and B" G (and conversely). Note that any complex in QT,W is semisimple and GT X Gw-equi variant. 9 . 2 . 2 . We assume that we are given V , T , W in V, that W is a subspace of V and that T = V / W . We also assume that the obvious maps W —> V and V —• T preserve the I-grading. Let Q be the stabilizer of W in Gv (a parabolic subgroup of Gv)- We denote by U the unipotent radical of Q. We have canonically Q/U = GT X GWLet F be the closed subvariety of E v consisting of all x G E v such that Wh" for all h G H. We denote by i : F —• E v the inclusion. Note that Q acts on F (restriction of the Gv-action on Ev)If x G F, then x induces elements x' G ET and x" G E w ; the map x • (x', x") is a vector bundle k : F —> ET X EW- NOW Q acts on ET X EW through its quotient Q/U = GT X GW- The map K is compatible with the Q-actions. We set G v = G, Q/U = G, E v = E, E T x E w = E. We have a diagram

E^-F-^E. Let E" = G x P F, E' = G xv F. We have a diagram E&-E'

^ E "

^ E

where pi(g, / ) = «(/); p2(g, f ) = (g, /)_; p3(g, / ) = g{i(f)). Note that px is smooth with connected fibres, p2 is a G-principal bundle and is proper. Let A be a complex in QT,W and let B be a complex in QV- We can form k\(l*B) G T>(E). NOW p\A is a G-equivariant semisimple complex on E'\ hence (p2)\,p\A is a well-defined semisimple complex on E" (see 8.1.7(c)). We can form {p3)\(p2%p\A G V(V).

72

9. Quivers and Perverse Sheaves

L e m m a 9.2.3. (^3)1(^2)^1^ G Q v The general case can be immediately reduced to the case where A is a simple perverse sheaf in P T , w and this is immediately reduced to the case where A = Lv* Lu». (Note that a direct summand of a complex in Qv belongs to Qv-) Thus, it suffices to prove that (ps)\(P2)\>Pi(LU' <S> Lu») G G X and 1/" = { y ' { , v G X Q v , where «/' = ( i / J , ^ , • • • satisfy dim T j = J2i dim W i = v" 1 for all i G I. Let v'v" be the sequence of elements in N[I] formed by the elements of the sequence v' followed by the elements of the sequence v". Recall that T v ' v " consists of pairs (x, / ) where x G E v and / is a flag of type v ' v " in V which is x-stable. Now the subspace with index ra' in / = (V = V° D V I D . . . ) is in the G-orbit of W . The pairs (x, / ) for which this subspace is equal to W form a closed subvariety TV'v",0 of fV'u"\ for such (x, / ) we have x G F, hence (x, / ) —• x defines a (proper) morphism Tv'v»$ —• F. This morphism is Q-equivariant (for the natural actions of Q). Hence it G XQ F = E". Since induces a proper morphism u : G XQ TV'V>' ,O G XQ J~I/'V"$ is smooth, the complex L = u|1 G T>(E") is semisimple. (See 8.1.5.) It is clear from the definitions that p^L = p\(Lu' 0 Lu») and ip2)\>V*i(Lv>®lv") = L. It remains to show that (pz)\L G Q v , or equivalently, that (p^u)\l G Qv— TV'V"\ then P^U = IRV>U». We may identify in a natural way G XQTU>V»$ It follows that (p^u)\l = LU'U" which is in Q v by definition. The lemma is proved. L e m m a 9.2.4. K\(L*B) G

QT,W-

We may assume that B is a simple perverse sheaf in Vv- Since a direct summand of a complex in QT,W belongs to QT,W we see that it suffices to prove that k\(l*Lu) G QT,W, where v G X satisfies dim V | = Y^i F°R all iG I. Let F C Tu be the inverse image of F C E under Let n : F —> F be the restriction of iru. We have i*Lu = 7hl; hence k\(L*Lv)

Let 1/ = (i/ 1 , z / 2 , . . . , vm).

= /cml = (k#)J1.

For any r , u G X of the form

such that T1 +UL = vl for all /, we define a subvariety F ( r , cj) of F as the set of all pairs (x, / ) where x G F and / = (V = V° D V 1 D • • • D V m = 0) G

9.2. The functors Ind and Res

73

f u is x-stable and is such that the graded vector space (V l _ 1 flW)/(V*flW) belongs to V^i for / = 1 , 2 , . . . , m. If (x, / ) is as above, then there are induced elements (x', / ' ) G T t and {x", f") G J^o,; here x" is deduced from x by restriction to W and x' is deduced from x by passage to quotient; f is given by the images of the subspaces in / under the projection V —> T and f" is given by the intersections of the subspaces in / with W. Thus we have a morphism A : F(T,LJ) —• T-R x We have a commutative diagram F ( T , LJ)

•

F

• E .

f-R X

where the upper horizontal arrow is the obvious inclusion and the lower horizontal arrow is TTT X TT^. It is not difficult to verify (as in [9, 4.4]) that a is a (locally trivial) vector bundle of dimension M(r,u>) — V \ ,, It is clear that the locally closed subvarieties F(T, LJ) form a partition of F. Let FJ be the union of all subvarieties F(T, U>) of fixed dimension j. Let Zj be the disjoint union of the varieties T r x (union over those ( r , u>) such that F(r,u>) C Fj). The maps a above can be assembled together to form a vector bundle Fj —• Zj. The maps NT x TT^ can be assembled together to form a (proper) morphism Zj —> E. We have a commutative diagram Fj • F

I -1

Zj

>E

We may therefore use 8.1.6 to conclude that (k7t)\1 is a semisimple complex and that, for any i and j , there is a canonical exact sequence (in M(E)): (a)

0

Hn(fM -

Hn(f<M -

^ n ( / < j - i ) i l -» 0

where f j : Fj —> E and f<j : Uj>-.j'<jFj> —> E are the restrictions of KTT. The earlier arguments show that (b)

(/,). 1 - ®(Z T L t f ) [ - 2 M ( r , « ) ]

where the direct sum is taken over all (T,U>) such that F(R,A;) C Fj.

74

9. Quivers and Perverse Sheaves

From (a),(b) we see by induction on j that all composition factors of H (f<j)|1 are in P y Taking j large enough we see that all composition factors of Hn(K7r)\l are in Vw- Since («7r)il is semisimple, it follows that (ktt)\1 G QT,W- The lemma is proved. n

9.2.5. By Lemmas 9.2.3, 9.2.4, we have well-defined functors ~ v I n d x w : QT,W and

~ v Res T w

Qv

(p3)\(p2)\>P*iA)

: Q V —• Q T , W

(B

K\(L*B)).

~ v Since Ind T w is defined using a direct image under a proper map and inverse images under smooth morphisms with connected fibres, it commutes with Verdier duality up to shift (see 8.1.1, 8.1.4); more precisely, D(IndJ fW (>*)) = Ind£ w (D(i4))[2di - 2d2] where d\ is the dimension of the fibres of p\ and d2 is the dimension of the fibres of p2. We have di-d2 = ^2 h We set

dim T

h ' dim W h » +

dim Ti dim Wi. i

v ~ v Ind T -vv = Ind T t wMi ~~ d2].

Then D(Ind¥ t W (i4)) =

Ind%jW(D(A)).

The functor Ind^f w is called induction. 9.2.6. From the proof of 9.2.3 and of 9.2.4, we see that (a)

(b)

_ v Ind T

Res^w^ -

® L w )[-2M(r,w)]

where the sum is taken over all r = (r 1 , r 2 , . . . , r m ) , u = (a;1, a ; 2 , . . . , u m ) such that d i m T | = ^ r / j d i m W i = [ for all i and t 1 + u l = vl for all I.

9.2. The functors Ind and Res

75

9.2.7. We have / ( i A / " ) = / ( u ) + / ( i / ) + dx - d2; hence from 9.2.6(a) we deduce that ® Lu") = Lu'u". Ind T 9.2.8. Let T be a smooth irreducible variety with a free action of G. Let f = U\T. Then f is a smooth irreducible variety with a free action of G induced by that of G. Consider the diagram E^-TxE-^

G\( r x E)

with the obvious maps s,t. As in 8.1.9, s*A is a semisimple G-equivariant complex on r x E and, since t is a principal G-bundle, the semisimple complex t\,s*B G Z>(G\(r x E)) is well-defined. In particular, we can ~ v replace B by I n d x WA and we obtain the semisimple complex t \ , s *

w

A ) G

P(G\(r x

E)).

Replacing E, T, G by E, f , G, we obtain a similar diagram E ^ - t x E - U G \ ( f x E) and we can consider the semisimple complex t\,s*A G T>(G\(T x E)). In - v particular, we can replace A by Res T WB and we obtain the semisimple complex < b r(ResT jW J5) G X>(G\(f x E)). Let u : G \ ( r x E) —> {point} and u : G \ ( r maps. L e m m a 9.2.9 (Adjunction).

We have a natural

Hnm(tbs*(Inc%^A)®tbs*(B))

^ nnui(hs*(A)

{point} be the obvious isomorphism ®

tbs*{Re^wB))

for n G Z. Hence, for any j € Z, we have (a)

Dj, (E, G; Ind^ ^A,

B) = D^ (E, G; A, Res^

wB)

where j' = j + 2 dim G/Q. The proof (which uses 8.1.6) is given in [2]; we will not repeat it here. The shift from j to j' comes from the formula dim(G\r) = dim(G\f)—dim G / Q .

76

9. Quivers and Perverse Sheaves

9.2.10. In order to eliminate the shift from j to j' in the previous lemma, we define ResT,W(£) = R e s ^ w M i -
D j ( E , G\ Ind^ ) W ,4, B) = D j ( E , G; A, R e s ^ > w £ ) . Note that dim G/Q =

dimTi dim Wi; hence

di - d2 - 2 d i m G / Q = ^ d i m T h < d i m W h - - ^ d i m T i d i m W | . h i The functor Res^ w is called restriction. 9.2.11. We can rewrite 9.2.6(b) as follows: R e s -

©(^T

where the sum is taken over all r = (r 1 , r 2 , . . . , r m ) , lj — (a;1, a ; 2 , . . . , um) such that dim T | = r/, dim W i = Yhi vi f° r * a n d t 1 + u l = vl for all I; we have A f ; ( r , u>) = d1-d2-

2 dim G/Q + /(u) - f ( r ) - / ( w ) - 2 M ( r , a;).

We show that the last expression is independent of the orientation of our graph. From the definitions we have M'(T,U>)

=

+E

dim T V dim

h

T ,fj

-

^T

dim T | dim W |

i

+ E li+ E

+

2

- E

h,l'
-2 E

It follows that A/'(T,U>) = —

h,l'
+ ^ ( d i m T f c / dim W/^" + dim T^" dim h

~i ;/
+i;f>{'E

which is clearly independent of orientation.

)

"i EdimTidimWi'

9.3. Categories

TV;I';>7 o,nd TV;I';7

77

9 . 3 . T H E CATEGORIES T V ; I ' ; > 7 AND T V ; I ' ; 7

Let F be a discrete subset of I (see 9.1.3). Let 7 = E i ^ 1 € N [ I ] be such that 71 = 0 for all i E I —F. Let TV;i';>7 he the full subcategory of TV consisting of perverse sheaves which are direct sums of simple perverse sheaves L that have the following property: there exists a graded subspace W C V and an object A E QT,W such that T = V / W satisfies dim Tt > if i E F, and Ty = 0 for i' $ I'; moreover, L is a direct summand of Ind T v/A. Clearly, (a) 'Pv;i/;>7 D ^>V;i/;>7/ if l ' € N[I] has support contained in F and 7i < 7i for all i E F. Any object of TV is in TV;i';>o- Moreover, TV;i',>7 is empty if 71 > dim Vi for some i E F. Let TV;i';>7 be the full subcategory of TV consisting of the objects which are in TV;i ; > 7 ' f° r some 7' E N[I] with support contained in F such that 7 ; (i) > 7(i) for all i E F and 7'(i) > 7(i) for some i E F. Let TV;i';7 be the full subcategory of TV;i;>7 consisting of those objects of ?V;i;>7 which are not in TV;i;>7- If K is a simple object of TV a n d V 0, then K is a direct summand of some shift of Lu where v starts with i/1 = 7 which may be assumed to be of form ni for some i E I and some n > 0 (see 9.1.4); we see then that K E 7V;{i};>ni- Thus: (b) if K is a simple object of TV a n d V ^ 0, then there exists i E I such that K E TV;{i};>i9.3.1.

We now assume that W C V and T = V / W are such that for any h E H we have T^' = 0 (hence Wh> = V/^). It follows that ET = 0. Moreover, we have a natural imbedding t : E w E v ; if x = (xh) € Ew? then the /i-component of t(x) = x' is the composition V ^ = "WV Wh" C Vh". (In our case we have k : F = E w and the imbedding 1 above may be identified with the imbedding F —• Ev, see 9.2.2.) From our assumption it follows that the set {i E I|T| ^ 0} is discrete; let F be a discrete subset of I containing {i E I|Ti ^ 0}. We consider the locally closed subset G of E v consisting of all x E E v such that dimV|/(YlheHh , , =i x h0^h')) = d i m T | for all i E F, and the open subset S of E w consisting of all x E E w such that = Wi for all i E F. Let p : G XQ EW —• E v be the unique G-equivariant map such that (l,x) •-> i(x) for all x E E w ; let po : G XQ S —> © be the restriction of p. Note that po is an isomorphism. The inverse map can be described as 9.3.2.

78

9. Quivers and Perverse Sheaves

follows. Let x G ©. The I-graded subspace ©l€r(

J2 xh(Vh>))®(® i e i - r V i )

heH:h"=i

of V has the same dimension in each degree as W; hence it is equal to g(W) for some g G G. The element gx is equal to i(x') for a well-defined x' G E w - Then PQ1(X) = (g,x'). In particular, G, G XQ S , G XQ E W are irreducible of the same dimension. Since p is a proper map, its image p(G XQ EW) IS a closed subset of EV containing ©. Hence dimG XQEW > d i m p ( G x Q E w ) > dimG. It follows that these inequalities are equalities; hence © is open dense in P(GXQEW)We have a commutative diagram n s,Xn G

•s a

G Xq Ew

Po

.• aW ,< ^

—~—• Ev

<

—-—

Ew

where Lo,j,jo,m denote the inclusions. Both squares in the diagram are cartesian. Let V ^ be the full subcategory of whose objects are those perverse sheaves A such that any simple constituent of A has support which meets E. Let "Pw be the full subcategory of whose objects are those perverse sheaves A such that the support of A is contained in E w — S. Let P v be the full subcategory of Vv whose objects are those perverse sheaves B such that any simple constituent of B has support which meets G and is contained in the closure of 0 . Let be the full subcategory of Vv whose objects are those perverse sheaves B such that the support of B is disjoint from 0 and is contained in the closure of G. Clearly, any object A G ? w has a canonical decomposition A = A0 0 A1 where A0 G V^ and A1 G Moreover, any object B G Vv with support contained in the closure of G has a canonical decomposition B = B° © B1 where B° G V^ and B1 G V^. Proposition 9.3.3. (a) Let A G V / / n ^ O , we have HnInd% wA G Vyf. I f n = 0, then HnInd%wA G Vv has support contained in the closure dimG of G; hence = (H ^Ind^^A)0 G V% is defined. (b) Let B G then HnRes% wB defined.

If n ^ 0, we have HnRes% wB G ? ^ If n = 0, G hence p(B) = (H~dimG/QRes% WB)° G V^ is

9.3. Categories Vv;i>;>>y and Vv j'w

79

(c) The functors £ : Vw —• *Pv am^ P : ^ v ~~* ^ w establish equivalences of categories inverse to each other. Note that j*B is a perverse sheaf on 0 , since the support of B is contained in the closure of © and © is open in its closure. Moreover, j*B is a G-equivariant perverse sheaf. Since © = G XQ E, it follows that iQ(j*B)[—dimG/Q] is a perverse sheaf on S. But m*i*B = ioj*B, hence m*t*B[—dim G/Q] is a perverse sheaf on S. Since m is an open imbedding, we have m*(Hni*B) = Hn(m*i*B) for any n, and this is zero if n / — dim G/Q. Hence if n ^ - dim G/Q, the support of Hnt*B is disjoint from E. We have Res T = R e s x w . B [dim G/Q] = t*B and (b) is proved. Let A be the perverse sheaf on G XQ EW such that A = r£rM[dim G/Q] in the diagram EW G/U x EW —1• G XQ EW- By definition, „ y _ Ind T > W i4 = p\r'br*A = p\A[- dim G/Q}. - v This shows that Ind x y^A has support contained in the image of p, hence in the closure of ©. We have r(Ind^w^) =fpiA[-dimG/Q] = (p0)ij^A[-dim G/Q]. Since jo is an open imbedding, we see that j$A is a perverse sheaf on G XQE. Since PO : G XQ E —)• © is an isomorphism, it follows that {PO)\JOA ~ v is a perverse sheaf on ©. Thus j*(Ind T w v4) [dim G/Q] is a perverse sheaf on ©. ~ v has support contained in the closure of ©, it folSince Ind T lows that Hnlnd~T v WA has support contained in the closure of © and j*(Hnlnd^?wA)

= Hn(j*Ind^ -wA). This is zero if n ^ dim G/Q. Hence ~ v is disjoint from ©. This for n ^ dim G/Q, the support of i f n I n d T

proves (a), since I n d x W A = Ind x w >l[dimG/Q]. From the proof of (b), we have m*(p(B)) =

m*(H-dimG/Qt*B)

= m* (i*B[- dim G/Q]) = i*0j*B[- dim G/Q}. This implies that Jof>{B) = p$j*B. From the proof of (a), we have j*(«M)) =

j'(Hd^G^lndZwA)

= j*(Ind^ w A[dimG/Q]) = (p 0 ),j 0 M.

9. Quivers and Perverse Sheaves

80

Hence m*(p(((A)) = t y ( « i 4 ) ) [ - d i m G / Q ] =

dim G/Q] =

m'A

and j*(ap(B))

= (po)doP(B) = (po)\Poj*B = j*B.

Since A G V^ and B G The proposition is proved.

it follows that p(£(A)) = A and £(p(B)) = B.

9.3.4. Assume that I' is a subset of I such that h! ^ I' for any h G H, that is, i is a sink of our quiver, for any i 6 I'. Let V 6 V. For any 7 = J^j 7ii G N[I] with support contained in I', let E v ; 7 be the set of all x G E\R such that dim Vi/(

Y,

x

h{yh'))

= 7i

heH-.h"=i

for any i G I'. The sets E v ; 7 form a partition of E v with the following property: for any 7 as above, the union E v ; > 7 = U y E v ; 7 ' (with 7' running over the elements of N[I] with support contained in I' such that > 71 for all i G I) is a closed subset of E v - Hence for any simple object B of Vv, there is a unique element 7 = 7 s G N[I] with support contained in I' such that the support of B is contained in E v ; > 7 and meets Ev ; 7 - We have 71 < dim Vi for all i G V. L e m m a 9.3.5. Assume that B G 7V;i';7' where contained in V. Then = 7'.

G N[I] has support

We write 7 instead of 7 s . Let W be a graded subspace of V such that T = V / W satisfies dimTi = 71 for all i G I' and T|> = 0 for all i' G I - I'. We may apply Proposition 9.3.3 to I' and B. With notations there, let A = p{B) G Vw; we have that some shift of B is a direct summand of - v I n d T W J4. Hence B G TV;i';>7From the definition of induction we see that any perverse sheaf in 7V;i';> 7 ' has support contained in E v ; > y - In particular, the support of B is contained in Ev ; > 7 '- By definition, the support of B meets E v ; 7 ; hence E V ; 7 meets E Y ; > 7 ' , so that j i > 7( for all i G I ' . Assume that 71 > 7.' for some i G I'. Since B G £V;r ; > 7 , it follows that B G Pv;i';> 7 ', which contradicts our assumption that B G TV;i'; 7 '- Thus, we must have 71 = 7.' for all i G I'. The lemma is proved.

CHAPTER

10

Fourier-Deligne Transform

10.1.

F O U R I E R - D E L I G N E T R A N S F O R M AND RESTRICTION

10.1.1. In addition to the orientation h —> h h —> hu in 9.1.1, we shall consider a new orientation of our graph. Thus, we assume we are given two new maps H —> I denoted h — i > 'h and h — i > "/i, such that for any h 6 H,

For V e V, we define ' E v like E v in 9.1.2, but using the new orientation: ' E v = ©^HHomfV'/j, This has a natural Gv-&ction just like E v

®hGHlHom(V/lSV^)0(©/lG/f2Hom(Vv,V^))®(®/i6H2Hom(V^,V/l/)).

Vw) where the two unnamed maps are components of e. Let us consider the Fourier-Deligne transform 3> : £>(E V ) —> £>('E V ) defined by = d i m S e e dimV t\(s*(K) <8> CT)[dv] where dv = Y^heH2 h' ( 8.1.11.) Now let T, W be as in 9.2.1. We may consider a diagram like (a) for T and for W instead of V; taking direct products, we obtain the diagram

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_13, © © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010

81

10. Fourier-Deligne Transform

82

On each of E t and E w we have a linear form like T above; the sum of these gives a linear form T : E T x E w —• k. The Fourier-Deligne transform $ : P ( E T x E W ) -»• £ > ( ' E T x ' E W ) is given by Q(K) = t\(s*(K) ® CT)[dT + c/w]. The following result shows the relation between the Fourier-Deligne transform and the restriction functor.

Proposition 10.1.2. For any K G Qv we have

where 7r = ^ (dimT^" dimW/i' — d i m T ^ dim W ^ ) . heH2 We consider the commutative diagram of vector spaces and linear maps

4 4

t x E W Et x Ew

E

'Et x 'E

-—

4

F * <—^—

W

F

—

^

S

4 'F

—

4 4

E

E

V

v

'EV

where the following notation is used. F is the set of all x € E v such that Xh(Wh') C Wh" for all h G H; p is the obvious surjective map and i is the obvious imbedding. 'F is the set of all x G ' E V such that xh(W>h) C W»h for all heH;'p is the obvious surjective map and 't is the obvious imbedding. F is the set of all x G E v such that sx G F and tx G 'F. S is defined by the condition that (Z, t, i,'L) is a cartesian diagram. ^ is defined by the condition that (s,p,p, s) is a cartesian diagram. q is such that sq and pq are the obvious surjective maps. C is such that LC and i(, are the obvious imbeddings. We have E = ' F 0 (®heH2Kom(Vh>, Vh»)). Let Z be the subspace of S consisting of the elements such that each component V^/ —• VV (h G H2) carries Wh" to 0 and all other components are zero. Let c : E —• E / Z be the canonical map. Let T : S —• k be given by T(x) = T ( L ( X ) ) . From

10.1. Fourier-Deligne Transform and

Restriction

83

definitions, it follows immediately that the restriction of T to a fibre (= Z) of c : S —• 2 / Z is an affine-linear function which is constant if and only if that fibre is contained in the subspace (,{F). Let S ' = S - C ( F ) , and let ( 3 / Z ) ' = 0(2'). We have Z C C(F); hence all fibres of d : S ' —> ( S / Z ) ' (restriction of c) are isomorphic to Z. Let T' : 2 ' —* k be the restriction of T. As we have seen above, the restriction of T' to any fibre of d : 2 ' —> ( 2 / Z ) ' is a non-constant affinelinear function. Hence the local system CT> on 2 ' satisfies C\(CT') = 0 (see 8.1.13). Using the distinguished triangle associated to the partition 2 = 2 ' U C(F), w e deduce that CiO(C*£J.) = c\Cf. It is clear that the composition si : 2 —• E v factors through 2 / Z ; hence i*s*K is in the image of c* so that the previous equality implies ctiOXCCf) ® i*s*K) = c\(Cf ® L*s*K). It is also clear that the composition 'pi : 2 —• 'ET X 'EW factors through a / Z . Hence the previous equality implies f

pM0.(C£f) ® i*s*K) = rp\ii(Cf <8> l*s*K). We have TL£ = Tpq\ hence p*q*Cj> = C'L*CT = C*£f • Since q is a surjective linear map with kernel of dimension m = ^ ^ dim T \ " dim W ^ , heH2 we obtain q\q*L = L[~2m] for all L G We have $(Res%wK)

= U(£T ® s*pa*K)[dT

+ dw]

= t\(CT 0p]S*L*K)[dT

+ dw]

= ti(£f ihqiq*s*i*K[2m])[dr + d w ] = t\p\q](p*q*(Cf.) 0 q*s*i*K)[2m + dT + d w ] = 'pd]0Xp*q*(CT) Ci*s*K){2m = 'pM0Xp*Q*^t)

+ dT + dw]

® i*s*K)[2m + d T + d w ]

= W K C K C ^ T ) ® i*s*K)[2m + d T + d w ] = 'p\i\(Cf

0 i*s*K)[2m

+ dT + dw]

and R e s ^ w ( $ W ) M = 'p\i*t\(CT = 'p\I\I*{CT

<S> s*K)[ir + d v ] ® S*K)[TT

+ dv]

= 'p\t\(Cf i*s*K)[7r + d v ] . It remains for us to observe that 7r+dv = 2ra-f d x + d w - The proposition is proved.

84

10. Fourier-Deligne Transform

10.1.3. We can reformulate the previous proposition using Res^ w instead - v of Res T w ; the shift by 7r will then disappear: (Res%wK) = Res%iW((K)).

1 0 . 2 . FOURIER-DELIGNE TRANSFORM AND INDUCTION

Let i/ = (i/ 1 , v2,..., i/ m ) 6 X be such that dim V J = "l FOR A 1 1 1 Recall that we have a natural proper morphism ir u : T v —> E v - The same definition with the new orientation for our graph gives a proper morphism 'TTu ' ' — • ' E v , where ' T v is the variety of all pairs ( x , f ) such that x G ' E v and / G Tu is x-stable; 'nu is the first projection. Recall the definition Lu = ( 7 G £>(Ev). Similarly, we set Lu — ('TT^I G P('Ev). 10.2.1.

= rLv[M]

Proposition 10.2.2. M

=

where

E

heH2;l>l'

~ Vh' VK" )'

Consider the commutative diagram

fv Ev

• s ——> 2 i

Ev

• Ev

where the following notation is used. E is the set of all (x, y, f ) where x G E v , / is an x-stable flag in Tv and y G ' E v is such that yu = Xh : V/»/ —• for any h G H\. E is the set of all ( y , / ) where 3/ G ' E v and / = (V = V ° D V 1 D ••• D V = 0) is a flag in Tv such that yh(Vlh') C Vlh„ for all I and all h G H\. The lower horizontal maps are as in 10.1.1(a); the other maps are the obvious ones. The left square is cartesian. We have s*(7r„)il = p\l. Hence M

$ ( L „ ) = t\(CT <8> P\l)[dv] = k is as in 10.1.1, f : S where T : E v 'Evt = tp : E

t\(Cf)[dv) k is given by t = Tp and

10.2. Fourier-Deligne Transform and Induction

85

The fibres of c are affine spaces of dimension N = J2heH2;Ki' ^h'^h'" (In the formula for N we have v l h , = 0 for / = since vl is discrete.) We have a partition S = Eo U Ei where Ho is the closed subset of S consisting of those (x, y, / ) such that / is y-stable. It can be verified that the restriction of T to the fibre of c at c(x, y, f ) is an affine-linear function and that this function is constant if and only if (x,y,f) G Eo- Note that Eo is a union of fibres of c. Using 8.1.13, it follows that ( c i ) ^ ^ ^ ) = 0, where d : Ei —• E is the restriction of c. Hence, if j : So —> E is the inclusion, we have c\j\(j*Cf) = c\Cf. From the commutative diagram above, it then follows that (tp)\(Cf = (*oM£f bo) where to : Eo —> ' E y is the restriction of tp. Let (x, y, / ) G E 0 with / as above. We have T[x, y, / ) - T{:r, y) = hEH2

tr (yhxh : Vh, - V fc /).

Since / is stable under both x and y, we have tr (yhxh : V * - Vh.) = £

tr

:V ^ M *

Vtf/V1*,).

i For any I, at least one of the vector spaces V ^ / V ^ , , Vj l 7/ 1 /Vj l „ is zero, since vl is discrete. Thus, tr (yh%h V^/ —> V^/) = 0 for each h G H2, so that T(x,y,f) = 0. Since T is identically zero on So, we have Cf |s 0 = 1 and we see that {tp)i(Ct = (t 0 )|l. Now to can be factored as a composition So —>'T v — ' E v , where the first map (restriction of c) is a vector bundle of dimension N . Hence (t0)a = (,7r1/)!l[-2TV] = 'LU[—2N\. It follows that ( t p ) \ ( C f = (to)\l = '!/„[—27V]. It remains for us to observe that d v — 2iV = M. The proposition is proved. 10.2.3. Using the proposition and the general properties of the FourierDeligne transform (see 8.1.11) we see that $ : 2>(Ev) —• P ^ E v ) defines an equivalence of categories Qv —• 'Qv and Vv —*• ' ^ v , where 'Qv,"Pv are defined as but using the new orientation of our graph. Hence $ induces a bijection between the set of simple objects in Vv and that in

86

10. Fourier-Deligne Transform

10.2.4. We have a natural action of (k*)H on E v (resp. on Tv ) given by (00 : (xh) »-> (ChXh) (resp. « f c ) : ((xh),f) ((ChX h ),f )• The map is n H compatible with these actions. It follows that H Lv is (k*) -equivariant for any n. Hence any K G Vv is (k*)H-equivariant. In particular, we have j*K — K, where j : E v —• E v is the involution which acts as —1 on the summands Hom(VV, Vh») for h € H2 and as 1 on the other summands. Hence for K G Vv, the Fourier inversion formula (see 8.1.11) simplifies to $($(#)) = K. 10.2.5. Let A G Qv and let A' G 'Qv- For any j G Z, we have a canonical isomorphism D, (E V , GV; A, $(A')) = D / E v , Gv; *{A), A'). This follows by applying 8.1.12 to the diagram Gv\(r

X

Ev)

Gv\(r

X

Ev)

Gv\(r

X

'Ev)

obtained from 10.1.1(a), where T is a suitable smooth variety with a free GV-action. Proposition 10.2.6. With the notations of Proposition 10.1.2, let L G QT,W- There exists an isomorphism in 'Qv: $(Ind%^wL) ^ 7 n d ¥ | W ( M ) . Since'Vv is stable under Verdier duality, we see from 9.1.6 that it suffices to check that (a) dimDjCEv^vl^^nd^wL),^) = dimD/Ev,GV;Ind£w($iO>^) for any K G Vv and any j G Z. By 10.2.5, the left hand side of (a) is equal to dim D j (Ev, Gv; I n d ^ w ^ K) and by 9.2.9, this is equal to dim DJ (ET X E

W

,GT X

By 9.2.9, the right hand side of (a) is equal to dimDj('ET x ' E W , G T x G w ; $ L , R e s ^ ) W ( $ / 0 )

10.3. A Key Inductive Step

87

and by 10.2.5, this is equal to dim DJ(ET

X E

W

L, $ ( R e s ^ ? w ($#)))•

, C T X GW;

Hence (a) is equivalent to d i m D j ( E T X E W , G T X G W ; L, R E S ^ w ^ ) = dim DJ (ET X E

,G

W

But this follows from Res%wK proposition is proved.

X G

T

W

; L, $ ( R E S ^ W ( $ # ) ) ) •

= ®(Res%;w($K))

(see 10.1.3).

The

1 0 . 3 . A K E Y INDUCTIVE S T E P

Lemma 10.3.1. Let I', 7 be as in 9.3.1. The Fourier-Deligne transform $ : Vv —* 'Vv defines an equivalence of categories between and the analogous category 'Vv-,v^ defined as Vw,i'a with respect to the new orientation. This follows immediately from the definitions since the Fourier-Deligne transform commutes with Ind. Proposition 10.3.2. Let I', 7 be as in 9.3.1. Let W be a graded subspace of V such that T = V / W satisfies dimTi = 71 for all i 6 I7 and T^ = 0 for all i' € I - F. (a) Let B be a simple object ofVvj'w Res^wB

^AB

where A is a simple object ofVw;i';o

We have (©jl/jb'l)

and Lj G ?w;i';>o for all j.

(b) Let A be a simple object o/?w;i';0- We have B ® (®jCj[j}) where B is a simple object ofVvj'57

and

Cj G Vv;I';>*y for all j. (c) The maps B A in (a) and A i—• B in (b) are inverse bijections between the set of isomorphism classes of simple objects in Vv,v-^ and the analogous set forVw;F;0-

88

10. Fourier-Deligne

Transform

This statement is independent of the orientation of our graph: we use the previous lemma and the fact that the Fourier-Deligne transform commutes with Ind and Res. Hence it is enough to prove the proposition under the additional assumption that h' ^ I' for any h G H. We can achieve this by a change of orientation. Let A be as in (b). By Lemma 9.3.5 , the support of A meets Ew ; oHence Proposition 9.3.3 is applicable to A, I'; it shows that I n d x w > l = B © (0-,-Cj[j]) where B is a simple object of Vv such that the support of B is contained in E y ; > 7 and meets E v ; 7 ; Cj G Vv has support contained in E v ; > 7 and is disjoint from E v ; 7 for any j. By Lemma 9.3.5, we then have B G 7 Conversely, let B be as in (a). By Lemma 9.3.5, we have that the support of B is contained in E v ; > 7 and meets Ev ; 7 - Hence Proposition 9.3.3 is applicable to B and I'. It shows that Resx w ^ — A ® {®jLj\j}) where A is a simple object of ? w such that the support of A meets Ew ; o and Lj G has support disjoint from Ew ; o for any j. By Lemma 9.3.5 we then have A G ?w;i';0 and Lj G ?w ; r;>o- This proves (a), (b). Statement (c) follows from the last assertion of Proposition 9.3.3. 10.3.3. Remark. The previous proof shows that, given I' as above and a simple object B in Vv, there is a unique 7 G N[I] with support contained in V such that B G Vv,!'^The existence of 7 is obvious. To prove uniqueness, we may assume that the orientation has been chosen as in the previous proof; but then 7 is such that the support of B is contained in E v ; > 7 and meets E v ; 7 and these conditions determine 7 uniquely since the support of B is irreducible. 10.3.4. Passage t o t h e opposite orientation. Let V G V. For each i G I, let Vf be the dual space of V | and let V* = 0|Vj* G V. Assume now that the new orientation (see 10.1.1) of our graph is the opposite of the old one, that is, 'h = h" and "h = h! for all h G H. We have an isomorphism p : E v = 'Ev* given by p(x) = x' where x'h : V*h>> —• VJ, is the transpose of Xh '• VV —• Wh" • This induces an equivalence of categories pi : £>(E V ) = X>(Ev) with inverse p*. v\ for all i G I. Let v = (i/1, i / 2 , . . . , vm) G X be such that dim Vj = m m_1 1 Let 1/ = (i/ , i / , . . . , i/ ) G X. It follows immediately from definitions that p\Lv = Lu> G X>(Ev*). From this we deduce that p\ defines equivalences of categories Vv —> 'Vv* and Qv —^ 'Qv*-

CHAPTER

11

Periodic Functors

11.1.1. Let C be a category in which the space of morphisms between any two objects has a given Q/-vector space structure such that composition of morphisms is bilinear and such that finite direct sums exist. We say that A functor from one linear category to another linear category is said to be linear if it respects the Q/-vector space structures. 11.1.2. Assume that we are given an integer n > 1 and a linear functor a* : C —> C such that a* n is the identity functor from C to C. We say that We define a new category C as follows. The objects of C are pairs (A, ) where A is an object of C and <j> : a* A —> A is an isomorphism in C such

Let (A, <j>) and (A1, <j>') be two objects of C. There is a natural automorphism u : Rom(A,Af) —> Hom(yl,yl / ) given by u ( f ) = 0 / a*(/)<^~ 1 . From

The composition of morphisms in C is induced by that in C. The direct sum of two objects (A, (f>) and 0') of C is (A © A(j) © ), (A', (//), (A", <j>n) of

(a) There exist morphisms i" : A" —> A and pf : A —> A' in C such that

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_13, © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010

89

11. Periodic Functors

90

We show that {A',') © {A",<j)") ^ (A,) in C. Recall that un(i") = i" where u : Hom c (,4",,4) -»• Hom c (A // ,>l) is as in 11.1.2. Let i" = E " = i u>(i")/n : A" -> A. Then i" E H o r n & { A ' \ A ' ) and p"i" = \A„. We set the p' = p' - p'i"p" : A-> A'. Then p'i' =

p'i" = 0, p"i' = 0, i'ff + i"p" = 1A.

It follows that (i'ji") define an isomorphism (A',(f>') © (A", ") —• (A, ) (in C). Our assertion is proved. 11.1.4. An object (A,(j>) of C is said to be traceless if there exists an object B of C, an integer t > 2 dividing n, such that a*lB = B, and an under which

], one for each isomorphism class of objects (B, ) of C, subject to the following relations: (a) [B, cj>] + [B\ cj>'\ = [B © B\ cf> © <j>'\(b) [B, (/>] = 0 if (B, 4) is traceless; (c) [B, C<{>} = C [By 0] if C € Qi satisfies Cn = 1. This definition is similar to that of a Grothendieck group. 11.1.6. Now let C' be another linear category with a given functor a* : C' —> C' such that a*n is the identity functor from C' to C'. Let b : C C' be a linear functor. Assume that we are given an isomorphism of functors ba* — a*b : C —> C'. Then b induces a linear functor b : C —• C' by b(A,(f>) = (bA,<j>') where ' : a*bA —> bA is the composition a*bA = ba*A bA. It is clear that [A, 4] ^ [bA, (j)'] respects the relations of JC(C), JC(C') and hence defines an 0-linear map fC(C) —> K,(C'). 11.1.7. Assume now that C is, in addition, an abelian category in which any object is a direct sum of finitely many simple objects. Let B be a simple object of C. Let ts be the smallest integer > 1 such that ( a * ) t B B is isomorphic to B; let f s (a*)tBB —> B be an isomorphism. We have n = n'tB where n' is an integer > 1.

11.1. Periodic Functors

91

The composition (a)

B = (a*)n'tBB

-> • • • a*2tBfB> a*2tB B

a

*'B/B> a*tB B

B

is a non-zero scalar times identity (since B is simple); hence by changing f b by a non-zero multiple of we can assume that the composition (a) is the identity. Consider the isomorphism 4>b : a*(B © a*B © • • • ( a ^ ^ B ) = a*B © a*2B © • • • (a*)tBB which maps the summand a*^B onto the summand a*i B by the identity map (for 1 < j s) is an object of C. Let 5 be a set of simple objects of C with the following property: any simple object in C is isomorphic to a*^B for a unique B in S and some j > 0. For each B in S we choose 0B as above. It is easy to see that any object of C is isomorphic to (b)

©BGS((£ © a*B © • • • (a*)tB~1B)

® EB, 4>B ® 1>B)

where for each B, EB is a finite dimensional Qj-vector space with a given automorphism tps Eb —• Eb such that ipB = 1 a n ( ^ = 0 for all but finitely many B. Note that the summands corresponding to B such that t s > 2 are traceless. Let B be a simple object of C such that a* B = B. The isomorphisms (f> : a*B = B such that (B,(f>) G C, generate a free (9-submodule of rank 1 of Home( a *B, B)\ w e denote this (9-submodule by OB- It is easy to see that OB depends only on the isomorphism class of B. 11.1.8. From 11.1.7 it follows easily that (a)

K(C) =

®bOb

as O-modules (the sum is taken over the isomorphism classes of simple objects B such that a*B = B); to the element (j) G Ob such that (B, ) G C corresponds the element [B,(p] G JC(C).

Quivers with Automorphisms

12.1.1. An admissible automorphism of the graph (I, H,h > [h]) consists, by definition, of a permutation a : I —> I and a permutation a : H —> H such that for any h G H, we have [a(/i)] — a[h] as subsets of I and such that there is no edge joining two vertices in the same a-orbit. In this chapter we assume that we are given an integer n > 1 and an i > [ft]) in 9.1.1. We assume admissible automorphism a of the graph (I, H,h — that a n = 1 both on I and on H. From the definition it follows that (a) any a-orbit on I is a discrete subset of I, in the sense of 9.1.3. An orientation h —> h\h —> h" (see 9.1.1) of our graph is said to be compatible with a if, for any h G H, we have (a(h))' = a(h') and {a{h))n — a{hn). From property (a) we can deduce that there is at least one orientation of our graph which is compatible with a. This is seen as follows. Choose a set of representatives Ho for the a-orbits on H. For each h G HQ, choose one element h! of [h]\ let h" be the other element of [h]. Now let h G H. We can find n G Z such that h = an ho where ho G Ho is uniquely determined. We set h' — an(ho) and h" = an(hQ). We must prove that h h " are independent of the choice of n. We are reduced to verifying the following statement: if ra G Z satisfies a m /io = ho, then we have a m (/iQ) = h'0 and a711 (h^) = /IQ. If this is not the case, we have

12.1.2. Let Va be the category of finite dimensional I-graded A;-vector spaces V = 0 i e i V | with a given linear map a : V —> V such that a(V\) = V a (j) for all i G I and a 7 |vi = lvi f° r all i G I and all j > 1 such that a J (i) = i. The morphisms in V a are isomorphisms of vector spaces compatible with the grading and with a. Note that a : V —• V automatically satisfies

For each v G N[I] a , we denote by V® the full subcategory of Va whose objects are those V such that dimVi = v\ for all i G I. Then each object

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_13, © Springer Springer Science+Business Science+Business Media, 2011 Media, LLC LLC 2010

92

12.1. The Group fC(Qv)

93

of Va belongs to VJ for a unique v G N[I]° and any two objects of VJ are isomorphic to each other. Moreover VJ is non-empty for any v G N [ I ] A . We choose an orientation of our graph, compatible with a. If V G V a , then we may regard V as an object of V (by forgetting a); in particular, G v and E v are defined. There is a natural automorphism a : G v —>' Gv given by a(gz) = a(g)(a(z)) for all g G Gv and z G V; moreover, there is a natural automorphism a : E v —• E v (x •—• a(x)), such that for any x = (Xh) G E v and any h G H, the compositions VH' ^

V h „ A V A ( F C „) and V h . A V a{h>)

va{h,t)

coincide. Both of these automorphisms satisfy a n = 1. Note that a(gx) = a(g)(a(x)) for all g G G v and x G Ev- Taking the inverse image under a : E v —'• E v gives us a functor a* : £>(E V ) —>> ^ ( E v ) . v f o r a11 1 Let v = (i/1, i / 2 , . . . , i/ m ) G X be such that dim Vi = \ € here, vl G N[I] for each I. Let i / = (i/ 1 , i / 2 , . . . , i / m ) G * be defined by v'\ = ^(i)- We have natural isomorphisms a : Tv> —> Tv and a : fw — given by a ( f ) = (V = a(V°) D a ^ 1 ) D - •• D a ( V M ) = 0) for / = ( V = V ° D V 1 D • — D VTH = 0) and a(x,f)

= (ax, a / ) .

It

follows that a*Lu = LU'. From this we see that a* takes Qv into itself and TV into itself. Applying the definitions in 11.1.2, 11.1.5 to the linear categories Q v , ? V and to the periodic functor a* on them, we obtain the linear categories Q v , ^ V and the O-modules /C(Qv), JC(Pv). 12.1.3. We shall use the notation O' = 0{v,v~l] (v is an indeterminate). Now K(Qv) is naturally an O'-module by vn[B, ] = [B', '], where (B', ') is obtained from (B,) by applying the shift [n]. We define O'-linear maps fC(Qv)

O'

K(Vw)

IC(Qv)

as follows: p' sends a symbol [B,] (where B G Vv) to [2?,$]; p sends v~n[HnB, Hn()]. It is easy to a symbol [B,] (where B G Qv) to verify that p, p' respect the defining relations; hence they are well-defined. It is also clear that pp' = 1. / If K G Qv and we are given an isomorphism <j> : a* K —*• K, then <> induces isomorphisms (j)n : a*(HnK[—n]) = HnK[—n] for each n; moreover we have (K,4>) = ®n(HnK[—n\,(f)n) as objects of Qv- This follows from

94

12. Quivers

with

Automorphisms

8.1.3(a) by repeated applications of 11.1.3. This shows that p'p — 1. Thus, p' is an isomorphism
= £(Qv).

Using this and 11.1.8, we see that (a)

®BOf ®o Ob = fC(Qv)

where the sum is over a set of representatives B for the isomorphism classes of simple perverse sheaves in Vv such that a*B = B; OB is a free 0-module of rank 1 defined as in 11.1.7 (to (J> G OB such that (B,]EK{Qv)).

12.1.4. Now let T , W be two objects of V a . We can form E T , E W and their product ET X EW- This has an action of GT X GW (product of actions as in 9.1.2) and an automorphism a (product of the automorphisms a of the factors) such that an = 1. The functor a* : £ > ( E T X E W ) —*> T>(ET X E W ) takes the subcategories VT,W and QT,W into themselves. Applying the definitions in 11.1.2, 11.1.5 to these linear categories and to a*, we obtain linear categories VT,W, QT,W and (9-modules JC(VT,W)I)C(QT,W)These are special cases of the definitions of Vv, Qv and K(Vv), £(Qv) in 12.1.2; indeed, T x W and E T x E W are special cases of V and E v where our graph has been replaced by the disjoint union of two copies of itself. Thus, JC(QT,W) is naturally a O'-module and we have CT <8>o £(7>t,w) = K(Qt,W) and ®bO'

®O

Ob=JC(QT,W)

where the sum is over a set of representatives B for the isomorphism classes of simple perverse sheaves in VT,W such that a*B = B, OB is a free Omodule of rank 1 defined as in 11.1.7 (to G OB such that (B,(f>) G VT,W corresponds [B,<j>] G £ ( Q T , W ) ) From the definitions it is clear that any simple object B G "PT.W IS the external tensor product B' B" of two simple objects B' G VT and B" G Vw (and conversely); we have a*B = B if and only if a*B' ^ B' and a*B" ^ B" and then Horn(a*B,B) = Hom(a + B / , B') ® Hom(a*£", B") and OB = OB' ®O OBIt follows that £ ( Q T , W ) = K ( Q T ) ®O> / C ( Q W ) .

12.2. Inner Product

95

1 2 . 1 . 5 . Assume now that W is an I-graded, a-stable subspace of V and that T = V / W with the induced grading and a-action. The functors Indip W : QT,W —5- Qv and Res^ w : Qv —'• QT,W are linear and compatible with a*; hence by 11.1.6, we have induced linear functors QT,W —» Qv, Qv —• QT,W and CMinear maps

ind^W

:

£(QT,W)

fc(Qv)

and

res^W

:

£(Qv)

—>' £(QT,W)-

These last two maps are in fact O'-linear since Ind^ w commute with shifts.

and R e s ^ w

1 2 . 2 . INNER PRODUCT

12.2.1. Let (#',") be two objects of Qv. The vector space ~ D j ( E , v , G v , B ' , B n ) (see 8.1.9) has a natural automorphism a such that an = 1. This can be constructed as follows. Let V be an I-graded k-vector space such that for each i £ I, dim V\ is finite but large. Let T be the smooth, irreducible variety consisting of all injective linear maps V —> V which respect the I-grading. Then T has an obvious free Gv-action and its Qj-cohomology is zero in degrees 1 , 2 , . . . , m where m is a large integer. By the definition of V a , we have an automorphism a : V —• V. This induces an automorphism a : T —• T. Taking the product with the automorphism a : E v —>! E v , we obtain an automorphism a : T x E v —> T x E v This induces an automorphism a of the orbit space G v \ ( r x E v ) . The isomorphisms B' and <j)" : a*B" —> B" induce isomorphisms 4>' : a*B' B' and 4>" : a*B" B" where B' = t^B'.B" = t\,s*B" are semisimple complexes in G v \ ( T x E v ) defined as in 8.1.9. Now ft j>" : a*(B' B") -> {& <8> B") induces an isomorphism of Hj+2dim^r)(rn(B'®B")) onto itself, or equivalently, an isomorphism of D j ( E v , Gv; B\

B")

onto itself, denoted again by a. Here u denotes the map of G v \ ( r x E v ) into the point. It is clear that a : D j ( E v , Gw\B', B") -»• D j ( E v , G v ; B', B") satisfies a n = 1. We define {(B',4>'),(B",")}

= £ j

tr

( a - D j ( E v , G v ; B ' , B " ) ) f - J ' € 0((i;)).

12. Quivers with

96

Automorphisms

The last inclusion follows from 8.1.10(e). (In general, given a ring R, a n € R such that an — 0 for the ring of power series E n e z n 0) is denoted R((v)) (resp. /2[[v]]; or R((v~i)); or R[[v~'}]).) If (B', (f>f) is traceless, then we can write B' = C ® a*C © • • • © (where t > 2) and (pf acts like a matrix without diagonal terms. We have a corresponding decomposition Di(Ev,Gv;B/,B") = Dj(Ev,

C, B") © Dj(Ev, GV; a*C, B") © • • •

©Dj(E w . G w ' ^ - ^ C . B " ) Then a : r>j(EVjGv; B', B") D j ( E v , Gv; B") acts with respect to the above decomposition like a matrix without diagonal terms; hence it has trace zero. Thus, {(£', '), (B", ")} = 0 if either (£',') or {B",(f>") is traceless. It follows immediately that {[B',4>% \B\4,"}} =

(B",f)}

is a well-defined symmetric CX-bilinear pairing JC(Qv) x/C(Qv)x - 0 ( ( v ) ) . We can define in the same way a symmetric O'-bilinear pairing {, } : AC(QT,W) X £ ( Q T , W ) X

0({v));

indeed, as we have seen, /C(QT,W) is a special case of /C(Qv) in the case where our graph has been replaced by the disjoint union of two copies of itself. Using the definitions and 8.1.10(f), we see that {^i (8) A3, A2 ® A4} = for any A\,A2

6 JC(QT)

Ai

and AZ,Aa

e

£(Qw)- We then have

A3, A2®A4e

We have the following result.

Aa)

/C(QT,W)-

12.3. Properties of Lv

L e m m a 12.2.2. Let a G

£(QT)w),/? G K(QV).

97

We have

{a, re$T )W (/?)} = {«™J^ w (a),/3}. Let (A, 0) be an object of QT,W and let (£?, 0') be an object of Q Y We take the trace of the automorphism a in both sides of 9.2.10(a); the lemma follows. 1 2 . 3 . P R O P E R T I E S OF LU

12.3.1. Let XA be the set of all V = (i/ 1 , z/ 2 ,..., VM) G X such that vi e N [j]a for all I L e t u = (j/^ I/ 2 ) . . . ^ " i j e r be such that dim Vi = ^i for all i G I. We have natural automorphisms a : Tv —> Tv and a : Tv —> Tv defined as in 12.1.2. The obvious isomorphism a*l = 1 on T v induces an isomorphism 00 : a*{nv)\l = (7^)1 (o* 1) = (nu)\l in P ( E V ) . It is easy to see that (L„,0o) is an object of Qv- Applying to 0o a shift by / ( f ) (see 9.1.3(c)), we obtain an isomorphism a*Lu = Lu which is denoted again by 0o- Note that (Z/^,00) is an object of QvLet W C V and T = V / W be as in 12.1.5. Let i / = (i/ 1 , i / 2 , . . . , i / m ) G a A* A' and i/" = ( i / / ; 1 , i / / 2 , . . . , i / , / n ) G A"1 be such that d i m T j = dim W i = for all i G I. L e m m a 12.3.2. The following equality holds in /C(Qv)-' ind% W([LU>,(j>0] 0 [!/„",0O]) = [Lu*„»,<J. This follows from 9.2.7. L e m m a 12.3.3. Let v = (i/ 1 ,!/ 2 ,... ,i/ m ) G A"1 be such that dimVi = I// for all i G I. The following equality holds in J C ( Q T , W ) - ' re%iV,[LuM =

<S>

where the sum is taken over all r = ( t ^ t 2 , . . . , r m ) , u ; = (a;1, a ; 2 , . . . ,cj m ) in XA such that dimTi = dim W i = J^i for all i and T1 +ul = vl for

all I; M'(T,U)

is as in

9.2.11.

This follows from the decomposition 9.2.11. The terms in that decomposition corresponding to T,W with some coordinate not in N[I] a can be grouped according to non-trivial a-orbits and contribute traceless objects, hence they disappear from the final result.

98

12. Quivers with Automorphisms

L e m m a 12.3.4. Let i be an a-orbit on I and let 7 = ]T\ 6 i i € N[I]. Let V e Va be such that dimVi = n for all i € i and d i m V j = 0 for all i £ i. Let d be the number of elements of i. The following equality holds in n-1

(a)

[ Z 7 ) ( n _ l h , fa] = J 2 v ~ 2 8 d [ ^ 5=0

M

Using the definitions and the known structure of the cohomology of a product of d copies of a projective (n — l)-space, we see that the left hand side of (a) is represented by ( 0 1 [—2si — 2s2 — • • • — 2Sd],<j>), a sum over all sequences (s 1,32,... ,3d) with 0 < Sj < n — 1; here (f> maps the summand corresponding to («i, s2,..., Sd) to the summand corresponding to (S2, S 3 , . . . , Sd, s\). The summands corresponding to sequences whose terms are not all equal to each other, form traceless objects. The remaining terms give the right hand side of (a). 12.3.5. R e m a r k . In the previous lemma we have [L 7 ) ( n _i) 7 , o] = d n 1 hence v ( - )[Z 7 > ( n _ 1 ) 7 ,^ 0 ] and [Lni,0] = [ L n 7 > M n-1

(a)

[i-r,(n-i)7. -Ao] =

£

v~2sd[Lny,

0O],

s=0

L e m m a 12.3.6. We preserve the assumptions of Lemma 12.3.4, take n — 1. We have

an

d we

00

(a)

{[L 7 ,0o], [£-,,(>]} = Y , v 2 d S = 0- - " V s=0

Since i is discrete, we have E v = 0 and L 7 = 1. Using the definitions we see that the left hand side of (a) is computed as follows: we consider the product of d copies of an infinite projective space with the automorphism given by cyclically permuting the factors; we must find the trace of the induced automorphism on each cohomology space. This is given by the same computation as in the previous lemma. L e m m a 12.3.7. Let v = (1/1, i / 2 , . . . , i/ m ) and v' = (i/ 1 , i / 2 , . . . , v'n) be two elements of Xa such that dimVi = v\ = Yli v'\ for aM i € I- We have {[Lu,(j>o],[L^,o}} G Z((v)) n Q(v). Using 9.1.4, we see that we may assume that each vl and each v'1 is some a-orbit i in I. Using 12.3.5, we can further of the form N E i e i * assume that we always have N = 1.

99

12.4• Verdier Duality

We argue by induction on m + n. The case where m or n is zero, is trivial. The case where m = n = 1 follows from 12.3.6. Hence we may assume that m or n is > 2. Assume first that n > 2. Let W be an I-graded a-stable subspace of V such that T = V / W satisfies dimTi = v\ for all i € I. Let i/ 1 = (i/ 1 ), i/ 2 = (i/2, i / 3 , . . . , i/ m ). We have

= {[Lvi, 0O] <8>

, 0O], res^wI-^'> <M}

and this is in Z((v)) fl Q(v), by 12.3.3 and the induction hypothesis. We can treat similarly the case where ra > 2. The lemma is proved. 1 2 . 4 . V E R D I E R DUALITY

12.4.1. Let V € V a . Let {B,) be an object of Qv. Recall that (f> : a*B ^ B. Applying Verdier duality, we obtain D() : D(B) ^ D(a*B) = a*(D(B)). The inverse of this isomorphism is an isomorphism D{(f>)~1 : a*(D(B)) D{B). It is clear that (D(B), D(<j))~l) is an object of Qv. It is easy to see that we have a well-defined homomorphism of abelian groups D : K(Qv) £ ( Q v ) given by D[B,] = [D(B), D()~1]. This homomorphism has square equal to 1. Moreover, from the definitions, we see that it is semi-linear with respect to the involution : O' —• O' given n n 1 n and C -»• C" for C such that C = 1. by v v~ Let W C V and T = V / W be as in 12.1.5. We have a homomor£ ( Q T , W ) defined in the same phism of abelian groups D : )C(QT,W) way as D : K,(Qv) —• JC(Qv) (and in fact is a special case of it). From the definition we see that the following diagram is commutative: 12.4.2.

JC(QT)

® £(QW)

I

£(QT,W)

JC(QT)®JC(QW)

1

d •

£(QT,W)

where the tensor products are over O' and the vertical maps are the isomorphisms in 12.1.4. Lemma 1 2 . 4 . 3 . For any a mcQpW(Z)Q) in K,(Qv). This follows from 9.2.5.

G £(QT,W);

we have D(incE£w(a))

=

100

12. Quivers with Automorphisms

1 2 . 5 . SELF-DUAL ELEMENTS

L e m m a 12.5.1. Let B G Vv be a simple object such that a*B = B. (a) If i G I and n € N are such that B G Vv,{i},ni> then B G Vv,i\ni where i is the a-orbit of i and 7 = X^'ei*' £ N[I]. (b) Assume that B G Vv-,i-,y where i is as in (a) and 7' G N[I] has support contained in i. Then 7' = 717 /or some n > 0, where 7 is as in (a). (c) / / V / 0 , then there exists an a-orbit i in I such that B G w/iere 7 is as in (a,), /or some n > 1.

Vv-,i-,n7

We prove (a). Our statement is independent of the orientation, as in the proof of 10.3.2. Hence we may assume that the orientation is such that h' £ i for any h G H. Such an orientation exists since i is discrete. By our assumption and by Lemma 9.3.5 we have

dim Vi/( Y ,

MVh'))>n

heH:h"=i

for all x in the support of B and dim V | / (

Y

*h(Vh,))

= n

heH:h"=i

for some x in the support of B. Since the support of B is a-invariant, it follows that the same holds when i is replaced by any i' in i. Using Lemma 9.3.5 again, we see that the conclusion of (a) holds. We prove (b). Let i G i. We can find n G N such that B G 7V;{i);ni- By (a), we have B G Vv,i;n-y By 9.3.5, the inclusions B G Vv,i-,n-y,B G Vv,iyy' imply 7' = 717 and (b) follows. (c) follows immediately from (a) and 9.3.1(b). Proposition 12.5.2. Let B be a simple object of Vv such that a*B is isomorphic to B. (a) There exists an isomorphism <j> : a*B = B such that (B,(j>) G Vv and such that (D(B), D(0)_1) is isomorphic to (B,) as objects ofVv Moreover, (f> is unique, if n is odd, and unique up to multiplication by ±1, if n is even. (b) We have D{B) = B as objects ofVv (b) clearly follows from (a). We prove the existence of <j> in (a). This is trivial when V = 0. Hence we may assume that V ^ 0. By 12.5.1, there

12.5. Self-Dual Elements

101

exists an a-orbit i on I and an integer n > 1 such that B G Pv^rry where 7 = i. Hence it is enough to prove the following statement for any fixed a-orbit i in I. For any n > 1, and any simple object B G Vv,i-,n-y such that a*B = B, there exists 0 : a*B = B such that (B,<J>) G Vv and such that (.D(B),D(<j>)-1) is isomorphic to (B,). We argue by descending induction on n. (We have n < dimV|, for any i G i). Let S be the set of all simple objects C' G Vv;i-,>n-y (up to isomorphism) such that a*C' = C; for each such C", we denote by fc> some isomorphism a*C' ^ C' such that ( C ' , f c ' ) e Vv- By 12.5.1(b), for each C' G <S, we have C' G "Pv;i;n'7 for some n' > n, hence the induction hypothesis (on n) is applicable. Hence we may assume that f c is such that (.D{C'),D(fc')~l) is isomorphic to (C'Jc) for each C' G S. Let W be an I-graded a-stable subspace of V such that T = V / W satisfies dimTi — n for all i G i and Ti = 0 for all i G I — i. Such W exists since i is an a-orbit. By Proposition 10.3.2 (for I' = i) we can find a simple object A of V\v,i,o such that Ind^f = B © (©jCjbl) where Cj G Vv ;i;>n7 for all j. Since Ind^ w commutes with a* and a*B — B, we have also Ind^f w a M = B(B(®ja*Cj\j]). By the uniqueness of A in 10.3.2, we must then have a* A = A. By the induction hypothesis (on dim V) we can find an isomorphism 0' : a* A == A such that (A, (j)') G Vw and such that ( J D(^), J D(0 / )~ 1 ) is isomorphic to {A,'). Applying to [A, (f)'} the homomorphism i n d £ w : /C(Qw) = /C(QT>w) -> £(Qv), we necessarily obtain an element of the form [£, 0] + S e e s Pc[C\ fc] where <j> : a*B = B is some isomorphism such that (B,(J>) G Vv- The coefficients P c are in O'. Since D[A,'] = [A,'\, we have D(ind%wA) = i n d ^ w , 4 (see 12.4.3). Hence we have £>[£, 0] + ^

Pc>D[C', fc] = [B, 0] + Y i

(Pc as in 12.4.1). Substituting here D[C', fc>] = [C'Jc], D[B,4>] = \B10] + Y(pc> C'

~

we obtain

Pc)[C'Jc]-

Now the elements [B,0] and [C, fc] form a subset of a basis of the (9'-module JC(Qv) and D[B, 0] is a scalar multiple of some element in that

12. Quivers with Automorphisms

102

basis. Hence the previous equality forces D[B, ] to be equal to [B, 4>}. This proves the existence of . Assume that has the same property as <j>, where ( satisfies Cn = 1. We have C[B,4>] = [B,C4>] = D[B,C4>] = D(C[B,(f>}) = ClD[B,] = Cl[B,Y, hence C = C _ 1 and £ — ±1. If n is odd, it follows that £ = 1. The proposition is proved. Lemma 12.5.3. Let (B,), (£',') be objects ofVv simple objects of Vv.

such that B,B'

are

(a) If B, B' are not isomorphic, then {[B,], [#',']} G vO[[v\\. (b) If(B',') = (D(B),D((p)~1),

then {{B, <J>], [Bf, ')} el + vO[[v]].

(c) If (B, (p) is as in 12.5.2, then {[B, ], [B, <£]} G 1 + vO[[v]]. (a) and (b) follow from 8.1.10(d) and the definitions, using the fact that D(B) ~ B (see 12.5.2(b)). (c) follows from (b). 1 2 . 6 . Lv

AS A D D I T I V E G E N E R A T O R S

Let V G Va. Let M V be the ,4-submodule of / C ( Q V ) generated by the elements [Lu,(f>o] for various u = (i/1, v2,..., i/ m ) G Xa such that dimVi = A f o r a11 i € I. 12.6.1.

Lemma 12.6.2. For any a, a' € Mv we have {a, a'} G Z((v)) D Q(v). This follows immediately from 12.3.7. Proposition 12.6.3. The following two A-submodules ofJC(Qv) coincide: (a) M v ; (b) the A-submodule generated by the elements [B, (/>] where (B, (f>) is as in 12.5.2. We show by induction on dim V that (c) if (B,) is as in 12.5.2, then [B,(f>] G Mv. This is trivial when V = 0. Hence we may assume that V ^ 0. By 12.5.1, there exists an a-orbit i on I and an integer n > 1 such that B G ? v ; t w where 7 = J 2 i e i i. Hence it is enough to prove the following statement for any fixed a-orbit i in I. For any n > 1, and any (B,<j>) as in 12.5.2 such that B G TV;*;n7> w e have [B, 4>] G Mv-

12.6. Lv as Additive Generators

103

We argue by descending induction on n. We have n < dimVi, for any i 6 i. Let W be an I-graded a-stable subspace of V such that T = V / W satisfies dim Ti = n for all i G i and T | = 0 for all i G I — i. (Such W exists since i is an a-orbit.) As in the proof of 12.5.2, we can find a self-dual element (A,') G such that A G Vw,i,o and such that (with the notation there) ind¥ f W [i4,^] = [£,0i] + £ c , e 5 P c 4 < ? ' , / c ' ] where 0! = ±. By 12.5.1(b), for each C' G <S, we have C' G Vv,i-,n'-y for some n' > n; hence (d)

[C",/HGMV,

by the induction hypothesis. Replacing, if necessary $ by —(f)', we can assume that (f>\ = (f>. We now show that, (e) for any integer r < 0, the coefficient of vr in Pc is an integer, for any C' G Assume that this is not so. Then we can find an integer r < 0 which is as small as possible with the following property: there exists C' G S such that the coefficient c(C') of vr in Pc is in O — Z. For such C", we have

{ind¥iW[i4, '], \C', fc)} = {[B, 4>l [C, fc]} (f)

+ £ C"es

Pc"{[C",fc»],[C,fc-\}.

By the induction hypothesis, we have [A, <j>'] G M w and by 12.3.2, ind^p W carries M w into My; hence, i n d ^ w [ ^ 4 , G M y . Using now 12.6.2 and (d), we see that the left hand side of (f) is in Z((v)). In particular, the coefficient of vr in the left hand side of (f) is an integer. By 12.5.3(a), we have {[£,0], [C, fc]} G vO[[v]] since B,C' are not isomorphic. This implies that the coefficient of vr in {[B, 0], [C7, fc]} is zero since r < 0. If C" G S is not C' then, by 12.5.3(a) and 12.6.2, we have {[C", f c ] , [C, fc]} e vO[[v]] D Z((v)) = vZ[[v]], hence Pc"{[C", fc"], [Cr,fc]} £ .Pc"^Z[[v]]. Using the minimality of r, we see from this that the coefficient of vr in Pc»{[C", /c"], [C, fc]} is an integer. Similarly, using 12.5.3(c) and 12.6.2, we have {[C, fc], [C, fc]}

e (l + vO[Ml) n Z((v)) = l + vZ[[v]]-

hence Pc>{[C', fc], [C', fc]} G Pc>{l+vZ[[v]]). Using the minimality of r, we see from this that the coefficient of vr in Pc{[C', fc], [C',fc]} is c(C') plus an integer.

12. Quivers with Automorphisms

104

Combining these results, we see that the coefficient of vr in the right hand side of (f) is c(C') plus an integer; on the other hand, we have seen that the coefficient of vr in the left hand side of (f) is an integer. It follows that c{C') is an integer; this is a contradiction. Thus, (e) holds. Next we note that D[A, '] = [A, ft]. Using 12.4.3, we deduce that jD(ind^ w [i4, ']) = ind^ w [;4, <£']. Hence £>([£,<£]+ J2 Pc»[C",fc»]) C"es

= [B,4>]+ E C"es

p

C"[C"Jc>']

or equivalently, [B,4>] + Y

C"es

Pc»[C",fc»]

= [B,]+ ]T

C"es

Pc»[C",fc>]

where P c is as in 12.4.1. Comparing coefficients on the two sides, we obtain Pc» = Pc" for all C". Thus, the coefficients of vr and of v~r in P c coincide. Since either r or —r is < 0, we see from (e) that the coefficient of vr in P c is an integer, for any r and any C" G S. In other words, we have Pc G A. In the identity =

Y.

C"es

p

c>[C",fc»],

all terms except [B, \ are in Mv, as we have seen. It follows that so is [B,]. Thus (c) is proved. The remainder of the proof will be similar to that of (c). Consider the set T of all simple objects C G Vv (up to isomorphism) such that a*C = C; for each such C we select fc : a*C = C such that (C,fc) G Vv and D[CJc] = [CJc]- Let a G M v . Using 12.1.3 and 12.5.2, we can write uniquely a = YtCerPclC, fc] with pc G O'. To complete the proof, it remains to show that pc G A for all C. Assume that this is not so. Then we can find an integer ro which is as small as possible with the following property: there exists Co G T such that the coefficient 6(Co) of vr° in pc0 is in O — Z. For such Co we have (g)

{«, [Co, fCo]} = J2 Pc{\C, fc], [Co, fc.]}. cer

Using (c), 12.5.3, and the definition of ro, we see that the coefficient of vr° in the right hand side of (g) is equal to b(Co) plus an integer. Using (c) and 12.6.2, we see that the coefficient of vr° in the left hand side of (g) is

12.6. Lu as Additive Generators

105

an integer. It follows that b(Co) is an integer. This is a contradiction. The proposition is proved. 12.6.4. Let M be a module over a ring R. A subset B of M is said to be a signed basis of M if there exists a basis B' of M such that B = B' U (—B'). Let By/ be the subset of K(Q\j) consisting of all elements [B, 4>] with (B,4>) as in 12.5.2 (if n is even) and of all elements ±[i?, ] with (B, ) as in 12.5.2 (if n is odd). Prom 12.1.3 and 12.5.2 we see that (a) Bv is a signed basis of the O'-module

£(Qv),

and from 12.6.3 we see that (b) By/ is a signed basis of the A-module My/.

13.1.1. We preserve the setup of the previous chapter. Given v E N[I] a , we may regard V /C(Qv) as a functor on the category V® with values in the category of ©'-modules. An isomorphism V = V ' in V® induces an isomorphism E v — E v ' compatible with the a-actions; this induces an isomorphism Qv — Q v which induces an isomorphism /C(Qv) — £ ( Q v ) that is actually independent of the choice of the isomorphism V = V ' by the equivariance properties of the complexes considered. Hence we may take the direct limit lim^ /C(Qv) over the category V®. This direct limit is denoted by o'^w By the previous discussion, the natural homomorphism The signed basis Bv of /C(Q V ) (see 12.6.4) (where V E V®) can be regarded as a signed basis of the ©'-module o'k^, independent of V; we 13.1.2. Let 0 / k = 0,y{o'K) {y runs over N[I] a ). Let B = V1UBU, a signed basis of the ©'-module Q> k. An element x E o'k is said to be homogeneous if it belongs to o'k„ for some v\ we then write |x| = v. The homomorphisms i n d ^ w can be regarded as ©'-linear maps o'kr<8>c>' (o'kw) o*ki/, defined whenever E N[I] a satisfy r -f- u = v. They define a multiplication operation, hence they define a structure of ©'-algebra on c>/k. For any u = (i/ 1 ,..., i/m) E Xa, we may regard Lu as Lemma 12.3.3 can be now restated as follows:

Since the elements Lu generate Q>k as a ©'-module (see 12.6.3), it follows that the algebra structure on eyk is associative. One can also see this more 13.1.3. The homomorphisms resip w can be regarded as ©'-linear maps O'k^ - c>'kr®c>/ (o'ka>), defined whenever r,u;, v E N[I] a satisfy r+u = v. By taking direct sums, we obtain an ©'-linear map f : o ' k —> (o'k)-

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_13, © © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010

106

13.1. The Algebra o'k

107

13.1.4. We have a symmetric bilinear pairing Z[I] x Z[I] —• Z given by v • v' =

+ Vh"Vh')i h This bilinear form is independent of orientation. Let o'k(8>c>'(c>'k) be the O'-module c>'k®c>' (c>'k) with the (9'-algebra structure given by (x 0 y){x' y') = for x,x',y,y'

® yy'

homogeneous.

L e m m a 13.1.5. f : Q/(c>'k) is a homomorphism algebras.

of O'-

We must check that r(xy) = f(x)f(y) for any x, y G c>'k. Since the elements Lu generate the (9'-module o'k (see 12.6.3), we may assume that j/m) and v" = ( i / ; / 1 , . . . , v"n) are x = Lut,y = Lu», where v' = (un,..., a elements of X . We have f(Lu,)

(a)

=

®

where the sum is taken over all r ' = ( r ' 1 , . . . , r / m ) and a;' = ( u / 1 , . . . , a / m ) in A"1 such that r n + u/ z = i/" for 1 < I < m; M ' ( r ' , u ; ' ) is as in 9.2.11. Similarly, we have

and u>" = where the sum is taken over all r " = ( r , / ( m + 1 ) , . . . , (^/(m+l)^ . >fa/'(m+n)) i n ^a g u c h t h a t T » l + J > l = ^ l - m for m + 1 < / < ra + n. Hence where the sum is taken over all r ' = ( r ' 1 , . . . , r ' m ) , CJ' = (a;' 1 ,... , a / m ) , r//

=

(r//(m+l)?

1

that r " + uj' = v 771 + 72. We have

)T//(m+n))?

n

^/z

=

(a;//(m+l)j . . .

for 1 < / < ra and r

f(Lu,Lv»)

nl

1

ja

/'(m+»))

l m

+ u" = v" -

i n

^a

s u c h

for ra + 1 < I <

= r(Lv>u») = J 2 v M ' ( t , " ) l T ® ^u,

where the sum is taken over all r = ( r 1 , . . . , rm+n) and w = (u1,..., cjrn+n) = j/"-™ for in ; t a such that rl + J = vn for 1 < / < m and rl + ra-fl
• |LR»| = M ' ( T ' T > V ) - M ' ( T ' V ) ~

M'(T"

This follows by a straightforward computation. The lemma is proved.

13. The Algebra o' k and k

108

13.1.6. The pairing {,} on /C(Q V ) (see 12.1.2) (where V G VJ) can be regarded as an (^'-bilinear pairing {,} : c>'k„ x 0 >k„ —• 0((v)), which is independent of V. This extends to an O'-bilinear pairing {, } : cvk x o>k —> 0((v)) such that for homogeneous x, y, {x,y} is given by the previous pairing if |x| = |y|, and is zero if |x| ^ \y\. 13.1.7. We define a (^'-bilinear pairing {,} on e>'k®o' (o'k) by {x'®x",y'®y"}

=

{x\y,}{x",y"}.

The identity {x,y'y"} for all x,y',y"

= {r{x),y'

®y"}

G o'k, follows immediately from 12.2.2.

13.1.8. The homomorphism D : K(Qv) ~> £ ( Q v ) (where V G VJ) can be regarded as a group homomorphism D : o'ku —> o'ku that has square 1 and is semi-linear with respect to the ring involution : O' —• O' given by vn t—> v~n and £ ^ C - 1 f° r C £ ^ C " = 1- By taking direct sums we obtain D :
(c>'k) as an O'-algebra with

(x 0 y)(x' y') = v^'^xx'

yy'

for x,x',y,y' homogeneous. This should be distinguished from the algebra (c>'k) be the ring isomorc>/kc>'(c>'k). Let D : o'kf&oKo'k) —> phism given by D(x y) = D{x) <S> D(y) for all x, y. Let r : o> k —• o' k <8>o' (c?'k) be the (9'-algebra homomorphism defined as the composition O'k

o'k

o'k<8>c>'(o'k)

o'k®c>' (c>'k).

13.1.10. Let (,) : 0 >k x 0 / k -> 0 ( ( v - 1 ) ) be the O'-bilinear pairing given by (x,y) = {D(x),D(y)}. Here, ~ : 0((v)) —• 0((v~1)) is given b y E n a n V n ^ E n a n V - n (an G O). From 13.1.7, we deduce the identity (a) for all x, y' ,y" G o' k.

{x,y'y")

= {r{x),y'

®y")

13.1. The Algebra o'k

109

13.1.11. From the definition we have (a) D(b) = b for all b G B. We have (b) {&, b'} G vZ[[v\] n Q(v) for any b, V G B such that b' ^ ±b. Indeed, {6,6'} is in vO[[v]] by 12.5.3 and in Z((v)) by 12.6.2 and 12.6.3, and hence in vZ[[v]]. We have (c) {6, b}e 1 + vZ[[v]] n Q(v) for all b G B. Indeed, {b,b} is in 1 + vO[{v\) by 12.5.3 and in Z((v)) by 12.6.2 and 12.6.3, and hence in 1 + vZ[[v]]. From (a),(b),(c) we deduce: (b') (6, b') G v^ZKt;- 1 ]] D Q(v) for any b, b' G B such that b' ^ ±b. (c') (b,b) G 1 + v-lZ[[v-1}}

fl Q(v) for all b G B.

13.1.12. Let i be an a-orbit on I and let 7 = i. For any n > 0, c k n 7 has a distinguished element denoted l n j ; it corresponds to [1,1] G JC(Qv) where V G V" 7 . This element forms a basis of the (9 ; -module o>\i n i . When n = 0, this is independent of i and is denoted simply by 1 G o' ko- Note that (a) 1 is the unit element of the algebra o' k. (b) The elements Lu G c>'k are precisely the elements of o'k which are products of elements of form 1ni for various i, n. Hence the elements 1 ni generate o ' k as an O'-algebra. (for n > 1), where d is (c) We have l i l ( n _ i ) i = ^ " ^ ( E ^ o the number of elements in the orbit i. (See 12.3.4.) From 12.3.6, we have (d) {li, 1 i} = (1 — i> 2 d ) - 1 (where d is as above) or equivalently, since £>(1*) = 1»: (do ( i i I i 4 ) = ( i - « - " ) - » . From (b) and (c) we see that (e) f(li)

= r(l<) = l i ® 1 + 1 <8) 1 4 .

This is obvious from the definitions. It is clear that (f) ( i , i ) = {1,1} = 1.

110

13. The Algebra o'k and k

From 10.3.4, it follows easily that there is a unique (^-linear map a : o>V. —> o ' k such that a(Lv) = Lv* for any v = (i/ 1 , i / 2 , . . . , i/m) G Xa, where v' = (i/ m , i / m _ 1 , . . . , i/1) G moreover, we have 0. 13.1.13.

1 3 . 2 . T H E ALGEBRA k

Let ^ k be the ,4-submodule of o ' k spanned by B, or equivalently (see 12.6.3), by the elements Lu for various v G Xa. Thus, on the one hand, ^ k is the .4-subalgebra of o ' k generated by the elements 1n i as in 13.1.12(b), and on the other hand, B is a signed basis for the .4-module ^ k . We have ^ k = ©^(^k^) where A\Lu is the ,4-submodule generated by Bv. 13.2.1.

From 13.1.5(a), we see that f restricts to an .4-linear map ^ k —> ^ k <S>A (^k), denoted again by f; this is an ,4-algebra homomorphism if Ak <8)^4 (^k) (which is naturally imbedded in o'k<8>o'(e>'k)) is given the induced ,4-algebra structure (see 13.1.5). 13.2.2.

By 13.1.11(a), the ring homomorphism D : o ' k —• o ' k restricts to a ring homomorphism D : ^ k —» ^ k which has square 1 and is semi-linear with respect to the ring involution ~ : A —> A. 13.2.3.

From 13.2.3 and 13.2.2, it follows that the O'-algebra homomorphism r : o ' k —> o ' k ® o ' (c?'k) (see 13.1.9) restricts to an «4-algebra homomorphism ^ k —> A^QA (^k), denoted again by r. This is an .4-algebra homomorphism if Ak®A (^k) (which is naturally imbedded in o'k(g>o' (o'k)) is given the induced ,4-algebra structure (see 13.1.9). 13.2.4.

13.2.5. The pairing (,) : o ' k x o ' k C>((v -1 )) (see 13.1.10) restricts to an ^-bilinear pairing (,) : ^ k x ^ k Z ( ( v - 1 ) ) fl Q(v) (see 13.1.11(b / ),( c/ ))The equation analogous to 13.1.10(a) continues of course to hold over A. 1 3 . 2 . 6 . Let k be the Q(v)-algebra Q(v) (^k). Note that B is a signed basis of the Q(v)-vector space k. We have a direct sum decomposition k = where \au is the subspace spanned by Bv. From 13.2.1 and 13.1.12(c), we see by induction on n that k is generated as a Q(i>)-algebra by the elements 1{ for the various a-orbits on I.

111 13.

The Algebracvkandk

13.2.7. The homomorphism r in 13.2.4 extends to a Q(u)-algebra homomorphism k —• kQ(v) k (denoted again by r) where k®Q( v ) k is regarded as a Q(v)-algebra by the same rule as in 13.1.9. 13.2.8. The pairing (,) on ^ k extends to a Q(v)-bilinear pairing (,) : k x k —• Q(i>). Prom 13.1.11(b/),(c/), we see that the restriction of this pairing to k u is non-degenerate, for any v. (Its determinant with respect to a basis contained in B u belongs to 1 -I- v - 1 Z[[v - 1 ]] D Q(v) and hence is non-zero.) 13.2.9. Let I be the set of a-orbits on I. We identify Z[/] with the subgroup Z\l]a = {ueZ[l]\u^va{i)

Vi G 1}

of Z[I] by associating to each v G Z[J] the element of Z[I] (denoted again i/), in which the coefficient of i is v^ where i is the a-orbit of i. For i/, v' G Z [I] we define v • v' G Z by regarding z/, v' as elements of Z[I] as above and then computing v-v' according to 13.1.4. (This is a symmetric bilinear form). According to this rule, we have, for i,j G I: i • j = minus the number of h G H such that [h] consists of a point in i and a point in j, if i j and i • i = twice the number of elements in the orbit i. Note that, Hi then —244 G N; indeed, this is the number of h G H such that [/i] consists of a given point in i and some point in j. Hence we have obtained a Cartan datum (/,•)• 13.2.10. Let f be the Q(v)-algebra, defined as in 1.2.5, in terms of the Cartan datum (/, •) just described. Recall that f = ' f / J where 'f is the free associative Q(i>)-algebra on the generators 9i(i G I) and J is a two-sided ideal defined as the radical of a certain symmetric bilinear form (,) on 'f. Let x 'f —^ k be the unique homomorphism of Q(v)-algebras with 1 such that x{Qi) = l i f° r e a c h ® £ IT h e o r e m 13.2.11. x induces an algebra isomorphism f = ' f / J = k. The homomorphism x is surjective, since k is generated by the l i as a Q(v)-algebra (see 13.2.6.) The homomorphism r : 'f —> 'i 0 'f (see 1.2.2) and the homomorphism r : k —• k k (see 13.2.7) make the following diagram commutative: r

k

T

* k k

112

13. The Algebra cvk and k

Indeed, first we note that \ ® X 1S a n algebra homomorphism, since v • v' on Z [I] has been defined in terms of the pairing on Z[I]. Hence the two possible compositions in the diagram are algebra homomorphisms; to check that they are equal, it suffices to do this on the generators But they both take 0; to U <8> 1 + 1 (see 13.1.12(e)). (the right hand side is as in For x,y G 'f, we set ((x,y)) = (x(x),x(y)) 13.2.8). We have (a) ((ft, = (li, li) = (1 - v-"'2)-1 (1i 5 1 , ) = 0, if i ^ j (trivially);

(b)

(see 13.1.12(d')); ((ft,*;))

-

((*,vV')) = ( x ( * ) , x ( y W ) ) = (r(x(x)),x(y')®x(y")) r x = (X ® x)( ( )> x(y') ® x{y")) = («*),
we have used 13.1.10(a) and the commutativity of the diagram above. By (b) and the symmetry of ((,)), we obtain (C)

((xx',3/)) = ((x ®

x',r(y))).

We have ((1,1)) = 1 (see 13.1.12(f)). Thus, ((x,y)) satisfies the defining properties of (,) in 1.2.3; hence it coincides with (,). Since X is defined as the radical of (,) on 'f, we also get (d)

I = {x€'f|(x(x),x(j»)) = 0

fye'f}.

Hence, if x G 'f satisfies x(x) — 0, then x 6 I , so that Ker x C X. Conversely, assume that i E l Let z £ k. We have z = x(y) f° r some y G 'f (recall that x is surjective). We have (x(^) 5 z ) = (x(x)> x(y)) — 0 by (d). Thus x{x) is i n the radical of the form (,) on k. But this radical is zero (see 13.2.8). Hence x{ x ) = 0. Thus we have proved that Ker x — The theorem follows.

CHAPTER

14

The Signed Basis of f

14.1.

C A R T A N D A T A AND G R A P H S W I T H A U T O M O R P H I S M S

14.1.1. There is a very close connection between Cartan data and graphs with automorphisms. Given an admissible automorphism a of a finite graph [/i]) (see 12.1.1), we define I to be the set of a-orbits on I. For (I, Hyh i,j G J, we define i • j G Z as follows: if i ^ j in / , then i • j is —1 times the number of edges which join some vertex in the a-orbit i to some vertex in the a-orbit j] i • i is 2 times the number of vertices in the a-orbit i. As shown in 13.2.9, (/, •) is a Cartan datum. Conversely, we have the following P r o p o s i t i o n 14.1.2. Let ( / , •) be a Cartan datum. There exists a finite graph (I, H,h\-> [/i]) and an admissible automorphism a of this graph such that ( / , •) is obtained from them by the construction in 14-1-1. For each i G / , we consider a set Di of cardinal di = i • i/2 and a cyclic permutation whose restriction to each Di is the permutation a : Di

Di

For each unordered pair i,j of distinct elements of / , we choose an aorbit p of the permutation a x a : Di x Dj —* Di x Dj. Then p has cardinality equal to the lowest common multiple l(di,dj) of di and dj, which by the definition of a Cartan datum, divides —i • j. Hence we may consider a set Hij which is a disjoint union of —i • j/l(di, dj) copies of p with a permutation a : Hij —> Hij whose restriction to each copy of p is the permutation defined by ax a. We have a natural map Hi y j —> Di x D j whose restriction to each copy of p is the imbedding p —» Di x Dj. Let H = UHi j (union over the unordered pairs iy j of distinct elements in I). This has a permutation a : H —> H (defined by the permutations a : Hij —> Hij) and a map H —• UDi x Dj (union over the unordered pairs i,j of distinct elements in I) induced by Hij —• Di x Dj. This defines a structure of a graph on I, H. This clearly has the required properties.

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_14, © © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010

113

114

14- The Signed Basis of f

14.1.3. Remark. In general, the graph with automorphism whose existence is asserted in the previous proposition is not uniquely determined by (/, •). However, if the Cartan datum (/, •) is symmetric, the construction in the previous proposition attaches to (/, •) a graph (I, H,h [/i]), called the graph of (I, •), which is unique up to isomorphism; in this case, I = / , H i j is a set with - i - j elements and a is the identity automorphism. 14.1.4. Classification of symmetric Cartan data of affine or finite type. The symmetric Cartan data of affine type are completely described by their graphs. We enumerate below the graphs that appear in this way. An(n > 1); a polygon with n + 1 vertices; for n = 1, this is the graph with two vertices which are joined with two edges.

Dn(n > 4) (a graph with n + 1 vertices):

EQ:

E 7:

14-1- Cartan Data and Graphs with Automorphisms

115

According to McKay, these graphs are in 1-1 correspondence with the various finite subgroups of SL2(C), up to isomorphism. Certain vertices of these graphs are said to be special : namely, all vertices for An, the four end points for Z)n, the three end points for EQ, the two end points furthest from the branch point for E 7 , the end point furthest from the branch point for Eg. The group of automorphisms of any of the graphs above acts transitively on the set of special vertices. Therefore, by removing a special vertex from one of the graphs above, we obtain a graph which is independent of the special vertex chosen. The resulting graphs are denoted An, Dn, EQ, £7, Eg. We get in this way the various graphs corresponding to irreducible, simply laced Cartan data of finite type. 14.1.5. Classification of non-symmetric Cartan data of affine type. Let us consider one of the graphs An,..., Eg, together with an admissible automorphism a of order n > 1, which has at least one fixed vertex. We enumerate the various possibilities (up to isomorphism). (a) An (n > 3, odd), n = 2 and a has 2 fixed vertices:

(b) Dn, n = 2 and a has n — 1 fixed vertices:

<

(c) Dn, (n > 5), n = 2 and a has n — 3 fixed vertices.

116

14-

The Signed Basis of f

(d) Dn (n even), n = 2 and a has 1 fixed vertex:

(e) Dn (n even), n = 4 and a has 1 fixed vertex:

(f) L>4, n = 3 and a has 2 fixed vertices:

14-1. Cartan Data and Graphs with

Automorphisms

117

(g) EQ, n = 2 and a has 3 fixed vertices:

(h) Eq, n = 3 and a has 1 fixed vertex:

(i) Ej, n = 2 and a has 2 fixed vertices:

In each case (a)-(i), we may define a Cartan datum as in 14.1.1. We then obtain exactly the various affine non-symmetric Cartan data, up to proportionality (see 1.1.1) which were classified by Kac, Macdonald, Moody and Bruhat-Tits. 14.1.6. Classification of irreducible, non-symmetric Cartan data of finite type. We consider one of the graphs An,..., Eg, together with an admissible automorphism a of order n > 1. We enumerate the various possibilities (up to isomorphism), (a) An (n > 3, odd), n = 2.

118

14- The Signed Basis of f

(b) Dn, n = 2. (c) D4, n = 3. (d) E6, n = 2. In each case (a)-(d), we may define a Cartan datum as in 14.1.1. We then obtain exactly the irreducible non-symmetric Cartan data of finite type, up to proportionality. 1 4 . 2 . T H E SIGNED BASIS B

14.2.1. Let V be a Q(v)-vector space with a given basis B and a given symmetric bilinear form (,) : V x V —> Q(v). We say that B is almost orthonormal for (,) if (a) (b, h') G < V + v- 1 Z[[v~ 1 l] H Q(v) for all b, h' G B. Let A = Q[[v _ 1 ]]nQ(v). Let by B and let L(V) = {x G V\(x,x)

be the ,4-submodule of V generated G A}.

Lemma 14.2.2. In the setup above, the following hold. (a) L(V) is an A-submodule of V and B is a basis of it. (b) Let x G aY be such that (x,x) G 1 + v~*A. such that x = ±b mod v~1L(V). (c) Let x G V be such that (x,x) G v~xA.

Then there exists b G B

Then x G v~1L(V).

Let x G V. Assume that x ^ 0. We can write uniquely x = YlbeB0^ with Cb G Q(v). Since only finitely many Cb are non-zero, we can find uniquely t G Z and pb G Z (zero for all but finitely many b, but non-zero for some b) such that, for all b, we have v~tCb~ Pb G i> -1 A. We have (x, x) = (52bPb)v2t moc* v2t~l A. Note that JZbPb 1S a rational number > 0. Hence (x, x) G A if and only if t < 0; this is equivalent to the condition that G A for all b and (a) follows. If (x,x) G f - 1 A , then we must have t < 0; hence q> G v - 1 A for all 6; (c) follows. If x G j y and (x, x) G 1 + v A , then we must have t = 0 and ^2bPb = 1 Pb G Z; hence pb = ± 1 for some b and pb = 0 for all other b; thus, (b) follows. The lemma is proved. In the remainder of this chapter we fix a Cartan datum (7, •). Let f, ^f be defined in terms of (I, •) as in 1.2.5, 1.4.7. Theorem 14.2.3. Let B be the set of all x G f such that x G ^f, x = x and (x,x) G 1 + v - 1 Z[[i; - 1 ]]. (The last condition is equivalent to {x,x} G 1 + vZ[[v]] since x = x.)

14-2. The Signed Basis B

119

(a) B is a signed basis of the A-module ^ f and of the Q(v)-vector space f. (b) Ifb,b' vZ[[v]].

G B and b' ^ ±b, then (b,b') G v ^ Z Q t r 1 ] ] and {6,6'} G

By 14.1.2, we can find a finite graph ( I , H , h [h]) and an admissible automorphism a of this graph such that (/, •) is obtained from these data by the construction in 14.1.1. Let n > 1 be such that an = 1. Then, by 13.2.11, f has a natural isomorphism, say x> o n t ° the corresponding algebra k (see 13.2.6). Under the pairings (,) and (,) on k and f correspond to each other. This has been seen in the proof of 13.2.11. Moreover, the involutions D : k —> k and : f —• f correspond to each other (they both map the generators and ft to themselves). Note that x carries l n j € k to 9^ for any i e I and any n > 0 (this follows from 13.1.12(c)); hence it carries the ,4-subalgebra ^ k (which is generated by the 1n i ) onto ^ f (see 1.4.7). Moreover, x carries the signed basis B of k (see 13.1.2) onto a signed basis of f, which we denote by the same letter. By the already known properties of the signed basis of k, it remains to prove the following statement: let x G ^f be such that x = x and (x,x) 6 1 + v - 1 Z[[v - 1 ]]; then x G B. Let B be a basis of f such that c B = B U (—B). We can write uniquely x = bb where b runs over B and ei G A are zero except for finitely many b. Using 14.2.2(b), we see that there is a unique b G B such that Cb G ±l + v~1Z[v~1] and Cb> G v - 1 Z[v _ 1 ] for b' ^ b. From x = x and b' — b' for all b' G B, it follows that cb> = C& for all b' G B. It follows that cb = ± 1 and cb> = 0 for all b' ± b. Thus, x G B U (—B). The theorem is proved. 14.2.4. Definition. B is called the canonical signed basis of f. Although the proof of the existence of B requires a choice of a graph with automorphism, which is not unique in general, the resulting signed basis is independent of any choice, hence the word canonical 14.2.5. The following properties of B follow immediately from the definitions. (a) We have B = UvBu where BU = BC\ iu. (b) We have G B for any i G / and s > 0; in particular, 1 G B. Using 1.2.8(b), we see that

(c)


14- The Signed Basis of f

120

Proposition 14.2.6. (a) r and f map ^f into the A-submodule ^f <8u(.4f) of f f. (b) For any x,y G ^ f we have (x,y) G Z[[v -1 ]] fi Q(v) and {z,?/} G Z 1H1 n Q(w). This follows immediately from the analogous properties of ^ k (see 13.2.2, 13.2.4, 13.2.5) which are already known. 1 4 . 3 . T H E S U B S E T S Bi;n

OF

B

14.3.1. Given i G / and n > 0, we set Bi;>n = Bn6?f and aBi;>n = BOfQf. Let Bi-n = Bi.> n - Bi.> n+ 1 and a B ^ n = a B i ] > n - a Bi ; > n +i. Thus, we have partitions H i; > n = U n /> n 5 i ; n / and aBi;>n = U n '>n a B^ n >. Since a(B) = B and a(0?f) = f0J\ we have aBi,>n = (r(Bi;>n) and aBi;n = a{Bi;n). Theorem 14.3.2. (a) Bi;>n is a signed basis of the Q(y)-vector space 9™f and of the A-module a

(b) B^>n A-module

Yln':ri>n

is a signed basis of the Q(v)-vector

space f0™ and of the

M

£n,:n,>nUf0, ).

(c) If b € Bi-o, then there is a unique element b' G Bi]n such that b' plus an A-linear combination of elements in #i;>n+i. Moreover, b is a bisection 7rijTl : Bi]0 —• Bi-n.

= b'

(d) If b G aBi;o, then there is a unique element b" G aB^n such that W\ = b" plus a an A-linear combination of elements in Bi->n+Morea a a over, b b" is a bijection iTi}n ' Bi;o —> Bi-n. For the proof we place ourselves in the setup considered in the proof of Theorem 14.2.3. Thus i is now regarded as an a-orbit in I. Let V G VJ. For any n > 0, let #v;i;n be the set of all ±[B,] G Bv (see 12.6.4) such that i. By 10.3.3 and 12.5.1, we have a partition B G Vv,i-,n-y where 7 = Bv = L-ln>o#V;i;n- By our identification By/ = Bu, this becomes a partition Bu = Un>0#iy;i;nLet B[ = Uj,B v .^ n . We will show below that B\ n just defined is the same as B^n in 14.3.1; see (h) below. We then have a partition B = U n>0 B'i.n. Translating the geometric properties of By/ti;n expressed by 10.3.2(c) we obtain the following property of B[.n. (e) For any n > 0, there is a unique 1-1 correspondence b b' between an d B' such that O^b = b' plus B{. vn an .4-linear combination of elements in 0Un'>nBi. ,. n)

n

113. The Subsets

of B

121

an Let Mn = En':n'>n ^ d let M'n be the .4-submodule of f generated by Un':n'>nB<.n" We show that for fixed i/,

(f) any b G Bu;i;n is contained in Mn. We argue by descending induction on n; note that Bu;i;n is empty unless n < v\ for any i G i. By (e), we have beeln\4f+

£ n':n'>s

by the induction hypothesis, we have C M n / and it follows that b G Mn. Thus (f) is proved. Thus we have B[.n C Mn. If n' > n, we have B(;n, C Mn> C M n . It follows that, for any n > 0, we have M'n C Mn. Next we show that (g) for any b G Bv, we have 0j n) 6 G M'n. where £ > 0, We argue by induction on c(b) = Ei^i- If & £ then as we have seen, we have b G Mt; hence b is an ,4-linear combination of elements Q^bi with m > t and with bi G B such that the induction hypothesis applies to 61. Then is an ,4-linear combination of elements 61 G with 61 as above. By the induction hypothesis, we have M G Em ^ n C M'n, as required. M'n. Hence Next we assume that b G Bu.i;0. Then O^b G M'n by (e). Thus, (g) is proved. It follows that, for any n > 0, we have M

n =

E n':n'>n

Y.

K'CM'n-

n':n'>n

We have proved that Mn C Mn> C Mn. Thus, M'n = Mn. It follows that (a) holds and Bi ; > n = Un':n'>n&i.n>- In particular, we have (h) Bi-n = S{.B. We now prove (c). The existence of b' asserted in (c) follows immediately from (e). We now prove uniqueness. Assume that b[ G Bi ;n has the same property as that asserted for b' in (c). Then b' — b[ is on the one hand a linear combination of elements in U n '>n#t;n' and on the other hand it is a linear combination of elements in B^n. It follows that b' = b[ and (c) is proved. Now (b) and (d) follow from (a),(c) by applying a. The theorem is proved.

122

14-

The Signed Basis of f

14.3.3. By 12.5.1(c), we have B - { ± 1 } = UIG/;N>OBi,n.

1 4 . 4 . T H E CANONICAL BASIS B OF f

14.4.1. We would like to find in a natural way a basis of f contained in its canonical signed basis. If (/, •) is symmetric, such a basis is given by geometry. In this case, a is the identity automorphism of our graph and we can take n = 1. Hence we have O' — A and /C(Qv) = a^(Qv) (for V G V) has not only a natural signed basis, but a natural basis consisting of the elements [B, 1] where B is a simple object of Vyj and 1 is the identity isomorphism 1 : B = B. 14.4.2. In the general case, the descent from a signed basis to a basis will be non-geometric. We lay the groundwork with some definitions. For any v € N[/] we define a subset T$u of Bv by induction on tr v as follows. If v = 0, we set B^ = {1}. If tr v > 0, we set B„ = UieI,n>0,vi>n'Ki,n(J$u-ni H B{;Q). Let B = U„B„ C B. By 14.3.3, we have that B = B U ( - B ) . We can now state the following result. Theorem 14.4.3. Let v e N[I]. Then (a) B u n ( - B „ ) = 0; (b) B„ fl ( - a ( B „ ) ) = 0; (c) *(B V ) = B U . (d) B is a basis of the A-module ^ f and a basis of the Q(v)-vector space f. (e) For any v, B^ is a basis of the A-module Q (v)-vector space f„.

and a basis of the

14.4.4. Proof of the theorem, assuming that (/, •) is symmetric. In this case, as in the proof of Theorem 14.2.3, we have a natural choice for the graph (with identity automorphism a), see Remark 14.1.3. Moreover since a = l , the corresponding algebra k has a natural basis inside its natural signed basis, defined as in 14.4.1. From the definitions, it is clear that this basis (transferred to f) coincides with B and has all the required properties. This completes the proof (in the symmetric case).

14-4• The Canonical Basis B o / f

123

14.4.5. The proof of Theorem 14.4.3 in the general case will be given in 19.2.3; in the remainder of this section we shall assume that the theorem is known in general. 14.4.6. Definition. B is called the canonical basis of f. fl B for any i G I and We shall use the following notation: B i ; n = i;n = a(Bi;n). n G N; note that 7ri)n defines a bijection B i ; 0 = Bi ; n . Set a Then r;)Tl defines a bijection Bj ; o = ^Bj ; n . 14.4.7. We can regard B as the set of vertices of a graph colored by I x { 1 , 2 , . . . } in which 6,6' are joined by an edge of color (i,n) if b G Bijo,^ G B i ; n and b' — 7r;)n(6). This is called the left graph on B. Similarly, we can regard B as the set of vertices of a graph colored by I x { 1 , 2 , . . . } in which 6,b" are joined by an edge of color (i,n) if b G a B; ; o, b" G ^B^n and b" = ^71-^(6). This is called the right graph on B. 14.4.8. Let us choose a finite graph (I, H, h [/i]) and an admissible automorphism a of this graph such that (/, •) is obtained from them by the construction in 14.1.1. We define a new (symmetric) Cartan datum (/, •) associated to the same graph and to its identity automorphism, as in 14.1.3. More precisely, we have I — I, i • i = 2 and, for i ^ j G I, we have that i • j is —1 times the number of edges joining i to j. Let f be the algebra defined like f, in terms of the Cartan datum (/, •) and let B C f be its canonical basis. Similarly, let 7r-hn : Bi ;0 —> Bi ; n and a TTitn : ^Bijo —^ a B | . n be the bijections analogous to 7Ti>n : B i ; 0 —> B i ; n and a7ri?n : CTBi;o —> i;n in 14.4.6. Now a : I —• I induces an algebra automorphism a : f —> f which restricts to a bijection a : B —> B whose fixed point set is denoted by B a . The I x {1,2,... }-colored left graph structure on B (as in 14.4.7) defines a I x {1,2,... }-colored graph structure on the subset B a as follows. We say that 6, b' G B a are joined by an edge of color (i, n) if they can be joined in the left graph on B by a sequence of edges of colors (ii, n), (i 2 , n ) , . . . , (i s , n) where ii, i 2 , . . . , i s is an enumeration of the elements of i in some order. This is called the left graph on B a . By replacing "left" by "right" we obtain a I x {1,2,... }-colored graph structure on B a , called the right graph on B a . Theorem 14.4.9. There is a unique bijection \ B —*• B a compatible with the structures of I x {1,2,... }-colored left (resp. right) graphs and such that x(l) = 1. The two bijection corresponding to "left" and "right" coincide.

14- The Signed Basis of f

124

The inverse bijection is obtained geometrically by attaching to a pair (B,) where B is a simple object of a suitable Vv and (j> is an isomorphism a*B = B, the simple object B without specifying . The fact that this bijection is compatible with the colored graph structures is also clear geometrically (using, in particular, 12.5.1(a)). 14.4.10. Remark. This theorem shows that to describe the left or right graph structure on B it is enough to do the same in the case where the Cartan datum is symmetric. Theorem 14.4.11. Assume that the root datum is Y-regular. Let A G X+ and let Ax = f / E i f0 +1 be the U-module defined in 3.5.6. As in 3.5.7, let T) G Aa be the image of 1. Let B(A) = niG/(Un;0
W<<»A>+1. This follows immediately from Theorem 14.3.2.

14.4.12. Definition. B(AA) is called the canonical basis of AATheorem 14.4.13 (Positivity). Assume that ( / , •) is symmetric. (a) For any b, b' G B, we have bb' =

c

J2

WV',nVnb"

b"eB;neZ

where ct,^',b",n € N are zero except for finitely many b",n. (b) For any b G B we have dbM',nVnti®b"

r(b) = &',6"<EB;n<EZ where

G N are zero except for finitely many

b',b",n.

(c) For any b,b' € B we have (b,b')=Y,h,b>,nV~n n€ N

where

fb,b',n

€ N.

The theorem asserts the positivity of certain integers; in our definition in the framework of perverse sheaves, these integers are the dimensions of certain Qj-vector spaces. The theorem follows.

14-5. Examples

125

14.4.14. R e m a r k . For non-symmetric (7,-), the integers in question are not dimensions, but traces of automorphisms of finite order of certain Qjvector spaces and it is not clear whether they are positive or not. 14.4.15. In the case where (7, •) is symmetric, the set B„ is (conjecturally) in natural 1 — 1 correspondence with the set Xu of irreducible components of a certain Lagrangian variety naturally attached to (I,-) and to v (see [9, 13.7]). The union UuXu has a natural colored graph structure (defined as in [8]) and one can hope that the previous bijection respects the colored graph structures. 14.5.

EXAMPLES

14.5.1. Assume that (7, •) is a simply laced Cartan datum of finite type. Let (I, i f , . . . ) be the graph of (7, •) (see 14.1.3); note that 1 = 7. We choose an orientation of this graph. Let V G V and let G v , E v be as in 9.1.2. From the results in [9], it follows that there is a 1-1 correspondence between the set of orbits of G v on E v (a finite set, by Gabriel's theorem) and the set of isomorphism classes of objects of Vv (see 9.1.3): to an orbit of G v corresponds the GV-equivariant simple perverse sheaf whose support is the closure of that orbit. This is well-defined since the action of G v has connected isotropy groups. Assume that (7, •) is such that 7 = {z, j} and i-i = j -j = = j • i = — 2. Then (7, •) is a symmetric Cartan datum of affine type. Let (1,77,...) be the graph of (7, •) (see 14.1.3); note that 1 = 7 and 77 has two elements. We orient this graph so that h! = i for both h G 77. Let V G V and let G V , E V be as in 9.1.2. Note that E v consists of all pairs T, T' of linear maps V; —• V j . Assume that both V* and V j are ndimensional and n > 2. Then Gv acts on E v with infinitely many orbits. Let v = (2n terms). Then ir„ : —> E v (see 9.1.3) is a principal covering with group Sn (the symmetric group) over an open dense subset of Ev- This gives rise to irreducible local systems over an open dense subset of E v , and hence to simple perverse sheaves on E v , indexed by the irreducible representations of Sn. These simple perverse sheaves belong to Vv14.5.2.

14.5.3. Assume that (7, •) is such that 7 = {i} and i-i = 2. The canonical basis B of f consists of the elements (a G N).

126

14- The Signed Basis of f

14.5.4. Assume that (I, •) is such that I = {i,j} and i • i = j • j = 2 and i • j = j • i = — 1. The canonical basis B of f consists of the elements O^O^O^ (a,b,c € N , 6 > a + c) and of the elements c) a) (a,b,c € N , 6 > a + c) with the identification 00< 0V>eW0(*) for 6 = a + c . 14.5.5. Assume that (J, •) is as in 14.5.2. Oi Oj, 9j Oi. The elements of B2»+ij are:

The elements of Bf_j_j are

/)(2)/,(2) zi(2)/3(2) /) /d(2)z) /a p(2)fi p p p p aWqW p p p p Vj,ViVjViVj — Vj fVjViVjVi — Vj V{ .

For further examples, see [11].

Notes on Part II 1. The canonical basis of f has been introduced by the author in [7], assuming that the Cartan datum is symmetric, of finite type. In fact, in [7] two definitions for the canonical basis were given: an elementary algebraic one, involving braid group actions, and a topological one, based on quivers and perverse sheaves. (The elementary definition applies essentially without change to not necessarily symmetric Cartan data of finite type.) The topological definition in [7] was in terms of intersection cohomology of certain singular varieties arising from quivers by a construction reminiscent of that in [4] of the new basis of a Hecke algebra (which used the intersection cohomology of Schubert varieties). One of the main observations of [7] was that the canonical basis of f gives rise simultaneoulsy to a canonical basis in each U-module A\, which had rather favourable properties. 2. After [7] became available, Kashiwara announced an elementary algebraic definition of the canonical basis which applied to an arbitrary Cartan datum. Kashiwara's paper [3] contains an inductive construction of the canonical basis, both of f and of A^, which advances like a huge spiral. His construction agrees with that in [7], as shown in [8]. On the other hand, the author [9] extended the topological definition [7] of the canonical basis to arbitrary (symmetric) Cartan data. (The case of not necessarily symmetric Cartan data was only sketched in [9].) The definition of [9] resembles that of character sheaves [5]. While the method of [9] is not elementary, it has the advantage of being more global and to yield positivity results which cannot be obtained by the elementary approach. The agreement of the definitions in [3], [9], was proved in [2]. 3. The exposition in Part II essentially follows the treatment in [9], with two main differences. First, in order to include not necessarily symmetric Cartan data in our treatment, we have to take into account the action of a cyclic group, which is a complicating factor, not present in [9], where only symmetric Cartan data were treated. In addition, we make use of the geometric interpretation of the inner product on f given in [2]; this simplifies somewhat the original proof in [9] and provides the link with [3]. 4. The basic reference for the theory of perverse sheaves on algebraic varieties is the work of Beilinson, Bernstein, Deligne and Gabber [1]. 5. The representation theory of quivers (which is implicit in the constructions in Chapter 9) has a long history going back to Kronecker. In Ringel's work [12], the connection between the representations of a quiver of finite type over a finite field Fq and the plus part of the corresponding Drinfeld-Jimbo algebra at parameter y/q was observed for the first time. This work of Ringel was an important source of inspiration for the author's work on the canonical basis. In particular, the definition of the induction functor in 9.2 was inspired by Ringel's definition of the Hall algebra arising from quivers over Fq. On the other hand, the definition of the restriction functor in 9.2 was inspired by the analogous concept for character sheaves [5]. 6. The geometric definition of the inner product in 12.2 is taken from [2] where,

128

References

however, the cyclic group action was not present. 7. The idea that the canonical basis can be characterized by an almost orthonormality property for the natural inner product, has originally appeared in Kashiwara's paper [3] and has been later used in [2]. This is analogous to the orthogonality properties of character sheaves [5]; it is a hallmark of intersection cohomology. 8. The description of non-symmetric affine Cartan data given in 14.1.5 is different, as far as I know, from the ones in the literature. 9. The ingredients for the definition of the colored graph in 14.4.7 were introduced in [9]; it turns out that that graph contains the same information as the colored graph defined by Kashiwara (but the two graphs are different). 10. The statement 14.4.13 appeared in [2]. 11. The example in 14.5.2 is a special case of the results in [10] where the perverse sheaves which constitute the canonical basis in the affine case are described explicitly. (The results in [10] dealt with symmetric affine Cartan data; but in view of Theorem 14.4.9, the same results can be applied in the case of non-symmetric affine Cartan data.) 12. The geometric method used here to construct canonical bases can be applied, more or less word by word, to quivers in which edges joining a vertex with itself are allowed. (This includes, for example, the classical Hall algebra with its canonical basis.) We have not included this more general case in our discussion (but see [11]). REFERENCES 1. A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Asterisque 100 (1982), Soc. Math. France. 2. I. Grojnowski and G. Lusztig, A comparison of bases of quantized enveloping algebras, Contemp. Math. 153 (1993), 11-19. 3. M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 6 3 (1991), 465-516. 4. D. Kazhdan and G. Lusztig, Schubert varieties and Poincare duality, Proc. Symp. Pure Math. 36 (1980), 185-203, Amer. Math. Soc., Providence, R. I.. 5. G. Lusztig, Character sheaves, I, Adv. in Math. 56 (1985), 193-237 II, Adv. in Math. 57 (1985), 226-265. 6. , Cuspidal local systems and graded Hecke algebras, Publ. Mathematiques 6 7 (1988), 145-202. 7. , Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498. 8. , Canonical bases arising from quantized enveloping algebras, II, Common trends in mathematics and quantum field theories (T. Eguchi et. al., eds.), Progr. Theor. Phys. Suppl., vol. 102, 1990, pp. 175-201. 9. , Quivers, perverse sheaves and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365-421. 10. , Affine quivers and canonical bases, Publ. Math. I. H. E. S. 76 (1992), 111-163. 11. , Tight monomials in quantized enveloping algebras, Quantum deformations of algebras and representation (S. Shnider, ed.), Israel Math. Conf. Proc. (Amer. Math. Soc.), vol. 7, 1993. 12. C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583-592.

Part III KASHIWARA'S OPERATORS A N D APPLICATIONS In the author's elementary algebraic definition [4] of the canonical basis of f, there were three main ingredients: (a) the basis was assumed to be integral in a suitable sense; (b) the basis was assumed fixed by the involution (c) the basis was assumed to be in a specified Z[v -1 ]-lattice C and had prescribed image in C/v~1C. Of these three ingredients, the last one is the most subtle; in [4], C and the basis of C/v~lC were defined in terms of a braid group action. This definition does not work for Cartan data of infinite type. Kashiwara's scheme [2] to define a basis of f involves again the ingredients (a),(b),(c) above, but he proposes a quite different way to construct the lattice C and the basis of C/v~1C, which makes sense for any Cartan datum. The main ingredients in his definition were certain operators €i, (f)^ '. f —> f and some analogous operators Ei,Fi on any integrable Umodule. (The last operators were already introduced, in a dual form, in an earlier paper [1].) Part III gives an account of Kashiwara's approach and its applications. (The results in Part III will be needed in Part IV.) Our exposition differs from that of Kashiwara to some extent. In particular, we will make use of the existence of canonical bases (up to sign) established in Part II, while for Kashiwara, that existence was one of the goals. The algebra IL in Chapter 15 is defined in a different way than in [2], but eventually, the two definitions agree. The operators 6i,(f>i, Ei, F{ are defined in Chapter 16. Chapter 17 contains a proof of a crucial result of Kashiwara on the behaviour of Ei,Fi in a tensor product. Chapters 18 and 19 are concerned with various properties of the canonical basis of AA, in particular with the fact that this basis is almost orthonormal for the natural inner product. Chapter 20 deals with bases at oo (or crystal bases in Kashiwara's terminology). Chapter 21 deals with the special features which hold in the case where the Cartan datum is of finite type. Chapter 22 contains some new positivity results. In the remainder of this book we assume that, unless otherwise specified, a Cartan datum ( / , •) and a root datum (Y, X,...) of type ( / , •) have been fixed. The notation f, U, etc. will refer to these fixed data.

L e m m a 15.1.1. The algebra homomorphism x : 'f —• U given by Oi »—• E[ = (Vi — v~l)K-iEi (i G / ) factors through an algebra homomorphism Let x' ' ; f U be the algebra homomorphism given by 0i »-» Ei /). A simple computation shows that, if / G then

(i G

where TV depends only on v and not on / . Hence if / is a homogeneous element in 1 (so that x ' ( f ) ~ 0), then x ( / ) = 0- The lemma is proved. 15.1.2. Let U+ be the image of x- Using the previous lemma we see that Let U ° be the subalgebra of U generated by the elements for % G I. From the triangular decomposition of U, we can deduce that multiplication U. These defines injective maps U_U°U+ -> U and U+ ® U ° ® U " maps have the same image, which is a subalgebra U of U; this follows from the identity E^Fj = v^FjEt + S{J( 1 - K^) for all i j . Note that the elements K - i which are invertible in U are not invertible in U. The left ideal generated by them in U coincides with the right ideal generated by them in U. The quotient of U by this ideal will be denoted by il. We have obvious algebra homomorphisms U~ —> il and U ^ —> il and it is clear that (a) multiplication defines isomorphisms of vector spaces U~ U"1" = il (b) the algebra il is defined by the generators ei,i (i G I) and the relations e ^ j = vl'i(/>j€i + 6 i j for all together with the relations f(ei) = f{(j>i) = 0 for any homogeneous / = f(9i) G I. Here, e^,^ are the images 15.1.3. There is a unique Q-algebra homomorphism u : il —» il such that u(ei) = Vi(pi, iv(cf)i) = —Vi€i, u(v) = v~l. We will not use it. Note that

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © © Springer Springer Science+Business Science+Business Media, LLC 2010 2011

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15. The Algebra U

131

Lemma 15.1.4. For each i G I we define <j>i : f —• f to be left multiplication by Qi and e^ : f —> f to be the linear map ^r in 1.2.13. (a)

make f into a left ii-module.

(b) Ei : f —> f is locally nilpotent for any i € / . The identity e ^ = v l 'i(frjCi+Sij (as maps f —• f) follows from ir(6jy) = ir(9j)y-\-vjl0j(ir(y)). Let / = / ( f t ) be a homogeneous element of J . Prom the definition we have f(i) = 0 as a linear map f —• f. We must show that f(ei) = 0. Let / ' = a ( f ) G J . From the definition we have (x,€i(y»

= (1 - v~2)(iX,y)

for all x,y G f. It follows that ( x 1 f ( e i ) ( y ) ) = c(f((f>i)(x)1y) where c G Q(v). From f'( f. Thus the relations 15.1.2(b) of il are verified; (a) is proved. If x G then Ci(x) G iu~i if Vi > 1 and €i(x) = 0 if Vi = 0. It follows that €i : f —• f is locally nilpotent. The lemma is proved. L e m m a 15.1.5. There is a unique algebra homomorphism d : It —> It U (vi-Vil)l®K-iEi such that d((f>i) = A'_i + l.Fi andd(ei) = €i®K-i + for all i e I. The homomorphism A : U —• U ® U , satisfies A {Fi) = F i ® K - i + l ® F i , A (El) = Et®K-i + (vi-v~1)l®K-iEi, and A (K-i) = K-i® K-i. Hence A restricts to an algebra homomorphism U —> U ® U and this induces an algebra homomorphism d : 11 —• It U which has the required properties.

Kashiwara's Operators in Rank 1 16.1.

D E F I N I T I O N O F T H E O P E R A T O R S <^,6* AND

FI,EI

16.1.1. In this chapter we fix i G / . Besides the category C[ (see 5.1.1), we shall consider another category T>i which shares some of the properties of Let T>i be the category whose objects are Q(v)-vector spaces P provided with two Q(f)-linear maps e;,i : P —> P such that €i is locally nilpotent

the morphisms in the category are Q(t>)-linear maps commuting with P be defined as For P G Vi and s G Z, let \a) : P and as 0, if s < 0. From (a) we deduce by induction on N:

if 5 > 0

This is well-defined, since e^ is locally nilpotent. For N > 0, we define a subspace P(N) of P by P(0) = {x G P\ei(x) = 0} and P(N) = [N)P(0). L e m m a 16.1.2. (a) We have e;IIt = 0 for all t> 0. (b) We have Ylt>o

— 1- The sum is well-defined since,

(c) We have a direct sum decomposition P = ®n>oP(N) as a vector space. Moreover, for any N > 0, the map restricts to an isomorphism

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011

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16.2. Admissible Forms

133

and (a) is proved. Now (b) follows immediately from 1.3.4. We prove (c). If x G P, we have by (b): x = fi^x^ where xn = N ( N ~ I Y 2 H N ^ X Y By V ~ w e h a v e XN € -P(O). It remains to show the uniqueness of the x^; it is enough to prove the following statement. If 0 = Y2N>o ifri^BNi where XN G P(0) are zero for all N > Nq (for some N0 > 0),~then xNo = 0. We argue by induction on No. For No = 0 there is nothing to prove. Assume that No > 1. Applying and using 16.1.1(b), we obtain 0 = v N 1 ( ) N 1 x 2AT>O \ ~ ^ t \ ~ ^ N- The induction hypothesis is applicable to this equation and gives XN0 = 0 . This proves (c). (d) follows immediately from (c). 16.1.3. We define linear maps fa, e^ : P —> P by *MN)y)

AND

= 4>iN+1)v

FOR A11

*i(iN)v) =

v E P(O).

L e m m a 16.1.4. Let M G C[ and let x G Ml. (a) We can write uniquely x — X)s s > o s + t > o ' w h e r e xs G ker(£'i : fy[t+2s m) and xs = 0 for large enough s; we can write uniquely x = £s;s>0;s-t>0

E S) x

\

's

wher

e

X

s €

ker(F;

enough s. (b) We have £a;s>0;a+(>0 note either of these sums by Fi(x).

: M t _ 2 s -> M) and x's = 0 for large = £

a ; s

>

0 ; s

_

(

>

0

W e

(c) We have E . , . > o i . + t > o ^ ' " 1 ) « . = £ s ; s > 0 ; 3 - t > 0 ^ + 1 ) < note either of these sums by Ei(x).

We

dede

~

This follows from 5.1.5 (we are reduced by 5.1.4 to the case considered there.) The operators
=»

Ei(x) G Mn+2,Fi(x)

:M

M

G Mn~2.

1 6 . 2 . ADMISSIBLE FORMS

16.2.1. We fix P G T>i,M G C[. We will study the properties of the operators fa : P P, e{ : P -> P and : M M, Ei : M M in parallel.

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134

16.2.2. A symmetric bilinear form (,) : P x P —> Q(v) is said to be admissible if (a) (x,c<(y)) = (1 - v^2)(ix,y) for all x,y € P. A symmetric bilinear form (,) : M x M —• Q(v) is said to be admissible if (a') ( M n , Mn') = 0 for n^n' (b') (EiX,y) = v?-l(x,Fiy)

and for all x G Mn~2,y

G Mn.

16.2.3. Besides the subrings A = Z[v,v~l) and A = Q[[v -1 ]] fl Q(v) of Q(v) we shall need the subrings A(Z) = Z[[v -1 ]] fl Q(v) and A = Z((i> -1 )) H Q(t>) of Q((u - 1 )). 16.2.4. Let B be a basis of the Q(v)-vector space P (resp. M). We say that B is integral if (a) the ^4-submodule of P generated by B is stable under Ci,<j)^ : P —> P for all t > 0 (resp. the ,4-submodule ^Af of M generated by B is stable under E^.F^ : M -> M for all* > 0); in the case of M, it is further assumed that B fl Mn is a basis of Mn for all n. Assume that we are given an admissible form (,) and an integral basis B for P (resp. M) which is almost orthonormal (see 14.2.1). Let C{P) = {x e AP\{x,x)

e A}

C(M) = {x e AM\(x,x)

e A}.

and

Lemma 16.2.5. (a) C(P) is a Z[v~ l ]-submodule of AP and B is a basis of it. (b) Let x G aF be such that (x, x) G 1 + v~x A. Then there exists b G B such that x = ±b mod v~lC(P). (c) Let x G aP be such that that (x,x) G v~lA.

Then x G

v~lC(P).

(d) C(M) is an A(Z)-submodule of ^M and B is a basis of it. (e) Let x G be such that (x,x) G 1 + u _ 1 A . Then there exists b G B such that x = ±b mod v~1£(M). (f) Let x G

be such that that (x,x) G v _ 1 A . Then x G

This follows from Lemma 14.2.2.

v~lC(M).

16.2. Admissible Forms

135

Lemma 16.2.6. Let y G aP (resp. y G jM fl Ml with t > 0) be such that Ciy = 0 (resp. Eiy = 0,); let n > 0 (resp. 0 < n < t). We have = *n{y,v) (resp. {F^y.F^y) = Kfav)) where 7rn G 1 + v- 1 Z[[v- 1 ]] (resp. < e 1+ v~1Z[[v~1]]). It suffices to show that

(resp. (F^n+1)y,F^n+1)y) = ^'(F^y, F^y)) where f € l + tr'ZUtT 1 ]] _1 _1 (resp. 7r' e 1 + i> Z[[u ]]) and n > 0 (resp. 0 < n < t). We have Wl n + l ) ?/,0l n + 1 ) !/) = «n + i ] " V i t P y J ^ y ) = (1 - v-'r'ln = (i - v^)-1

+1 l - H ^ e ^ y ) [»+i]r

V(*1B)».

4n)y)-

Similarly, ( F ^ y , F ^ y ) = ([„ + l]"1^"

V

F?+»y)

= v 4 - t + 2 B + , [ - n + t] < [n+ l l r ' ^ t f , ^ ) . It remains to observe that [n + i l r V e i + ^ ' z i b " 1 ] ] for 0 < n and

for 0 < n < t. The lemma follows. Lemma 16.2.7. (a) Let x G aP; write x = J2n>O Vn where y^ = and XN G P(0) are zero for large N (see 16.1.2(c)). Then XN G aP for all N. (b) If x G C(P), then each xn and yx above is in C(P). I f , in addition, (x, x) G 1 + v - 1 A, then there exists No > 0 and b G B such that XN0 = ±b mod v~1C(P) and XN = 0 mod v~1C(P), YN = 0 mod v~lC(P) for all N^N0.

16. Kashiwara's Operators in Rank 1

136

(c) Let x G M* f l ^ A f . We write x = J2s-,s>0;s+t>0 V* where Vs = F^xs and xs G ker(Ei : Mt+2s —> M) are zero for large enough s; then xs G A M for all s. (d) If x G Ml fl C(M), then each xs and ys above is in C(M). I f , in addition, (x, x) G 1 + v~*A, then there exists sq > 0 and b G B such that xSo = ±b mod v~1C(M) and xs = 0 mod v~xC(M), ys = 0 mod v~1C(M) for all s so. ls We prove (a). We have XN = V^N('N~1^2IIN(X). Since stable under €{ and for all t, we see that A P is stable under IIAT : P • P. Hence xn G .4P. This proves (a). Next we show that the subspaces (N^P(0), < ^ ^ ' ^ ( 0 ) are orthogonal to each other under (,), if N ^ N'. We argue by induction on TV + N'. If N > 1, we have for z,z' G P(0) that z,(f>\N ^z') is equal to a scalar N times Ci(j>\ V ) , hence to a scalar times ^N so that it is zero by the induction hypothesis. We treat similarly the case where N' > 1. If N < 0 and Nf < 0, the result is trivial; our assertion follows. Now let x G AP be non-zero. We have (x, x) = (2/iv, 2/iv)- We can l t+1 find t G Z such that v- yN G C(P) for all N and v~ yN (£ C(P) for some N. Then there exist integers a ^ > 0, not all equal to 0 such that v 2t ~ (VN,yN) ~ &N G V - 1 A for all N. Hence

(e) v~2t(x,x)

~J2NaN

£ v~lA

and ^2NaN

> 0.

If x G £ ( P ) , then (e) shows that t < 0; hence yN G C(P) for all N and, using the previous lemma, we see that xn G C(P) for all N. If now x G £ ( P ) satisfies (x,x) G 1 + v _ 1 A , then (e) shows that t = 0 and a^o = 1 for some No and a^ = 0 for all N ^ No. In other words, we have {yNoiVNo) G 1 + v~1A and (vn^Vn) £ v _ 1 A for all N ± No. Using 16.2.6, we deduce that ( x n 0 , x n 0 ) G 1 + v _ 1 A and (x;v,x;v) G v _ 1 A for all N ^ No and the second assertion of (b) follows from 16.2.5. We prove (c). If xs = 0 for all s, then there is nothing to prove. Hence we may assume that xs 0 for some s and we denote by N the largest index such that XN ± 0. We have N > 0, N + t > 0. We argue by induction on N. If N = 0, there is nothing to prove; hence we may assume that Since N > 0. We have E^x = = E\N)aM C AM, we have p " * ' ] .sat G AM. We have [ ^ J " 1 G A, hence xn G a M . Then x' = x — F^xn G AM. The induction hypothesis is applicable to x' and (c) follows.

16.2. Admissible Forms

137

Next we show that (F$N)z, F ^ z ' ) = 0 if N + N' and z,z' are homogeneous elements in the kernel of Ei. We argue by induction on N + N'. If N > 1, we have that (F^N) z,F^N'] z') is equal to a scalar hence to a scalar times (F^'^z, F$N'~l)z) times (F^N~1)z1EiF^N,)zf)i so that it is zero, by the induction hypothesis. We treat similarly the case where N' > 1. If N < 0 and N' < 0, the result is trivial; our assertion follows. The remainder of the proof is entirely similar to that of (b). Lemma 16.2.8. (a) fa,€i : P —• P map C(P) into itself. (b) Fi,Ei : M

M map C(M) into itself.

Let x G C(P). We must show that fax G C(P),€IX G C(P). By 16.2.7, we may assume that x = for some xn as in that lemma. But then G C(P), by 16.2.6. We argue fax = 0, we denote by TN(P) the set of all x G aP such that x = [N)xf for some x' G P(0) D AP with (x',x') = 1 mod v~xA. For any 5, t such that 5 > 0, s + 1 > 0, we denote by T S)t (M) the set of all x G A M such that x = F^x' for some x' G ker (Ei : Mt+2s M)n^M l with (x',x') = 1 mod v~ A. From the definitions we see that (a)

fa(TN(P))cTN+1(P);

( b ) ii(TN(P))

C T

N

^ ( P ) f o r N > 1, Ci(T0(P))

(c) if N > 0, then fa : TN(P) -> TN+1(P) are inverse bijections;

= 0;

and u : TN+1(P)

TN(P)

(d) Fi(T8tt(M)) C T a + i, t _ 2 (M) if s > 0,5 + 1 > 1, and Fi(Ts,t(M)) if 5 > 0,5 + t = 0;

= 0

C rs_1)t+2(M) if s > 1,5 + t > 0, and Ei(Ts>t(M))

=0

(e) Ei(Ta,t(M)) i f s = 0,« > 0 ;

(f) if 5 > 0,s + t > 1, then Fi : TSyt(M) —i• TSit(M) are inverse bijections.

T s + i, t _ 2 (M) and Et :

Lemma 16.2.10. (a) Case of P. We have ±B + v~lC(P)

= UN>0TN(P)

+

Moreover, the sets BN — BC\(TN(P)-\-v~1 C(P)) of B.

v~lC(P). (N > 0) form a partition

16. Kashiwara's Operators in Rank 1

138

(b) Case of M. We have ±B + v~1C(M)

= Us,t-s>o,s+t>oTSjt{M) +

Moreover, the sets BSft = B D (T A , T (M) + v~lC(M)) form a partition of B.

v^CiM). (s > 0,S + t > 0)

By 16.2.6 and 16.2.5, we have TN(P) C ±B + v~lC(P). Conversely, let x G ±B. We have (x,x) G l + v - 1 A . Hence, by 16.2.7, we have x = y' + y" where y" G v~lC(P) and y' = <j)\N)x'' for some x' G P(0) D AP such that x' G ±B 4- v~lC(P) and some N > 0. Thus x G TJV(P) + v~xC{P) and the first assertion of (a) follows. To prove the second assertion of (a), it is enough to show that TNl(P) fl (TN2(P) + v~lC{P)) is empty for NX ^ N2. Assume that (f>\Nl)xi = x2+v~lz where z G C(P) and xx,x2 G P ( 0 ) n ( a P ) satisfy (x\,xi) = 1 mod v _ 1 A and (x2,x 2 ) = 1 mod v - 1 A . By 16.2.7, we can write 2? = £at>o \N)zN where zN G C(P) D P(0). We have \Nl)Xl = + V"1 Eiv>0 4N)*N. This implies, by 16.1.2(c), that x X ZN = 0 for N ^ N I , N 2 , V~ ZNX = X\ and V~ ZN2 = — x2. From the last 2 equality we deduce that ( x 2 , x 2 ) = V~ (ZN2,ZN2) £ v~2A, a contradiction. Thus (a) is proved. The proof of (b) is entirely similar. 16.2.11. Using the previous lemma and the results in 16.2.9, we deduce the following. In the case of P we have: (a)

fc(±BN

+ v~lC(P))

C ±Bn+1

+

v~1£(P);

(b) €i(±BN + v - 1 £ ( P ) ) C ± £ j v - i + v _ 1 £ ( P ) for N > 1, and ii(±B0 v~ C(P)) C v~1C(P);

+

1

±BN+1 + v~lC(P) (c) if N > 0, then fc : ±BN + v~lC(P) -1 _1 €{ : ± B N + 1 + v £ ( P ) —• + v £ ( P ) are inverse bijections.

and

In the case of M we have: (d) Fi(±Ba,t + v~1C(M)) C ± B a + i f t _ 2 + v - 1 £ ( M ) ) if s > 0,5 + t > 1, and Fi(±Ba,t + v~lC(M)) = 0 if s > 0,5 + t = 0; (e) £i(±B s ,t + f _ 1 £ ( M ) ) C 2 + v _ 1 /:(M)) if s > 1, s + t > 0, 1 and Ei(±Ba,t + v~ C(M)) = 0 if 5 = 0, t > 0; ±BS+M_2 (f) if s > 0,s + t > 1, then Ft : ±Ba,t + v~1C(M)) v~ C(M)) and Ei : ± P S + M _ 2 + t r ^ M ) ) ±BSjt + v~1C(M)) inverse bijections. l

+ are

16.3. Adapted Bases

139

1 6 . 3 . ADAPTED BASES

16.3.1. P, M, (,) are as in the previous section. We say that a basis B of P is adapted if there exists a partition B = Un>o-B(n) and bijections ?rn : B(0) B(n) for all n > 0 such that (a) for any N > 0, B(N)UB(N

+ 1)U£(AT + 2 ) U . . . is a basis of (f>\N)P;

(b) for any b G B(0) and any N >0 we have (f>[N)b - <jrN(b) G ^N+1)P. We say that a basis B of M is adapted if there exists a partition B = Us,t;s>o,s+t>oB(s,t) and bijections

for all s, t as above, such that (a) B(s, t ) U B ( s + M ) U B ( a + 2 , t ) U . . . is a basis of Ml n f\s) (b) for any b G 5 ( 0 , 2 s + t), we have

M\

- 7rs>t(b) G i ^ ( s + 1 ) M .

In this section it is assumed that B is integral, almost orthonormal (with respect to (,)) and adapted. Lemma 16.3.2. Let be

B.

(a) Case of P. We have b G Bo if and only ifb£

fa(P).

(b) Case of M. We have b G Ut>o#o,t V and only if b £ FiM. We prove (a). Assume first that b G Bn with N > 0. Then b = <j>^x' + v~lz where z G C(P) and x' G P. Since B is adapted, we can write 4>\N^x' = J2 c b'b' where b' runs over B fl <j>iN^P and Cb> G Q(v). We can also write z = J2b" db"b" where b" runs over B and dy> G Z[v - 1 ]. If b £ (f>[N^P, then by comparing the coefficients of 6, we obtain 1 = v~xdb, Since N a contradiction. Thus, we have b G 4N)p> 0, we have b G fa P. Conversely, assume that b G faP and b G Bo. Then b = x' + v~lz where z G £{P),x' G P(0), and b G <j>mP(0); using the equation N) (P(0), (f>\ P(0)) = 0 for N > 0, we deduce that (x\ b) = 0, hence (6, b) = v~1(z,b) G v~1A, a contradiction. This proves (a). The proof of (b) is entirely similar.

16. Kashiwara's Operators in Rank 1

140

Lemma 16.3.3. (a) Case of P. Let b G Bo,N > 0 and let b' be the unique element of ±B such that f{b) = b' mod v~lC(P). Then b' = npjb. (b) Case of M. Let b G Po,«+2* where s > 0. If s +1 > 0, then there is a unique element b' G ±B such that F?(b) = b' mod v~1C(M) and b' = tr 3ft b. If s + t < 0, then F?(b) = 0 mod v~lC(M). We prove (a). We write b = x + v~xz where z £ C(P) and x G P(0)C\aP satisfies (x,x) = 1 mod v - 1 A . Using 16.2.7, we write z = YIN' Z n ' w ^ e r e zN> G £ { P ) H (T><-N')P( 0) for all N ' . Replacing z b y x + V~1ZQ and z by z — zo, we see that we may assume that z satisfies in addition z G iP. The z and \N^b = ( f ) ^ x + v~ l ^z, together equalities = \N^x + N) imply with 4>?z G C(P) and \ z G 4>?b = 4>\N)b mod v~1C(P) +
By assumption we have b' = 4N)b

mod v~lC{P) +

Hence 4N+1)P.

Moreover, we have ^ ' 6 = 6,

mod^w+1)P

where b\ = n^b G B. We must prove that b' = b\. We have b\ + c\ = b' + c' where C\ G and d G v~1C(P). We have h <£ ^N+1)P. (Otherwise, we would 4N+1)P P; hence b G fcP, contradicting the previous lemma.) have \N^b G Hence, if we express b\ + ci as a Q(u)-linear combination of elements of P, the element b\ G B will appear with coefficient 1. On the other hand, if we express b' + c' as a Q(v)-linear combination of elements of B, then all coefficients are in v~1Z[v~1] except that of ±b'. This forces b\ = b' or b\ = —b'. If b\ = — b', then we have 2b\ + c\ = c' and ±&i appears in the left hand side with coefficient 2 and in the right hand side with coefficient in v~l A, a contradiction. Hence we have b\ = b' and (a) is proved. The proof of (b) is entirely similar. 16.3.4. The following result shows that the action of the operators <j>i,€i (resp. Pi, Ei) on the elements of B is described up to elements in v~lC(P) (resp. v~1C(M)) in terms of the bijections irn (resp. 7rSjt) in 16.3.1.

16.3. Adapted Bases

141

Proposition 16.3.5. (a) Case of P. Let b E B(N). Let b0 E B(0) be the unique element such that 7 = b. We have = 7r/v+i60 mod v~1C(P). We have 1 1 ei(b) = tt n _I6 0 mod v~ C(P) if N > 1 and e^b) = 0 mod v~ C(P) if N = 0. In particular, we have B^ = B(N) for all N. (b) Case of M. Let b E B(s,t). Let b0 E £ ( 0 , 2 s + t) be the unique element such that 7rSjt60 = b. We have Fi(b) = 7r s+ i )t _ 2 6 0 mod v~1C(M) if s 4-1 > 1 and Fi(b) = 0 mod v~lC(M) if s + t = 0. We have Ei(b) = 7r s _ M+2 fco mod v~lC(M) if s> 1 and Ei(b) = 0 mod v~lC(M) if s = 0. In particular, we have BS}t = B(s,t) for all s,t. This follows from 16.3.3.

17.1.

FIRST APPLICATION TO TENSOR PRODUCTS

17.1.1.

In this chapter we shall give three applications of Proposition

17.1.2. Let M, M G C[. Then MM is an object in C[ (see 5.3.1). Now let P eVt and M G C-. We define Q(v)-linear maps : P (g> M P ®M

where Ki : M —> M is the linear map given by Kiy — v™y for y G Mn. It is easy to check that (P(g)M, 6*) is an object of T>i. (This also follows from 15.1.5.) Hence the linear maps (j>i,€i : P M —* P 0 M are well-defined. From the definitions we deduce (using the quantum binomial formula)

for all x G P, y G M and t > 0; the sum is taken over all tr ,t" G N such L e m m a 17.1.3. ( a ) / / ( , ) : P x P Q(v) and (,) : M x M Q(v) are admissible symmetric bilinear forms in the sense of 16.2.2, then the symmetric bilinear form on P M given by (x (g> y, x' y') = (x, x')(y, y') (b) / / ( , ) : M x M -> Q(^) and (,) : M x M -> Q(v) are admissible symmetric bilinear forms in the sense of 16.2.2, then the symmetric bilinear form on M ® M ^wen by (x y, x' ® y') = (x, x')(y, y') zs admissible. We prove (a). Let x,x' € P,y € Mn,y'

G M n \ We have

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011

142

17.2.SecondApplication to Tensor Products

143

On the other hand, we have (<i(x ®y),x'(g> y') = (fa(x) <8> K~xy + rr (8) F<(y),x' 2/')

This proves (a). The proof of (b) is entirely similar. 17.1.4. We consider the following example. Let Po be the Q(i>)-vector We define Q(v)-linear maps fa, ei : Po —s• Po space with basis /?0, by = [a + l]iA+i for s > 0 and c<(A) = for s > 0 (with the convention = 0). It is easy to check that this makes Po into an object of Vi. We have # ( & ) = for s > 0,* > 0, and cf(ft) = v ~ t ( t + 1 ) / 2 + s < & _ t , for s > 0,t > 0, (with the convention /3_i = /?_2 = • • = 0). Let (,) be the symmetric bilinear form on Po given by s

t=l It is easy to check that this bilinear form is admissible. 17.1.5. We fix an integer n > 0. Let Mn be the Q(v)-vector space with basis bo,bi,... ,bn with Z-grading such that bm has degree n — 2m. It will be convenient to define bm = 0 for ra > n and for ra < 0. Let Ei, Fi : Mn —> Mn be the linear maps given by Ei(bs) = [ra — s + l]i&s_i and Fi(bs) = for all s. It is easy to check that in this way Mn is an object of C-. Note that for t > 0, we have E^fa) = [n~i+t]fi8-t s and F^fis) = P t T i ^ + t f° r • Let (,) be the bilinear form on Mn given 9{n s) by (bs, ba>) = 6s^v; ~ for 0 < s < n and 0 < s' < n. It is easy to check that (,) is an admissible form on Mn. 17.1.6. Let P 0 be as in 17.1.4 and let Mn be as in 17.1.5 (ra > 0). Then, as in 17.1.2, P = P 0 Mn is a well-defined object of Vi. Note that P has a basis {bSjSt = /3S 6 s /|s > 0,0 < s; < ra}. For any t > 0 we have t' + s t'

r

t"

Vs+t',s'+t"

where the sum is taken over all t', t" 6 N such that t' +1" = t, and U&.,, = «rB+a''+*"1fc.-1,3'

+ vrn+2s'~2{vi

- Ol"

-

+ 1 }ib,,s'-i-

17. Applications

144

By convention, we set bStS> = 0 if either s < 0 or s' < 0 or s' > n. For any m e Z, we define P m to be the Q(v)-subspace of P spanned by the vectors with S + S' = M. We have P = © m P m , <^ t } P m C P m + t and EIPM C P

m _ 1

.

By definition, the operator fc : P —> P (resp. LI : P —• P ) is an (infinite) linear combination of operators (resp. it follows that (a)

fc{PM)

C P m + 1 and EI(Pm) C P™"1.

17.1.7. Let a*'= XXt(n+t~s'} t=0

5+ t t

Vs+t,s'-t

for s > 0, s' > 0, s + s' < n, n -f t - sr t

-t(s+t) t=0

t>s+t,s'-t

for s > 0, 0 < s' < n, s + s' > n; the two definitions agree if s + s' = n. Note that C*,*' E PS+S'. For s + s' > n, we have (a)

= J2 » R t'=o

, W

-1 - n + s'

Indeed, the right hand side of this equality is, by definition, s' Y^

s'-t' ^

-+"<

V^"^'+t")~t's-t'

t'= 0 t"= 0

4- t" t" — —ft'sf 4-+ t.' I I- 1—1- n—+n s'-4- V n -Ibs+t'+t",s'-t'-t"t'

The coefficient of bs+t,s>-t (where 0 < t < s') is n+ t-

E t'+t"=t

t"

[ " - 1 - n + s'" \A

.

We replace the exponent -t"(s + t' + - t' by (n + t - s')t' ( - 1 - n + s')t" + / where / = t(-n - t - s + s' - 1) depends on t', t" only through their sum. Hence the coefficient of bs+ttS'-t is s')t'-(-l-n+s')t"

=

t'+t"=t t- 1 yf = vf8t, o = <5t,o; t

'n + t - sr t"

"-1

i

-n + s r t'

17.2.SecondApplication to Tensor Products

145

(a) is proved. Prom the definitions and from (a), we see that the subset B of P consisting of the vectors £SiS> (with s > 0 and 0 < s' < n) is a basis of P. For ra G Z, let £ m be the Z[t; -1 ]-submodule of P generated by the vectors ba,s> with s + s' = ra. Lemma 17.1.8. (a) For any s > 0 and 0 < s' -bSy8<

ev~lCs+3'.

(b) Form > 0, Cm is the Z[v l]-submodule of P generated by the vectors £8tS' with s + s' = ra. is in Assume first that s + s' < n. The coefficient of bs+t}s>-t in _t(n + t - s ' ) + st = + Here t > 0; hence 2 t(s + s' — n) — t < 0 ; the inequality becomes an equality only for t = 0. Assume next that s + s' > n. The coefficient of bs+t,S'-t in (s,s' is in

Here t > 0; hence — t(s + t) + t(n — s') = t(n — s — s') — t2 < 0; the inequality becomes an equality only for t = 0. This proves (a). The previous proof also shows that the matrix expressing the vectors Cs,s' in terms of the vectors bs>s' (with s + s' = ra fixed) is upper triangular, with diagonal entries equal to 1 and with off-diagonal entries in v~1Z[v~1]. This implies (b). The lemma is proved. Lemma 17.1.9. The A-submodule j^P of P generated by B is stable under ^ : P —> P for all t > 0. (In other words, the basis B of P is integral) The formulas in 17.1.6 show that Ci(6S)S/) 6 j^P and <^(& S)S /) € ^ P for all t > 0. The lemma follows. Lemma 17.1.10. Assume that 0 < s < n and t > 0. We have ^ K f i = CS,t if s + t < n and i%,0

t-\- S — n

£ u;u>0;u>t—n;u<s+1—n

U

Cs+u,t—u

17. Applications

146

if s + t > n. Assume first that s + t < n. We have 4

Pi Ko =

'if + 5

2^vi

t'

t'= 0

bs+t',t-t' — Ca,t'

Assume now that s + t > n. We have r

d*)h

tf'b.,0

_

V^

=

n-t'(n-t+t')

£

f

t'-,t >0;t'
E

v

t' + s if

i

Vs+t',t-t'

Vi

K

'

i',t"€N;t'+t"
-1 -n + t"

t-t'

= E(

'if + 5" i t'

Cs+t'+t",t-t'-t"

E

-t"(s+t')-t"-t'(n-t+t') i

u = 0 t' ,t" ;t'-\-t"=u;t' >Q]t" >0;t' >t—n

'if + 5' Cs+u,t—u)• i t' t" Since the index t' satisfies t' > t—n and u>t', the index u must satisfy u > t-n. We substitute i " 1 " ? , = ( - i f ft"]* = (-^["t'^i and V7*"('+*')-*"-*'(»-*+*') = v7(n-t+u)t'+(-s-i)t\ — 1 — 71

t — t'

The condition on u implies n — t + u> 0; hence [n~5,+u] . is automatically zero unless n — t + u > t", i.e., if t' > t — n. Thus the condition t' >t — n can be omitted in the summation and we obtain

E E

0t—n

-(n-t+u)t' + (-s-l)t"

t',t">0

t'+t"=u

71 — t + UI

\—S —

i.rv] E (-1)"

CS + Ujt — U

t"

71 — t + U — S — 1

u;0t—n

y-v u;0

t—n

[—71 + t + S U

U

Cs+u,t—u

Cs+u,t—w

(We have used 1.3.1(e), 1.3.1(a).) Recall that s + 1 > n. It follows that i ~ ® u n l e s s u < t + s — n and then the condition u < t is automatic. Hence our sum becomes £ u ; u > 0 . y U > t - n - , u < s + t - n ['"Tl The lemma is proved.

17.2.SecondApplication to Tensor Products

147

Lemma 17.1.11. We consider the partition of B into the subsets B(t)

where t =

= {Ca,t|* + t < n} U { < t + 2 * - » , n - * | « + t >

n}

0,1,2,....

(a) For any t> 0, the set B(t) U 5 ( f + l ) U 5 ( t + 2 ) U - - is a basis of (b) The basis B of P is adapted. Prom 17.1.10, we see that CM G

if s + t
4>ft+"n)P

if s 4-1 > n. (The last inclusion is seen by induction on t.) It follows that (c) B(t) U 5 ( t + 1 ) U 5 ( H 2 ) U - C

4t)p-

Hence X(t) C (frf*P where X(t) is the subspace of P spanned by B(t) U B ( (d) H l ) ^Ub5c( fX + ( t2)) U - - - . We now prove the inclusion for any b G B fl Pm, by induction on ra > 0. Note that B(0) = {Cs,o|0 < s < n) = {fes,o|0 < s < n}. If b G B(0), then (d) follows from 17.1.10. If m = 0, then b = b0y0 G #(0), hence (d) holds. Assume now that ra > 1. If b G B(0), then (d) holds; hence we may assume that b G B — B(0). Then b G X(l) and by (c) we have b = fay where y G Pm~l. By the induction hypothesis we have (f>^+1)y G X(t + 1) C X(t), G X(t). This hence proves (d). Thus (a) is proved. Prom (a) we see that {6 G B\b $ faP} = B{0). Let Trt : B0 -»- Bt be the bijection given by (e) 7TtCs,o = Cs,t if s + t n. From 17.1.10, we see that ,o - ntCs,o G X(t + 1), hence t+1) P. (f) ^ C - , 0 = 7rtC*,o mod 4 The lemma is proved. Lemma 17.1.12. Consider the admissible form (,) on P defined as in 17.1.3, in terms of the admissible forms 17.1.4, 17.1.5, on Mn and Po. Then B is almost orthonormal with respect to (,). From the definition it is clear that the basis (bStS>) of P is almost orthonormal (actually different elements in this basis are orthogonal to each other). Since B is related to this basis as described in 17.1.8, it follows that B is also almost orthonormal.

148

17.

Applications

We now see that the hypotheses of 16.3.5 are verified in our case. Applying Proposition 16.3.5 to B, and taking into account 17.1.11(e),(f), we obtain the following result. Proposition 17.1.13. We have 4>i(Cs,s')

= Cs,s'+1

mod v~1C(P) if s + s' < n,

4>i(Cs,s')

= Cs+i,s'

mod v~1C(P) if s + s' > n,

uiCsy)

= Cs,s'~i

mod v~1C(P) if s + s' 1,

e»(C«,«') =

Cs-i,s'

mod v~xC(P) l

€i(Ca,o) = 0 mod v~ C(P)

if

if s + s' > n, s
Using Lemma 17.1.8, we can restate the proposition as follows. Corollary 17.1.14. ii{b3,s.) = bs,s>+x mod v ~ 1 £ ( P ) n P H V + 1 i f s + s' < n, 4>i(bStS') = bs+i>s/ mod v~lC(P) fl PS+S'+L u(bs,s>)

= bs,s>-1

l

m o d v~ C{P)

D P ^ ' "

l

h(ba,s.) = b8-its> mod v~ C(P) fl PS+S'~L

if s + s' > n, 1

i f s +

s ' < n ,

if s + s' > n.

What we actually get are the statements of the corollary with C(P) fl ps+s'±i replaced by C(P). But ba,a* G Ps+s'; hence from 17.1.6(a), 4>i(bs,s>) E PS+A'+L and €I(BS^) G P*+*'-\ The corollary follows. Corollary 17.1.15. Let P eVt, M € C[ and let (P ® M,<j)i,€i) G Vi be defined as in 17.1.2. Let x G P and y G Mn be such that CiX = 0, Eiy = 0. (Then n > 0.) For any m >0, let Cm be the Z[v'^-submodule of P ® M generated by the vectors Vs)x (8) Fls )y with s + s' = m. We set C-\ = 0. We have M(f>[s)x F^y) = (f>[9)x ® Ft(3'+1)y mod t r *£,+.'+1 if s + s' < n; tiiti^x ® F ^ y ) = ii(48)x

® Ft{s,)y) = 4s)x

e i l ^ W ^ ^ ^ "

® F^y Fls'~l)y 1

^ ^

mod v~lCs+s,+1 if s + a' > n; mod v^C^-x 1

mod v~ £s+s'-i

if s + s' < n; i f s + s'> n.

We may identify Po ® Mn with the subspace of P ® M spanned by the vectors (j) \s)x ® F^s )y with s > 0 and 0 < s' < n. It is in fact a subobject in T>i. Therefore the result follows from the previous corollary. 17.2.

SECOND APPLICATION TO TENSOR

PRODUCTS

17.2.1. We consider two integers p > 0 and n > 0 and form the tensor product M = Mp Mn. This is again an object of C[\ hence the operators

17.2. Second Application to Tensor Products

149

Ei,Fi : Mp Mn —> Mp Mn are well-defined. Now Mn has a basis {bs'\0 < sf < n} as in 17.1.5; similarly, Mp has a basis {bs|0 < s < p} as in 17.1.5; Then {bs,s> = bs®bs> |0 < 5 < p , 0 < < n) is a basis of M. As in 17.1.7, we define s+t t

t=o

for 0 < s < p, s' > 0, 5 + s' < n, "ra +

C

=

t=o

t - sr Ds+ty-t t

for 0 < 5 < p, 0 < s' < n, 5 + s' > n; the two definitions agree if 5 + s' = n. The vectors just described form a basis B of the vector space M, which is related to the basis (bSfS>) by a matrix with entries in Z[v - 1 ] whose constant terms form the identity matrix. (This is seen as in 17.1.8 or can be deduced from that lemma, using the natural surjective map P —> M which takes bSiS' to bSfS> if s < p and to zero if s > p; that map also takes Cs,8' to if s < p and to zero if s > p.) Hence the A(Z)-submodule of M, generated by the elements (6S,S')> coincides with the A(Z)-submodule generated by the elements (Ca,s')» w e denote it by C(M). As in 17.1.9, we see that the ^4-submodule of M generated by B is stable under hence B is an integral basis. As in 17.1.12, we see that B is almost orthonormal with respect to the form (,) on M defined as in 17.1.3 in terms of the forms (,) on Mp,Mn (see 17.1.5). As in 17.1.11, we see that the basis B of M is adapted. (Again, this could be deduced from the corresponding result for P.) We now see that the hypotheses of 16.3.5 are verified in our case. Applying Proposition 16.3.5 to J3, we obtain the following result, analogous to 17.1.13. Proposition 17.2.2. We have Fi(ts,sf) = Cs.s'+I

m o d

Pi(Cs,s') = Cs+its'

m

V~1C(M) if s + s' < n;

° d v~1C(M) 1

Fi(Cp,s') = 0 mod v~ C(M)

if s

Ei((s,s') = Ca,s'~i m°d v~lC(M) f

Ei(Cs,s ) = G - m ' mod v 1

_1

Eiits,o) = 0 mod v~ C(M)

if s < p and s + s' > n; sf > n; if s + s' 1;

£ ( M ) if s + s' > n; if

s
17. Applications

150

As in 17.1.4, we can restate the proposition as follows. Corollary 17.2.3. Fi(bSja') = bSyS<+1 mod v _ 1 £ ( M ) if s + s' < n; Fi(bSysf)

=

bs+i,s'

mod v~lC(M)

Fi(bPiS>) = 0 mod v~1C(M) Ei(bSyS') = bSiS'-1

if s + sf > n;

mod v~lC(M)

Ei(bs,s') = bs-it8> mod v~lC(M) l

Ei(ba,o) = 0 mod v~ C(M)

if s < p and s + sf > n;

if

if s + s' 1; if s + s' > n; s
Corollary 17.2.4. Let M £ C[, M e C[ and let M ® M € C* be defined as in 5.3.1. Let x e Mp and y e Mn be such that EiX = 0 , E i y = 0. (Then p > 0,n > 0.) For any m > 0, let C be the A(Z)-submodule of M M generated by the vectors We have Fi(F^s)x ® F^y)

= F$a)x ® F^+1)y

mod v~lC if s + s' < n;

Fi(F^s)x ® F^y)

= Ft(3+1)x ® F^y

mod v~lC if s + s'> n;

Ei{Fls)x

F^y)

= F^s)x F^'^y

mod v~lC if s + a' < n;

= F^~l)x ® F^y mod v~lC if s + s' >n. Ei{Fls)x F^y) We may identify Mp Mn with the subspace of M <S> M spanned by ^y with 0 < s < p and 0 < s' < n. It is in fact a the vectors F^x subobject in C[. Therefore the result follows from the previous corollary. 1 7 . 3 . T H E OPERATORS

: f —• f

17.3.1. We shall regard f as a il-module as in 15.1.4. Thus, for each i € / , <j>i : f —• f acts as left multiplication by 0i and €i : f —• f is the linear map ir in 1.2.13. For any i G I, f with the operators i,ei : f —> f is then an object of T>i. (See 16.1.1.) Hence the operators : f —• f (see 16.1.3) are well-defined. Note that the form (,) on f is admissible in the sense of 16.2.2 for any i. 17.3.2. For a fixed i G / , we define a Q(v)-basis B% of f as follows. By definition, B% = U t >oB*(t) where B%(0) is any subset of B^q such that Bi-o = B'(0) U (-£*(())) and, for t > 0, B*(t) is the image of £ f ( 0 ) under TIM : Bi-o ^ Bi-t (see 14.3.2(c)). By definition, we have B = B{U (-B1) and Bl is adapted (in the sense of 16.3.1) to f € Vi. By definition of B, we see that Bl is almost orthonormal for (,) and the ^4-module it generates is ^ f .

17.3. The Operators $iy u : f -»> f

151

17.3.3. Let £ ( f ) = {x e Af\(x>x) G A}. From Theorem 14.2.3 and Lemma 14.2.2, it follows that £ ( f ) is the Z[v -1 ]-submodule of f generated by B\ Lemma 17.3.4. (a) ^f is stable under the operators any i G I.

: f —> f, for

(b) £ ( f ) is stable under the operators fa,ii : f —• f, for any i G I. The stability under is clear from definitions. The stability under e^ follows from 13.2.4. This gives (a). Now (b) follows from Lemma 16.2.8(a) applied to f, (,) and Bl. Applying Proposition 16.3.5 to our case, we see that the following holds. Proposition 17.3.5. Let b G Bl(t). Let bo G Bl(0) be the unique element such that 7riftbo = b. We have fa(b) = 7ri)t+i&o mod v - 1 £ ( f ) . We have ii(b) = 7Tij-ibo mod v _ 1 £ ( f ) if t > 1 and c,(6) = 0 mod v~1C(f) if t = 0. 17.3.6. The following result shows that the endomorphisms of the Zmodule £ ( f ) / v _ 1 £ ( f ) induced by fa, ii act, with respect to the signed basis given by the image of B, in a very simple way, described in terms of the bijections 7Ti>n in 14.3.2(c). Corollary 17.3.7. Let i G I and let b G 0<;t- Let bo G B^o be the unique element such that 7ri)t&o = b. We have (a) 4>i(b) = 7r i)t+ i6 0 mod

v~lC(f);

(b) €i(b) = 7rM_i&o mod v~lC(f) i f t = 0.

if t > 1 and ii(b) = 0 mod v _ 1 £ ( f )

(c) If i G / and b G B, then we have i(b) = b' mod v - 1 £ ( f ) for a unique b' G B. Moreover, lib' — b mod v~1C(f). (d) If i G / and b G Bi;n for some n > 0, then we have ii(b) = b" mod v~lC(f) for a unique b" G B. Moreover, fab" = b mod v~lC(f). We apply 17.3.5 to b if b G B\ or to -b if -b G B\. This gives (a) and (b). Let b' = 7Ti>n+ibo G B j ; n + W e have fa(b) = b' mod v XC(f) by (a) and e i (6 / ) = b mod v _ 1 £ ( f ) by (b). This proves (c). Assume now that b G Bi]n with n > 0. Let b" = 7TI>N_I6O G Bi-n-1. We have ii(b) = b" mod v~lC{i) by (b) and fa(b") = b mod v~lC{f) by (a). This proves (d).

Study of the Operators Fu Ei on AA

18.1.1. In this chapter we assume that the root datum is ^-regular. Let A E X + . As in 3.5.6, we set A a = f f 0 t < < , A > + 1 . S i n c e A w i l 1 b e fixed in this chapter, we shall write A instead of A\. As in 3.5.7, we denote the Recall that there is a unique U-module structure on A such that EiJ] = 0 for all i E / , K^rj = for all /i E Y, and Fi acts by the map obtained from left multiplication by di on f. From the triangular decomposition for U, we see that A can be naturally identified with the U-module

by the unique isomorphism which makes rj correspond to the image of For any v E N[/], we denote by (A)^ the image of under the canonical map f —> A. We have a direct sum decomposition A = 0i/(A)i/. Note that (A)„ is contained in the (A — v)-weight space A A _ l / (the containment may 18.1.2. By Theorem 14.3.2(b), the subset U i>n;n >(i |A )+i a S»;n of B is a signed basis of the Q(t;)-subspace of f. Hence the natural projection f —> A maps this subset to zero and maps its complement B(A) = ^2G/(un;0
for all x E f and y E A. (See 15.1.5.) Hence for each i E / , we have

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Media, LLC 2010 2011

152

18.1. Preliminaries

153

Lemma 18.1.4. There is a unique Q(v)-linear map S : f —> f ® A such that (a) E(l) = 107?; (b) E ( f a x ) = fa(E(x)) for all x G f and all i G I; (c) E(e{x) = Ci(3(x)) for all x G f and all i G I. By 3.1.4, there is a unique algebra homomorphism f —> f ® U such that Oi •—• 9i K - i -f 1 <8> Fi for a l i i G I. Composing this with the linear map f ® U —> f ® A (identity on the first factor, the map u i—• urj on the second factor) we obtain a linear map E : f —> f <8> A which clearly satisfies (a) and (b). We show that it satisfies (c). For x = 1, (c) is trivial. Since the algebra f is generated by the various Oj, it is enough to show that (c) holds for x = Ojx', assuming that it holds for x'. We have E(eix) = E(ei(l)jx') = E(v1^jEix' 4- Sijx') = vimjj€iE(xr) +

6itjE(x')

and

e,(S(x)) = tiiStoj*)) = hence (c) holds for x. This proves the existence of S. The uniqueness of E (assuming only (a),(b)) is clear since f is generated by the Oj as an algebra. 18.1.5. Let £(f) be as in 17.3.3. We have £ ( f ) = ©„£(f)„ (sum over all v G N [ I ] ) where £(f)„ is the Z[v _1 ]-submodule of f generated by Bu. Lemma 18.1.6. (a) IfbeBis not equal to ±1, then there exist i G / and b" G B such that b - fab" G v " 1 ^ ) . (b) If v G N [I] is non-zero, then c(f)„ =

Yl i;ui>0

We prove (a). According to 14.3.3, if 6 is as in (a), then there exist i G / and n > 0 such that b G By 17.3.7, we then have fab" - be v _ 1 £ ( f ) for some b" G B. We prove (b). The sum j r ^ x ) faWfyv-i) 1S a Z[v _1 ]-submodule of C(f)u (by 17.3.4) and the corresponding quotient module is annihilated by v~l (by (a)). By Nakayama's lemma, this quotient is zero; therefore (b) holds. The lemma is proved.

154

18. Study of the Operators Fi, Ei on A \

Proposition 18.1.7. Let v G N [ I ] . (a) £(f)u coincides with the 7i[v~l\-submodule of f generated by the el• * * itl for various sequences ti,Z2,... iw / in which i ements appears exactly Vi times for each i E / . consisting of the images of the el(b) The subset of C(f)u/v~1£(f)l/ 05 ements (f>i2 • • • 1 for various ii,^--above, coincides with l the image of Bu in C{i)u/v~ C{i)u. This follows immediately from the previous lemma. 18.1.8. We will denote by L(A) the A-submodule of A generated by the signed basis {b~r]\b G #(A)} of A. We have a direct sum decomposition L(A) = ©„L(A)„ (v runs over N[/]) where L(A)l/ is the A-submodule of A generated by the elements {b~7]\b G 5(A) D Bu}. We have L{A)U C (A)„. Since A is integrable (see 3.5.6), A belongs to the category C[ for any i G /; hence the operators Ei,Fi : A —> A (see 16.1.4) are well-defined. For any v G N[/], we will denote by L'(A)U the A-submodule of A generated by the elements F ^ F ^ • • • Fi t rj for various sequences . . . , it in I in which i appears exactly Vi times for each i G I. Let L'{A) = L'(A)v C A. We have L'(A)U C (A)^. 1 8 . 2 . A GENERAL HYPOTHESIS AND SOME CONSEQUENCES

Until the end of 18.3.6, we shall make the following General hypothesis 18.2.1. N is a fixed integer > 1 such that, for any v G N[/] with tr v < N, we have (a) L{A)U = L'{A)„; (b) if i is such that Vi > 0, then Fi(x~rj) = (ix)~r) mod v~1L(A), all x G C(f)u-i; (c) if i is such that Vi > 0, then Ei(b~rj) = (€ib)~rj mod v~1L(A), all b G Bu such that b~r) ^ 0; in particular, Ei(L{A)u) c L{A)v-i.

for for

In this section and the next we will derive various consequences of the general hypothesis; we will eventually show that this is not only a hypothesis, but a theorem (see 18.3.8). Lemma 18.2.2. Let v be such that tr v < N, let i G I and let x G L'{A)u. x r w/iere the xr G (A)^-^ satisfy EiXr = 0 for all r Write x = and xr = 0 unless r + (i, A — v) > 0 (see 16.1.4)- Then (a) xr G L'(A)u-ri

for all r.

18.2. A General Hypothesis and Some Consequences

155

(b) I f , in addition, x = b~rj mod v~lL(A) for some b G Bv fl B(X), then there exist ro G [0,1/*] and bo G Bv-rQ% fl B(A) such that xro = b^rj and xr G v~1L'(A)l/-ri for all r ^ romod v~lL'(A)u-roi, Let t be an integer such that 0 < t < Ui and xr = 0 for r > t. We prove (a) by induction on t. If t = 0, then (a) is obvious. Assume now that t > 1. We have E{X = 2 r = o - F ^ W + i a n d xr+1 = 0 unless r + (i, A — v + i') > 0. (If we had simultaneously xr+i 0 and r + (i, A — v + i') < 0 then r + 1 + (i, A — v) < 0, a contradiction.) By the general hypothesis, we have EiX G L(A)„_;. By the induction hypothesis applied to EiX, we have xr G L'(A)„-ri for all r > 0. Hence F^r)xr = F[xr G L'(A)U for all r > 0. Since x G L'(K)U, it follows that x0 G L'(K)U. This proves (a). We prove (b) by induction on t as above. If t = 0, then (b) is obvious. Assume now that t > 1. By the general hypothesis, we have EiX = (iib)~r] to mod v~1L(A). By 17.3.7, we have that e ^ is equal modulo either 0 or to b' for some b' G Bu-i. If the first alternative occurs, or if the second alternative occurs with (ib')~r) = 0, then Ei(b~r)) G v _ 1 L(A); applying (a) to vEi(b~rj) we see that xr G v~1L(A) for all r > 0. We then have xo = b~rj mod v - 1 L(A), as required. Hence we may assume that (c) iib = b' mod v~lC{f),

where b' G Bv-i

fl B{A).

We have, by assumption, (d) EiX = Ei(b~rj) mod v^EiCtf),,

.

By the general hypothesis, we have EiC(i)u C £(f)„_i and Ei(b~rj) = (we have b~r) / 0, by assumption). Introducing (eib)~T] mod v~lL(K)u-i this in (d), and using (c), we obtain EiX = bf~rj

mod

v~1C(f)l/-i.

By the induction hypothesis applied to EiX, we see that there exist v~1L'(A)„-roi, r 0 G [1, Vi] and b0 G Bu-roiC\B(A) such that xro = b^r) mod l for all r such that r > 0 and r ^ ro- It follows that and xr G v~ L'(k)u-ri ^ x = Fl°~lxro

mod

v~lL{A).

By the general hypothesis, we have FiEiX = FiEi{b~r,) = Fi{{eib)~n) = (^€<6)-^

156

18. Study of the Operators Fi, Ei on A \

(equalities modulo v~lL(A).) Since iib = b' mod v~lC(f), we see from 1 It follows that FiEiX = b~7] = x 17.3.7 that fcub = b mod v~ £(f). 1 (equalities modulo v~ L(A)). We deduce that x = F i ( ^ r ° ~ 1 x r o ) = F[°xro (equalities modulo v~1L(A).) Since F[xr G v~lL{A) for all r > 0, r ± r0 and x = J2rFixr> w e deduce that xq G v~lL(A). This completes the proof. Lemma 18.2.3. Let i G I and let x G C{f). Write x = £r>o^i(C(f) © L'(A)V) c £(f) © L'{A) and Ci^flOL'lAJJcAflOi'tA). By Lemmas 18.2.2(a) and 18.2.3, the A-module £ ( f ) © L'(A)U is generated by elements cjy^x x' where x G £ ( f ) and x' G L'{A)v-.a'i satisfy €i{x) = 0 and Ei(x') — 0. The image of such elements under fc or li is contained in £ ( f ) © L'{A) by Corollary 17.1.15. The lemma follows. Lemma 18.2.6. Let x G L'{A)U where tr v < N. Assume that there exists b G Bu fl B{A) such that x = b~r] mod v~lL'{A). Assume also that Fix £v~lL'(A). Then fc(l ® x) = 1 ® Fix mod v~lC(f) © L'(A). By 18.2.2, we may assume that x = F^xf where x' G L'{A)U-Si satisfies Eix' = 0. Since x' / 0 and Eix' = 0, we have n = (i, A - i/ + si') G N; moreover, F^n+1)xf = 0. By Corollary 17.1.15, <^(1 ® F^s)x') is equal modulo v _ 1 £(f)©Z/(A) to 1 ®F^s+1)x' (if 5 < n) or to 0i^F^a)xf (if a > n). If the second alternative occurs, then Fix = F*+1x' = 0, contradicting our assumptions. Thus the first alternative occurs and the lemma is proved.

18.2. A General Hypothesis and Some Consequences

157

Lemma 18.2.7. For any v such that tr v < N, we have H(£(f)„) C £ ( f ) © L'(A). We argue by induction on tr v. If v = 0, the result is obvious. Assume that v ^ 0 and that the result is known when v is replaced by v' with tr v' < tr v. By 18.1.6(b), the Z[v -1 ]-module C(f)v is spanned by vectors fax with i e I and x G £(f)„_i, so it suffices to show that for such i, x, we have E(fax) G £(f)©Z/(A). Since E is a morphism in Vi, we have E(fax) = fa(E(x)). By the induction hypothesis, we have E(x) G £ ( f ) © L'(A). Prom the definition of E we see immediately that E(f„')c

I/"; tr u"< tr u'

f ® tr„77.

Combining this with the previous inclusion, we see that E(x)e v"\

tr

£(f)©£'(A)„". u"
Hence it is enough to show that fa(C(f) © L'{A)„„) c C{f) © L'{A) whenever tr v" < N. Now the A-module £ ( f ) © L^A)^" is spanned by vectors of the form (jy^x © F^y where x G £ ( f ) , y G L'(A)U"_ci satisfy €iX = 0, Eiy = 0 (see Lemmas 18.2.2, 18.2.3). Hence it suffices to show that fa(\a)x © F^c)y) \ belongs to the Z[v -1 ]-submodule generated by the vectors (j>[a ^x © for various a',c' > 0. But this follows from Corollary 17.1.15. The lemma is proved. 18.2.8. Consider the linear form f —• Q(v) which takes to zero for all v ^ 0 and takes 1 to 1; tensoring it with the identity map of A, we obtain a Q(t>)-linear map pr : f © A —• A. From the definitions, we see easily that (a) pr(E(x)) = x~r) for all x G f. Lemma 18.2.9. (a) We have pr(C{f) © L'{A)) C L'(A). (b) Let i G I. Let y G £ ( f ) © L'(A)U where tr v < N. pr(fa(y)) = Fi(pr(y)) mod v-lL'(A)„+i.

We have

Let x G £(f)j, and let x' G L'{A)„>. If v ± 0, we have pr(x © x') = 0; if v = 0, we have x — f l where / G Z[v - 1 ] and pr(x © x') = fx'. Thus (a) holds.

18. Study of the Operators Fi, Ei on A \

158

We prove (b). By Lemmas 18.2.2, 18.2.3, we may assume that y = 4>f*z © V where z G £ ( f ) (homogeneous) and z' G L'(A)U< satisfy €i(z) = 0, Ei(z') = 0 and a, a' G N. We may assume that z' ^ 0. Let n be the smallest integer > 0 such that F™+lz' = 0. By Corollary 17.1.15, we have that 4>i{y) is equal to (c) 4a+1)z

modulo t r 1 / ^ ) © L'{A), if a + a' > n, and to

<S> F^z'

(d) \a)z © F^a'+1)z'

modulo v^Ctf)

© L'(A), if a + a' < n.

If a > 0 or ^ ^ fo, then y and both vectors (c),(d) are in the kernel of pr, by the definition of pr; on the other hand, by (a), we have pr{v~lC{f) © Z/(A)) C v~lL'{A). Hence in this case the lemma holds for y. Hence we may assume that a = 0 and z = 1. We then h a v e p r ( y ) = F±a moreover, by the previous argument: MMv)) = F^'+l\z')

mod v ^ L ' i A )

if a! i(y)) = 0

mod

v~1L,(A)

if a' > n. On the other hand, by the definition of Fi, we have Fi{pr(y)) = f l ( j t f a V ) ) = = 0 if a! > n (by the definition of It remains to observe that F^a'+1)(zf) n). The lemma is proved. L e m m a 18.2.10. Let x G C(i)u Fi(x~rj) mod v~lL'(A).

with

trv
We have (4>ix)~rj =

Using 18.2.8(a) and the commutation of H with

we have

(•4>ix)~r] = pr(E(4>ix)) = pr(i(E(x))). Using again 18.2.8(a), we have

Fi(x~n) = #4(pr(S(s))). It remains to show that pr(ME(x)))

= Fi(pr(E(x)))

mod

v~lL'(A).

This follows from Lemma 1 8 . 2 . 9 ( b ) applied to y = E(x). (We have E(x) G f C(i) © L'{A) b y 18.2.7, and E(x) G tr „"< tr „ ® hence E(x)e

Y1 v"; tr

C(f)Q>L'(A)^ u"
so that Lemma 18.2.9(b) is applicable.) The lemma is proved.

18.3. Further Consequences of the General Hypothesis

159

Lemma 18.2.11. If tr v = N, we have L(A)„ C L'(A)„. By definition, L(A)l/ consists of the vectors of form x~rj where x G C(i)v. Since v ^ 0, the Z[v _1 ]-module C{f)u is equal to ) (see 18.1.6). Hence it suffices to show that (fa(x))~rj G L'(A)„ for any i G I such that Vi > 0 and any x G £(f)»/-». By 18.2.10, we have (fax)~r] = Fi(x~rj) mod v~1L'(A)u. Hence it suffices to show that Fi(x~r)) G 2/(A)„. By the definition of L(A)„_j, we have x~r) G L(A)u„i. Using our general hypothesis, we deduce that x~rj G L'(A)u-i. It remains to observe that Fi(L'(A)u-i) C L'(A)U (from the definitions). The lemma is proved. Lemma 18.2.12. If tr v = N, we have L'(A)„ C L{A)u +

v~lL'{A)u.

We have L\A)u

= J2

=

i\Vi>0

=

E i\Vi>0

Afi(£(f)^). i;j/»>0

The first and third equalities are by definition; the second one follows from our general hypothesis. Hence it suffices to show that Fi(x-r))eL(A)u+v-lL'{A)„ for all x G £ ( f ) „ _ ; (where Vi > 0 ) .

By 18.2.10, we have Fi{x~r]) = (4>ix)~r) mod v~lL'{A)u. On the other hand, we have fax G C(f)u (see 17.3.4); hence we have ( f a x ) ~ r j G L(A)u. The lemma is proved. Lemma 18.2.13. If tr v = N, we have L(A)U = L'{A)„. By 18.2.11, L(A)U is an A-submodule of L'(A)^. The corresponding quotient module is annihilated by v~l, see Lemma 18.2.12. This quotient is then zero by Nakayama's lemma. The lemma is proved. 1 8 . 3 . FURTHER CONSEQUENCES OF THE GENERAL HYPOTHESIS

Lemma 18.3.1. Let y = F^F^ • • • Fitr) G A where i\ + + • • • + it = v and t = tr v < N. Let x = fatfa2 • • • fat(l <8> rj) G f A. Assume that A). Thenx = l®y mod v~l£(f) © L'(A). yiv~lL'{ We argue by induction on t. If t = 0, there is nothing to prove. Assume now that t > 0 and that the result is known for t — 1.

18. Study of the Operators Fi, Ei on A \

160

Let y' = Fi2 - - • Fitrj G A and let x' = • • • it(l © rj) G f © A. Since F ^ t T 1 ! / ^ ) ) C u~ 1 L / (A), and y <£ v~lL'{A), we have 2/ £ ^ / / ( A ) . By the induction hypothesis, we have x' = 1 © y' mod v~lC{f) © Z/(A). Applying t ^ and using 18.2.5, we deduce that x = 4>ixx' = <^(1 © y') mod v~lC{f) © L'(A). By our general hypothesis, we have y' — • • • <j>itl)~r) mod v~lL{A); hence, by 18.1.7(b), we have y' = b~r) mod v~1L(A)l/-il for some b G Bu-i1. Since y' £ v~lL'{A)u-i1, we have 6 G 5(A). Applying Lemma 18.2.6, we see that <j>ix(\ © y') = 1 © F ^ ' = 1 © 2/ mod v _ 1 £ ( f ) © Z/(A). (That lemma is applicable since F^y' = y £ v~1L'(A).) Hence x = 1 © y mod v~~lC(f) © L'(A). The lemma is proved. L e m m a 18.3.2. If tr v < AT, then Ei{L(A)u)

C Z/(A).

We will prove, for any n > 0, that Ei(L(A)v) C vnL'(A), by descending induction on n. This is obvious for n large since L(A)„ is a finitely generated A-module. Hence it is enough to prove the following statement. (a) Assume that n > 1 and Ei(L(A)„) C vnL'{A); then F»(L(A)„) C v L'(A). n_1

We first show that (b) e~i(£(fV © L(A)„») C v n £ ( f ) © Z/(A) provided that 1/ -+ v" = 1/. In the case where tr v" < AT, this follows from 18.2.5. Assume now that tr v" = N; then 1/ = 0. It suffices to show that €i( 1 © x) G v n £ ( f ) © Z/(A) for any a; G L{A)U. We write z = Ylr>0where xr = 0 unless r + (z,A — f ) > 0 and EiXr = 0. By the assumption of (a), we have £ r > i i^ ( r _ 1 ) rc r G vnL'{A) and by 18.2.2, we deduce that G vnL'(A) for r > 1, or equivalently, F[~1xr G vnL'(A) for r > 1. Using the general hypothesis (r — 1) times, we have E l ~ \ F r l x r ) C v n L'(A); hence xr G vnL'(A) for all r > 1. We have e/(l © ar) = J2r>i ® since e ^ l © x0) = 0. By 17.1.15, this -1 belongs to the Z[v ]-submodule generated by the elements ®F^xr with r > 1 and ri + 7*2 = r — 1 and these elements belong to v n £(f)©Z/(A). Thus (b) is proved. To prove (a), it suffices to show that Ei(y) G vn~1L'(A) for all y of the form y = FixFi2 ... Fitr) where ii + i2 H h it = v. If y G v - 1 Z/(A), then our inductive assumption shows that Ei(vy) G vnL'{A); hence Fj(?/) G w n_1 Z/(A), as desired. Thus we may assume that y £ v~1L'(A). Using now Lemma 18.3.1, we see that (c) 4>h4>i2 • • • 4>it (1 © 77) = (1 © y) + v_1z

18.3. Further Consequences of the General Hypothesis

161

where z G £ ( f ) © L'{A). From the definition of the operators fa on f © A we see that we have necessarily

Hence, by (b), we have ii(z) G vnC(f) 0 L'(A). Thus applying ii to (c), we obtain (d) iifa,fa2 • • • fat (1 © vi) = €i{ 1 © y) mod ^ " ^ ( f ) © L'( A). We have (using 18.2.7) hfaxfa2

• • • fat(l ® T]) = hfaxfa2 •••
C H(£(f),) c £ ( f ) © L'{A) C ^ - ^ ( f ) © Z/(A). Hence from (d) we deduce that (e) € i ( l ® y ) G i ; n - 1 £ ( f ) © L , ( A ) . We write y = X)r>o Fi^Vr, where the y r G (A)„_ r j satisfy = 0 for all r and yr — 0 unless r + (i, \ — v) > 0 . By our inductive assumption, we have Eiy = ]C r >i F^'^Vr G vnL(A). n Using 18.2.2, we deduce that yr G v L(A) for r > l7 We have c f (l © ?/) = 2 r > i e i ( l © F / r V ) , since 6^(1 ©2/0) = 0. By 17.1.15, we have for any r > 1, yr plus a linear combination with coefficients in v~ Z[v~ ] of terms © F$ V2) y r where n + r 2 = r - 1. The last linear n l combination is in v ~ C{i) © L'(A) since yr G vnL(A) for r > 1. Taking sum over r > 1 we obtain 1

1

c<(l ® 2/) = 1 ® Eiy

mod vn~lC{i)

Using this and (e), we deduce that 1 © Eiy G v pr, we obtain

n-1

© L'(A). £ ( f ) © L'(A). Applying

p r ( l © EiV) G v n " V ( £ ( f ) © L'(A)); hence Eiy G i> n-1 Z/(A). The lemma is proved. L e m m a 18.3.3. Let v be such that tr v < N, leti G / and let x G L'(A)U. Write x = X l r L o w h e r e the xr G (A)^-^ satisfy EiXr = 0 for all r and xT — 0 unless r + (i, A — v) > 0 (see 16.1.4)- Then xr G L'(A)„-ri for all r. When tr v < N, this is just Lemma 18.2.2(a). In the case where tr v = N, we can use the same proof since the inclusion ei(L(A)„) C L'(A) is now known for tr v = N by the previous lemma.

162

18. Study of the Operators Fi, Ei on A \

L e m m a 18.3.4. Let x G L'(A)U where tr v < N. ei(l ®x) = 1 ®EiX

We have

mod v~1C(f) © L'(A).

Using the previous lemma we can assume that x = F^x' and Eix' = 0. We can assume that x' ^ 0. Then L'(A)u-ri

where x' G

n = (i, A — v + ri') G N; we have mod v~lC{i)

= 0. By 17.1.15, c<( 1 © f\S)X') © L'{A). The lemma follows.

L e m m a 18.3.5. Assume that tr v < N.

=

1©

F^~1]x'

Then

Ci^flOL'tAWcAflOL'tA). When tr v < N this is shown in 18.2.5. When tr v < N, the same proof applies since Lemma 18.3.3 is now available. L e m m a 18.3.6. Assume that tr v = N. Let i be such that Vi > 0. Then Ei{b~rf) = (€ib)~rj mod v~1L'(A), for all b G Bu such that b~r] ^ 0. We can find i\, i2, •. • , iN with i\ + i2 + • • • + IN = V such that b = 4>ix4>i2 • • • 4>iNl

mod i> - 1 £(f)

(see 18.1.7(b)). Then y = Fi1Fi2---FiNr)eL(A)1/ satisfies b~rj = y mod v~1L'(A)t/ (see Lemma 18.2.10) and y ^ v~1L,(A). By 18.3.1, we have (j>ix(j)i2 • • • 4>iN( 1 ®7?) = 1©2/ = 1© b~r) up to elements tr U'ix$i2 • • •

(1 ® *?) = €i( 1 © b~rj)

mod v _ 1 £ ( f ) © Z/(A).

We have li4>iAi2 '"iN{ 1®T)) = Ct^ti^ta

•••0tJV(H(l))

= H{ei<j>ix(j>i2 • "4>iNl) = 1

m w )

modulo v~ C(f) © Z/(A) (using Lemma 18.2.7). Using Lemma 18.3.4, we have €i( 1 © b~rj) = 1 © Ei{b~iij) mod v _ 1 £ ( f ) © L'(A). We deduce that E(ei(b)) = 1 <&Ei(b~7]) mod v~lC{f) ©Z/(A). Apply pr to this congruence and use pr(E(x)) = x~r) for all x G f. We deduce that (6i(b))~r] = Ei(b~rj) mod v~1L'(A). The lemma is proved.

18.3. Further Consequences of the General Hypothesis

163

18.3.7. Prom the lemmas above, we see that, if we assume the general hypothesis 18.2.1 for N, then the properties (a),(b),(c) in 18.2.1 also hold when N is replaced by N 4- 1. Since they are obvious for N = 1, we see that we have proved by induction the following result. T h e o r e m 18.3.8. Let v G N[/]. We have (a) L(A)„ = L'{A)„; (b) for any i we have Fi(x~rf) = (fax)~r) mod v~lL(K),

for all x G

(c) if i is such that i > 0, then Ei(b~rj) — (eib)~rj mod v~1L(A), all b G Bu such that b~rj / 0; in particular, Ei(L(K)u) C Prom now on, we shall not distinguish between L(A) and L'(A).

for

CHAPTER

19

Inner Product on A

19.1.

FIRST PROPERTIES OF THE INNER P R O D U C T

In this chapter, we preserve the setup of the previous chapter. In particular, we write A = AA where A G X+ is fixed, except in subsections 19.1.1.

Let p : U —* \Jopp be the algebra isomorphism given by the composition

Proposition

19.1.2.

There is a unique bilinear form (,) : A x A —» Q ( ^ )

This bilinear form is symmetric. If x

G (A)^,?/ G ( A ) ^

with v ^ v', then

For any u G U, we consider the linear map of the dual space A* = Hom(A, Q(f)) into itself, given by £ »—• u(£) where u(£)(x) = £(p(u)x) for all x G A. This defines a U-module structure on A*, since p : U —» \Jopp is an algebra homomorphism. Let £o £ A* be the unique linear form such that £ 0 iv) — 1 a n d £o is zero on (A)^ for v ^ 0. It is clear that E^q = 0 for all t G / and = for all /x G Y. We show that 0 = 0. It is enough to show that, for any x in a weight space of A, the vector E\1'X^1X cannot be a non-zero multiple of 77. This follows from Lemma 5.1.6, since E^r] = 0 and — 0. From the description 18.1.1(a) of

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164

19.1. First Properties of the Inner Product

165

A, we now see that there is a unique homomorphism of U-modules A —• A* which takes r) to Now it is clear that there is a 1-1 correspondence between homomorphisms of U-modules A —• A* which take rj to £o and bilinear forms (,) on A which satisfy (a) and (b). The existence and uniqueness of the form x, y (x, y) follow. The form x,y >—> (y, x) satisfies the defining properties of the form x,y i—• (x,y), since p2 = 1. By the uniqueness, we see that these two forms coincide; hence (,) is symmetric. P r o p o s i t i o n 19.1.3. Let v G N[/]. (a) We have (L(A)„,L(A)„) € A. (b) For any i G / such that Vi > 0, and any x G L(A)u-i,x' have (Fix,xf) = (x,Eix') modulo v_1A.

G L(A)„ we

When tr v = 0, (a) and (b) are trivial. We may therefore assume that tr v = N > 0 and that both (a),(b) are already known for v' with and Ei(L(A)„) C L(A)„-i tr i/ < N. Since L(A)„ = Zi;i,t>0FiL(A)l,-i, whenever ^ > 0, we see that (a) for v follows from the induction hypothesis. (We use (a) and (b) for tr v' = N - 1.) We now prove (b) for v. Let i,x,x' be as in (b). By 18.2.2, we may assume that x = F^y and x' = F^y' where y G L(A) l / _ i _ s i , y' G L(A) I/ _ s / i , Ety = Eiy' = 0 and s > 0, > 0, s + (i, A - v + i') > 0, s' + (z, A — v) > 0 . We must show that (c) ( F f + 1 ) 3 , , . F f V ) = ( f f ' W - V )

mod f _ 1 A .

For any r, r' we have from the definitions (if V i f V ) = = v'

(y,vfK.riE^F?y) r - r' + (i, A - u + s'i')

This is zero unless r' > r. By symmetry, it is also zero unless r >r'. (Ft)y,Fr,y')

<(r'),/\

-

= 6r,r>vri = $r,r>vl

(i, A — i/ + s'i'] (y,K.riyf) r "(i,A - v + s'i')

M y

Thus

166

19. Inner Product on A

Hence (c) is equivalent to r _(a+l)((a+l)+
'2(s + l ) +

5+ 1

(d) -8

2 ( 5 + 1) +

,

(t,A-i/>'

5

(i, A

—

v)

M )

(y, y') mod v

1

A.

We may therefore assume that s + 1 = s'. Now y, y' are contained in and tr (v — i — si) < N; hence, by the induction hypothesis, L{K)u-i-si we have (?/, y') G A. We see that to prove (d) it suffices to prove -( l)((s + l) + (t,A-i/» '2(5 + l ) + V- s +

(i,A-i/>

5+ 1

_

(e)

v-s(s+2+(i,\-v)) "2(5 +

1) + (z, A — v )

5

mod

- i li). v~lZ[v~

From the inequality 5 + 1 + (t, A — v) > 0 , we deduce 2(5 + 1) + (i, \ — u)> 5 + 1 and 2(5 + 1) + (i, A - 1/) > 5. Since v~pq [p+q], G 1 + v^Zlv1] for > 0, <7 > 0, it follows that we have _( +l)(( a+ l) + (t,A-i/)) '2(5 + 1) + (i, A -!/>" G 1 + v~1Z[v~1] v.- a 5+ 1 and "2(5 + l ) +

(t,A-i/>'

ei +

v-iziv-1]

and (e) follows. The proposition is proved. Lemma

19.1.4.

Let

6,6'

G

5(A).

(a) Ifb' ^ ±b then (6-77,^-77) G

(b) We have (b~7],b~rj) G 1 + v

_1

v~*A.

A.

We may assume that 6, b' G Bu for some v. We argue by induction on N = tr v. The case where TV = 0 is trivial. Assume now that N > 1. We can find i G I such that i/j > 0 and b\ G Bu—i such that (f>ib\ = b and ub = b\ (both equalities are modulo i ; - 1 £ ( f ) ) . By 18.3.8, we have Ei(b~rj) = 6]"77 and Fi(b^rj) — b~r) (both equalities are modulo v - 1 L(A)). It follows that b^rj / 0. (From 77 = 0 we could deduce by applying Fi that b~rj G v~lL(A) which contradicts b G B(A).) Thus bi G B(A). Using again 18.3.8, we have Ei(b'~rf) = (e.ib')~r) mod v _ 1 L(A).

19.2. Normalization of Signs

167

Using the previous proposition, we have (c) V ) = ( F ^ v l b ' - r , ) = (^77, E ^ - f i ) ) = equalities modulo v _ 1 A . Assume first that b = b'. Then (eib')~r] = b^r) mod v~1L(A) and (c) becomes (b~7],b~r)) = (6^?7,6f?7) mod v~1L(A); by the induction hypothesis, we have (6^77,6^77) G 1 + v _ 1 A so that (6-77,6-77) G 1 + v - 1 A , as required. Assume next that b' ^ ±b. There are two cases: we have either hb' = 62 mod v _ 1 £ ( f ) for some 62 £ B or lib' G v - 1 £ ( f ) . If the second alternative occurs, then (6^77, (eib')~rj) G v~lA by 19.1.3(a); hence (b~7],b'~r]) G v _ 1 A , by (c). If the first alternative occurs, then 62 G B(X) (by the same argument as the one showing that 61 G B(A)) and we have b2 ^ ±61 (if we had b2 = ±61, then by applying fc we would deduce b' = ±b mod v~lC(i)\ hence b' = ±6). Prom (c) we have 77, b'~rj) = (6^77,6^77) mod v~l A and from the induction hypothesis we have (6^77, b2rf) G v~x A. It follows that (b~rj, b'~rf) G f _ 1 A . The lemma is proved. 1 9 . 2 . NORMALIZATION OF SIGNS

19.2.1. Let B„ be as in 14.4.2. From 17.3.7 and 18.1.7, we see that the following two conditions for an element x G f„ are equivalent: (a) xeBv

+

v~lC{f)v

(b) x = fcxfc2 • • • fctl mod v - 1 £ ( f ) l / , for some sequence i\, i2,... , it in I such that i\ + 12 H \~H = f . For the proof of Theorem 14.4.3, we shall need the following result. We regard f A as an f-module with Oi acting as fc (see 18.1.3.) Lemma 19.2.2. (a) Let b G B„ and b' G B„'. The vector fc(b b'~rj) is equal modulo v~1C(f) © L{A) to fc(b) (g> b'~t] or to b<8> Fi(b'~rj). (b) Let bo G The vector &o(l ® vi) is equal modulo v~1C(f) © L(A) to bi ® 77 for some b\ G BUl, b2 G B„2 with v\ + v2 = v. We prove (a). By 18.2.2, (which is now known to be valid unconditionally) there exists ro > 0 such that v[ > ro and y-n = mod v _ 1 L(A) where a;' G L(A)„>_ roi , Eix' = 0, x' ^ 0. By 16.2.7(b), there exists 7*1 > 0 such that 1^ > ri and 6 = mod v - 1 £ ( f ) where a; G £ ( f ) „ _ r i i , eiX = 0 , x ^ 0 . By 18.2.5 (which is now known to hold unconditionally), we have mod v - 1 £ ( f ) © L(A). fc(b ® b'-rj) = fc(iri)x ®

168

19. Inner Product on A

By 17.1.15, 4>i{(f)\ri)x © i ^ ( r ° V ) is equal modulo i r ^ f ) © L(A) to © F[°x' or to (jf^x © F / * 0 + V or equivalently to ®b'~r] or b © Fi{b'~rj). This proves (a). We prove (b). From 19.2.1, it follows that

4>r{1+1x

60 = 4>n4>i2

mod i> - 1 £(f),

for some sequence i \ , . . . ,it in I such that i\ +

H

1-it = v. We have

6 0 (1 0r?) = 6 0 E(1) = H(60) = E(4> h fa 2 • • • j> it l) = 4>ix4>i2 • • • <ME(i)) = j>h4>i2

4>it(i ©77).

The third equality is modulo v _ 1 £ ( f ) © L(A). It remains to show that 4>ilfa2 • * • © 7?) is equal to b\ © b^rj and 62 G B„ 2 with v\ + 1/2 = f . mod v~lC{f) © L(A) for some b\ G We show that this holds for any sequence ii, i2,..., it, by induction on t. The case where t = 0 is trivial. We assume that t > 1. Using the induction hypothesis and (a), we have that ilit(l ©77) is equal to <j)ilb®b'~~,q or to 6 © F i l ( 6 / ~ 7 7 ) modulo v~lC(i) ©L(A), for some b G B ^ b ' G B„ 2 such that 1/1+Z/2 = v—i\. We have ix6 = 61 mod u _ 1 £ ( f ) for some b\ G B„ 1 + i x and Fil(b,~rj) = (^b'^r) = b^rj mod v~lC(f) for some b2 £ Bi>a+*i> (b) follows. 19.2.3. Proof of T h e o r e m 14.4.3. The theorem is obvious when v = 0. Thus we may assume that tr v = TV > 0 and that the result is true when N is replaced by N' G [0, AT - 1]. We first prove part (b) of the theorem. Recall that a(Bu) — Bv. (See 14.2.5(c).) Assume that 6,6' G B„ satisfy a{b) = — 6'. We will show that this leads to a contradiction. We can find i G / , n > 0 such that ^ > n, and b" G B„_ n in#i;o such that b = 7Ti,n&". By 17.3.7, we have <^6" = b mod v~lC(f). Since b" G Bi;0, we can find fi G £ ( f ) such that £<(/?) = 0 and 6" = (3 mod v - 1 £ ( f ) . This is a special case of the equality Bpj = B(N) in 16.3.5(a). By definition, we have = Since ^ preserves v _ 1 £ ( f ) , we have ^(3 = (fib" mod v~lC{i). mod v~lC{i)\ hence b = For any j G J, we define Cj G N by b" G . Thus, we have Ci = 0. Since the root datum is assumed to be K-regular, we can find A G X such that («, A) = 0 and (7, A) > cj for all j G I — {2}. These inequalities show, using the definition of the Cj, that (r(bf,)rj ^ 0, where 77 G A = A* is as in 3.5.7.

19.2. Normalization of Signs

169

In the f-module f © A, we have o\n\l © 77) = d\n) © rj since Fi77 = 0 and K-ir] = 0 (recalling that (i, A) = 0). By definition of the f-module structure on f © A, we have ®v) = <^moln)

®n) =

0 ®

omv)+*

where z is in the kernel of the obvious projection prn : f©A b = 0\n)/3 mod i r ^ f ) , we have -b' = a(b) = a((3)9\n) hence (a)

-b'( 1 © 77) = 0?n) © a(/?)(r/) + z

f n »® A. Since mod v~xC(i)\

mod v " 1 ^ ) © L(A).

x(l © 77) G £ ( f ) © L(A); since x(l © 77) = We have used that x G £ ( f ) this follows from Lemma 18.2.7, which is now known unconditionally. By 19.2.2(b), we have b'{ 1 © 77) = 61 © b~r) mod v _ 1 £ ( f ) © L(A) where 61 G B ^ , G B ^ and v\ +1/2 = Comparing with (a), we deduce that E\ N ) © * ( / ? ) " (77) + bX © 62 77 + * G v " ^ ( f ) © L ( A ) .

Since (3 = b" mod v~lC(f), we have
if 62 G B v - n i and b^77

t T ^ A )

0 and CT(6,/)_77 G I T ^ A )

if b2 £ B„_ n i or 62 77 = 0. Both alternatives are impossible, since, by the induction hypothesis,
19. Inner Product on A

170

by part (b). Thus the first alternative holds. But then b\ = — cr(6') and this again contradicts part (b). This proves part (a). We prove part (c). Let b G B^. We have cr(6) G B„ U (—B^) and c7(6) ^ (—B„) by (b), hence a(b) G B„. This proves part (c). Clearly, parts (d) and (e) follow from part (a) since Bv is a signed basis of iu. The theorem is proved. 1 9 . 3 . FURTHER PROPERTIES OF THE INNER PRODUCT

19.3.1. We shall denote by .4 A a the image of the canonical map ^ f —> A\. The canonical basis B(AA) of AA (see 14.4.11) is clearly an ,4-basis of A^XFor any v G N[/], let (AA\)V be the image of J$ V under the canonical map f —AA- We have a direct sum decomposition A A = ©I/(^AA)I/. Proposition 1 9 . 3 . 2 . ^ A A is stable under the operators x~, x+ : A A —S• AA, for any x G a?• For this is obvious. To prove the assertion about x + , we may assume that x = 6^ for some i,n. Let y G (^AA)^. We show that E^y G by induction on JV = tr v. If N — 0, the result is obvious. Assume that N > 1. We may assume that y = F^y' where 1 < t < Vj and y' G ( ^ A A ) „ - T J . By 3.4.2(b), the operator E \ n ) o n A a is an ^-linear combination of operators F^ ^ E \ n \ By the induction hypothesis, we have y G ^AA; hence F^E^y' G ^A A SO that E\n)y = E^F^y' ^AA- This completes the proof. The following result is a strengthening of Lemma 19.1.4.

G

Proposition 1 9 . 3 . 3 . Let b,b' G B(A). (a) Ifb' / ±b then (b~rj,b'-r)) G

v-lZ[v~l).

(b) We have (6-77,6-77) G 1 + V ^ i T 1 ] . We shall prove by induction on tr v that (c) (x,y) G A for any x,y G (^AA)^. When tr v = 0, (c) is trivial. We may therefore assume that tr v — N > 0 and that (c) is already known for v' with tr v' < N. We may assume that x = F^x' where 0 < r < Vi and x' G (A^x)u-ri- From the definitions we have (d) ( f * V , y ) =

(x>,vfK-riE\r)y).

By 19.3.2, we have K - r i E ^ y G (.aAa)^-™; hence the right hand side of (d) is in A, by the induction hypothesis. This proves (c). The proposition follows by combining (c) with Lemma 19.1.4, since A f l . 4 = Z[t>-1].

19.3. Further Properties of the Inner Product

171

19.3.4. From the description 18.1.1(a) of AA, we see that there is a unique Q-linear isomorphism : A* —j• A* such that urj\ = ur)\ for all u G U. It has square equal to 1. Theorem 19.3.5. Let be A\.

We have b G B(AA) if and only if

(1) b G ^AA, 6 = 6 and

(2) there exists a sequence t'i, i2,..., mod v~1L(A\).

ip in I such that 6 = F^F^ • • • Fiprj\

We have 6 G ±B(AA) if and only if 6 satisfies (1) and v~lZ[v-1].

(3) (6,6) = 1 mod

If 6 G ±B(AA), then 6 obviously satisfies (1); it satisfies (3) by 19.3.3. Assume now that 6 satisfies (1) and (3). Since the canonical basis is almost orthonormal, from Lemma 14.2.2 it follows that there exists b' G B(A a ) such that 6 = ±6' mod v " 1 ! , ^ ) . Since 6 - (±6') = 6 - (±6'), it follows that 6 - (±6') = 0. Assume now that 6 G B(AA). We show that 6 satisfies (2). We have 6 = (3~rjA for some /3 G B. We can find a sequence ii,i2,•..,ip in I such that (3 = fcx4>i2 ••'ip 1 mod v - 1 £ ( f ) . Using 18.3.8, it follows that 6 = FhFi2 • • • Fiprj\ mod v~1L(Ax), as required. Finally, assume that 6 satisfies (1),(2). Let i\, i2,..., ip in I be as in (2). Using again 18.3.8, we see that 6 = • • • fcpl)~r]\ mod V _1L(AA). Let (3 be the unique element of B such that =

mod v - 1 £ ( f ) .

We have 6 = /3~rfx mod v~lL(Ax). Let b' = (3~r)x. Then 6 - b' G ^A a , _1 Y^V = 6 - 6' and 6 - b' G V L(A a ). It follows that 6 - b' = 0. Thus, 6 G B(AA). The theorem is proved. 19.3.6. We will now investigate the relation between the inner product (,) on f (see 1.2.5) and the inner product (,) on AA, which we now denote by (, )A since A G X+ will vary. Proposition 19.3.7. Let x,y G f. When A G X+ tends to oo (in the sense that (i, A) tends to oo for all i), then the inner product (x~r)\,y~r)\)\ G Q{v) converges in Q ( ( v - 1 ) ) to (x,y). We may assume that both x and y belong to f„ for some i/. We prove the proposition by induction on N = tr v. When N = 0, the result is

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19. Inner Product on A

trivial. Assume now that N > 1. We may assume that Vi > 0 and x = Oix' for some i and some x' G fu-i- We have (x~VXi 2/_77A)A = (Fix'~r)\,y~r)x)\ = (x , ~r)\,ViK-.iE i y~7)x)\. Using the commutation formula 3.1.6(b) and the equality EiT)\ — 0 we see that the last inner product is equal to

Using the induction hypothesis, we see that in the last expression, the converges to 0) and first term converges to 0 for A —* oo (note that 2 1 the second term converges to (1 — v~ )~ (x', ir(y)) which by 1.2.13(a), is equal to (x,y). The proposition is proved.

CHAPTER

20

Bases at oo

20.1.1. Let M be an object of C. We define a basis at oo of M to be a (a) a free A-submodule L of M such that M = Q(v) 0 A L — M and it is required that the properties (c)-(f) below are satisfied. (c) L is stable under the operators E^ Fi : M —> M for all i; thus, Ei, Fi

(e) we have L = ®LA where Lx — L fl Mx and b = UbA where b A = (f) given 6, b' E b and z E / , we have Eib =

if and only if Fib' = b.

The definition given above of bases at oo is due to Kashiwara who calls L e m m a 20.1.2. Let x E L fl Mx

and let i E / . Let t = (i, A).

Write

(b) I f x mod v~lL belongs to b, then there exists so such thatxs E v~lL for s so, xSo mod v~lL belongs to b and x = F^s°^xSo mod v~lL. We prove (a) by induction on N > 0 such that xs = 0 for s > N. For N = 0, the result is clear. Assume now that N > 1. We have EiX = where xs = 0 for s > N. By definition, we have E 8 -s>i-s+t>o

hypothesis, we have xs E L for all s > 1. Since L is stable under Fi and

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Media, LLC 2010 2011

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20. Bases at oo

174

Fj;s)xs = Ffxa, it follows that F$s)xs G L for all s > 1. Since x G L, we deduce that xq G L. This proves (a). We prove (b) by induction on N > 0 as above. If N = 0, there is nothing to prove. Assume that N > 1. If EiX G then v

J2s>-,s'>o-s>+t'>i FiS 1 e L and b y (a) we have vxs'+1 G ^ for all s' > for all s > 1. As before we then have F^xs G v-1L 0. Hence x8 G x then E{X mod v~lL for s > 1 and x = xq mod v~ L. If EiX £ belongs to b. By the induction hypothesis, there exists So > 1 such that F^So~^xSo xs G v~lL for s ^ so and s > 1. Therefore we have EiX = mod v~xL. Equivalently, we have EiX = F*°~1xSo mod v~lL. Applying Fi to this and using 20.1.1(f), we obtain x = F ^ x = F*°xSo = F^So)xSo mod v~lL. The lemma is proved. 20.1.3. In the next theorem we assume that the root datum is y-regular. Let A G X+. Let L be the A-submodule of AA generated by the canonical basis B(AA) and let b be the image of the canonical basis in L/v~1L. T h e o r e m 20.1.4. (L,b) is a basis at oo of A\. Property 20.1.1(c) follows from Theorem 18.3.8. We prove that property 20.1.1(d) is satisfied. Let be b. There exists (3 G B such that b is /3~rj\ mod v~lL. From Theorem 18.3.8 we see that Fib is <j>i(/3)~ri\ mod v~xL and Eib is €i((3)~r)\ mod v~lL or zero. By 17.3.7, we have i(/3) = (3' mod v~lC(F) for some (3' e B and this is necessarily in B. Then 4>i(/3)~r)\ = and (3'~rj\ mod v~lL (3'~r)\ mod v~lL so that Fib is (3'~r)\ mod is in b U { 0 } . By 17.3.7, we have either c<(/?) = /?" mod v _ 1 £ ( f ) for some (3" G B (which is necessarily in B) or ii((3) = 0 mod t> - 1 £(f). Then ei(/3)~r}\ = j3"~r)\ mod v~xL or ei(/3)~r]\ = 0 mod v~lL so that Eib is (3"~r)\ \ mod v~lL is in b u { 0 } . This proves property mod v~lL or 0. Now 20.1.1(d). Property 20.1.1(e) is clearly satisfied. We prove that property 20.1.1(f) is satisfied. Let b,b' G b. We have b = (3~r)\ mod v~lL and b' = (3'~r]\ mod v~lL where /?,/?' G B. By 18.3.8, we have Eib = b' if and only if ( E ^ ) " ^ = /?/-T?A mod v~xL. This is equivalent to the condition that (a) u(3 = (3' mod

v~lC(f).

Similarly, the condition that Fib' = b is equivalent to the condition that (b) j>i(3' = (3 mod ^ - ^ ( f ) . Now conditions (a) and (b) are equivalent by 17.3.7. The theorem is proved.

20.2. Basis at oo in a Tensor Product

175

2 0 . 2 . BASIS AT OO IN A TENSOR P R O D U C T

20.2.1. Let M, M ' G C. Assume that M and M' have finite dimensional weight spaces. Assume that (L, b) (resp. (L', b')) is a given basis at oo of M (resp. M'). Consider the tensor product M © M' G C. T h e o r e m 20.2.2. The free A-submodule L ©A L' of M ® M' and the Q define basis bob 'of (L ©A L')/v~l{L ©A L') = {L/v~lL) ©Q (L'/v^L') a basis at oo of M © M'. Only properties (c),(d),(f) in the definition 20.1.1 need to be verified. In verifying these properties, we shall fix i G / and write Ll for the sum ©LA over all A such that (i, A) = t. The notation Ln has a similar meaning. Let Gl be the set of all z G Ll such that z mod v~xL belongs to b and such that E{Z = 0. Let Gn be the set of all z' G Ln such that z' mod v~lL' belongs to b ' and such that Eiz' = 0. From the definitions, all elements of the form F^z (z G GtJs G [0,t]) belong to b modulo v~1L and according to 20.1.2, all elements of b are obtained in this way. Similarly, all elements of the form F^s>)z' {z' G Gn\ s' G [0, t'\) belong to b ' modulo v~lL' and all elements of b ' are obtained in this way. Using Nakayama's lemma, which is applicable since the weight spaces are assumed to be finite dimensional, we deduce that the elements F^z (z G G*,s G [0, t]) generate the A-module L; similarly, the elements F^s ^z' (z' G Gn ,s' G [0, t']) generate the A-module L'. Let zeG\z' G Gn',se [0,£],s' G [0,t']. According to 17.2.4, we have FiiF^z

© Fl s > ) z') = F l s ) z © F l 8 ' + 1 ) z '

mod v~l{L ©A L')

© F\s'\')

= F^s+1)z

mod v~l{L ©A L')

© F^z')

= Ft(s)z © F^~l)z'

mod v~\L

©a L')

© F^z')

= F^s~l)z

mod v~\L

©a L')

if 5 + s' < t'\ Fi{F\s)z

© F^z'

if s + s' > t'\ Ei{Fls)z if s +

< t'\ EiiF^z

© Ft(s,)z'

lists' > t'. It follows that Ei, Fi map a set of generators of the A-module L ©A L' into L ©A L'\ hence they map L ©A L' into L © A L'. This verifies property 20.1.1(c) of a basis at oo. Properties 20.1.1(e),(f) of a basis at oo are also clear from the previous formulas. The theorem is proved.

20. Bases at oo

176

20.2.3. Assume that z G G\z' G Gn' and that s G [0,t],s' G [0,£'] are such that t + t' = 2(5 + s'). By the formulas in 20.2.2, the condition that Fi(F^ s ) z G v~l(L ®A L') is that either s' = t' and s + s' < t', or s — t and s -f s' > t'. The first case cannot occur since s > 0. Hence the condition is that s = t and s + s' > t'. But if s = t then t' = s + 2s' hence s + s' > s + 2s' so that s' — 0. Thus the condition is s = t = t', s' = 0. We can reformulate this as follows. P r o p o s i t i o n 20.2.4. Let (M, L, b), (A/7, £/, b') be as above. Let b G b, b' G b'. Assume that b G b A and b' G b , v and (i, A) + (z,A;) = 0. Then the following two conditions are equivalent: (a) Fi(b®b')

= 0 in ( L ® A L')/v~l(L®A l

(b) Fi(b) = 0 in L/v~ L

and Ei{b') = 0 in

L'); L'/v^L'.

CHAPTER

21

Cartan D a t a of Finite Type

In this chapter we assume that the Cartan datum is of finite type; then the root datum is automatically F-regular and X-regular. Let A' = —wq(\). Then A' E X+ and we may consider the U-module "AA' as in 3.5.7. Since = Av as a vector space, the canonical basis B(AV) of Ay may be regarded as a basis of "Ay. 21.1.1.

P r o p o s i t i o n 21.1.2. There is a unique isomorphism A\ ^AA' such that x maps B ( A A ) onto B ( A A ' ) -

of U-modules x

By 6.3.4, AA' is a finite dimensional simple object of C'. Hence uAy is a finite dimensional simple object of C . By definition, its (—A')-weight space is one dimensional and the (—A' — i')-weight space is zero for any i. By Weyl group invariance (5.2.7), it follows that the A-weight space is one dimensional and the (A -f z')-weight space is zero for any i (we have A = wq{—\')). Let x be the unique element of B(AA') in this A-weight space. Then EiX = 0 for all i. By Lemma 3.5.8, there is a unique morphism (in C') X : AA —• "Ay which carries rj to x. Since x is a non-zero morphism between simple objects, it is an isomorphism. We can regard x as a n isomorphism of vector spaces AA = AA' such that x(uv) — w(u)x(y) f° r all u E U and y E A A and such that x(v) € B ( A A ' ) We have (a) x U A a ) C

AAy.

Indeed, let y E ^Aa- Then y = g~r] for some g E ^ f ; hence x(y) = g^xiv)- ^ remains to use the fact that is stable under g+ (see 19.3.2). We have (b) x(x) = x(x) for all

X

E AA.

Indeed, we can write x = urj with u E U. We have x(ut]) = u(u)x( rj) = u(u)x(v)

= v(u)xW

=

=

as required.

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Media, LLC 2010 2011

177

21. Cartan Data of Finite Type

178

We have (c) ( x , x ' ) =

for

x,x' G AA.

Indeed, if we set ((£,#')) = ( x W , x(x')) w e obtain a form satisfying the defining properties of (x,x'), hence equal to it. Let 6 G B(AA). Let b' = x(&)- Using (a),(b), we see that b' G and 6' = b'. Using (c) and the fact that (6,6) and (x(r/), x(v)) a r e i n 1 + v~1Z[v~1], we see that (6', 6') G 1 + Since b' G ^Aa', we have also (b',b') G A; hence (6 / ,6 / ) G 1 + v~lZ[v~1}. Using Theorem 19.3.5, it follows that ±6' G B(AV)- This argument also shows that, if (L, b), (Lr, b') are the bases at oo of AA, AA' defined in 20.1.4, then x(L) = L'. - Fitr) We can find a sequence i\, i2, •.., it in I such that 6 = F^F^ — EiX f° r It follows mod v~lL. From the definitions we have that that b' = Ei1Ei2 • • • Eitx(rj) mod v~lL', so that b' mod v~xV belongs to b'. Since ±6' G B(AA')> it follows that b' G B(AA')- The proposition is proved. I n particular, 21.1.3. We shall identify the U-modules A A and ^AA' via U the generator £ = £_A> of A\> (see 3.5.7) is now regarded as a vector in the wo(A)-weight space of AA, which belongs to the canonical basis of AA-

CHAPTER

22

Positivity of the Action of FuEi in the Simply-Laced Case

22.1.1. In this chapter, the root datum is assumed to be F-regular. We fix A G X + and we set A = A\. The main result of this chapter is Theorem 22.1.7, which asserts, in the simply laced case, that the matrices of the linear maps Ei and Fi of A into itself, with respect to the canonical basis of A, have as entries polynomials with integer, > 0 coefficients. Using Theorem 18.3.8, we see that Lemma 18.2.7 is true unconditionally.

22.1.3. In the following corollary we shall use the notation

C o r o l l a r y 22.1.4. Let b G B. Write r(b) = Ylhb-,bub2h ® b2 and f(b) = S ® b 2 where hb.bub2 G A and gb.but>2 G A; here bi,b2 run over B .

It suffices to prove the statement about gb;bl,b2. By 3.1.5, we have A(b~) = ®

*

the definition of

By the previous theorem, if b2 G B(A), the coefficient of bi^b^rj is in A. This coefficient is clearly in A; hence it is in Z[v~x]. The corollary follows.

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011

179

22. Positivity of Action of Fi,Ei

180

Corollary 22.1.5. Let i G I. Let b G B. Let us write ir(b) where b' runs over B and dbtoiti/yn are integers. Ylb'inezdbjit^nV71^

=

(a) Let b' G B(A) and let n G Z be such that db,ei,b'yn ^ 0. Then to(A-|6'|) + n > 0 . (b) We have vio(A-|6'|)+n

E

_

v-io(A-|6'|)-n

db,0i,b',n

Ti

&

f]

where b' runs over B(A). We apply the previous corollary to hb;bi,b2 with b\ = Of, we obtain 5><0(A-I*'I)+B<W.» e ZH n for any b' G B(A); (a) follows. We now prove (b). By 3.1.6(b), we have Ei{b-rj) =

- ^ ^ ^ - r i W ^

+ i;-|6|,<+i ,,t

since Fit) — 0. By 1.2.14, we have r*(&) = also that K±iT) — v f ^ ^ r j . Thus, V

Ei(b~v) = £ < W , » -

-i

^

»r(6), since b = b. Note

*

b'~r,.

Using now |6| = |6'| + w e obtain (b). 22.1.6. Let b G B(A). For any 2 G / , we set Fib = J2b',nfa,b>,i,nVnb',E{b = S b ' n fb,b',i,nVnb' where b' runs over B(A) and n runs over Z; the coefficients fb,b' ,i,ni fb,b',i,n are integers. T h e o r e m 22.1.7. Assume that the Cartan datum is simply laced. Then fb,b',i,n € N and fb,b',i,n € N for any b,b',i,n. If (3, (3' G B are such that /3r) = b,/3'rj = &', then with the notation of Theorem 14.4.13, we have fb,b',i,n = C$i,(3,(3' ,n

22.1. Action of Fi,Ei in the Simply-Laced Case

181

and J 2 fb,b',i,nVn

n

=

n

o (A -

|6'|) +

n]dbfiuv,«

(we have used Corollary 22.1.5(b) and the equality Vi = v). By Theorem 14.4.13, the integers are > 0. Hence fb,b',i,n € N. Again by Theorem 14.4.13, the integers db,0i,b',n are > 0 and, by Corollary 22.1.15(a), we have io(\ — \b'\)+n > 0 for any n such that db,di,b',n / 0. Since [N] is a sum of powers of v if N > 0, we deduce that fb,b',i,n € N. The theorem is proved.

N o t e s on Part III 1. Most results in Part III are due to Kashiwara [2]. An exception is Theorem 22.1.7, which is new. 2. Although Theorem 22.1.2 does not appear explicitly in Kashiwara's papers, it is close to results which do appear; the same applies to the results in 17.1. The proofs in 17.2 are quite different from Kashiwara's. 3. The proof in 19.2.3 is an adaptation of arguments in [3]. REFERENCES 1. M. Kashiwara, Crystallizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 1 3 3 (1990), 249-260. 2. , On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 6 3 (1991), 465-516. 3. , Crystal base and Littelman's refined Demazure character formula, Duke Math. J. 7 1 (1993), 839-858. 4. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-^98.

Part IV C A N O N I C A L B A S I S OF U

The algebra U is a modified form of U in which U ° is replaced by the direct sum of infinitely many one-dimensional algebras, one for each element of the lattice X of weights. This is an algebra without unit; it has instead many orthogonal idempotents by means of which one can approximate the unit element. The U-modules in the category C can be regarded naturally as modules for U; on the other hand, more exotic U-modules, like those without weight decomposition, cannot be regarded as U-modules. Thus, U is an algebra more appropriate than U for the study of objects of C. One of its main virtues is that it has a canonical basis B with remarkable properties. In Chapter 23, we define and study the algebra U and its .A-form ^ U . In Chapter 24, we prove an integrality property for the quasi-7£-matrix and we use it to define a canonical basis in any tensor product A v , which is fixed by an antilinear involution obtained by composing the obvious involution ~~ with the action of the quasi-7£-matrix. In Chapter 25, we show that these bases have some rather nice stability properties with respect to certain transition maps (see 25.1.5). The proof is based on results in Part III. From this we deduce (Theorem 25.2.1) that these bases can be "glued together" to form a single basis B of U . In Theorem 25.2.5, we show that B is simultaneously compatible with many subspaces of U and in Proposition 25.2.6, we show that B is a generalization of B. In Chapter 26 we show that B is characterized (up to signs) by an almost orthonormality property with respect to an inner product. This is in the same spirit as Kashiwara's idea of characterizing B by an almost orthonormality property (see Notes on Part II). As an application, we show that B is stable (up to signs) under a and u T h e same should be true without signs.

184

Part IV. Canonical Basis of U

In Chapter 27 we assume t h a t the Cartan d a t u m is of finite type; we show t h a t a tensor product of form 0 Aa 2 • • A A „ has a canonical basis and t h a t this induces a canonical basis on the space of coinvariants. This last basis has an invariance property with respect to cyclic permutations, as shown in Chapter 28. In Chapter 29 we prove a refinement of the Peter-Weyl theorem (in U instead of the coordinate algebra) and we discuss an extension of the theory of cells in Weyl groups to the case of U . In Chapter 30 we define a canonical topological basis of (a completion of) U ~ <8> U + (in finite type) and show t h a t this basis gives rise simultaneously to the canonical bases of the various tensor products A\ "Ay.

CHAPTER

23

The Algebra U

23.1.

D E F I N I T I O N AND F I R S T P R O P E R T I E S OF

U

2 3 . 1 . 1 . The objects of C may be regarded as modules over a certain algebra U, in general without 1, which is a modified form of U. We prepare

Consider the direct sum decomposition U = J(i/)> where v runs G through Z[J], defined by the conditions U ( I / ) U ( I / " ) C U ( I / ' + U(0), Ei G U(«), Fi G U(—i) for all */,*/' G Z[J], i G I, /j, G Y. We have There is a natural associative Q(t;)-algebra structure on U inherited from that of U. It is defined by the following requirement: for any A^, A'/, \' 2 , X2 and any t <E U(Ai -• AJ), 5 e^U(A'2 - A'2'), the product is

The algebra U does not generally have 1, since the infinite sum Y^xex does not belong to U (if X ^ 0); however the family (Ia)agx is in some

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011

185

23. The Algebra U

186

23.1.2. A finer d e c o m p o s i t i o n of U . We shall need a direct sum decomposition of U which is slightly different from the one in 23.1.1(b). The images of the summands U(i/) under ir\>t\» form the direct sum decomposition A'UA" = ©^(A'UA"^))- We set U(I/) = 0A',A"(A'UA"(^)). We then have direct sums decompositions U = ©„U(i/) and U = © v , a « i / ( 1 a ' U M 1 a » ) where A', A" run through X and v runs through Z[I]. Actually, the summand 1A'U(^)1A" = A'UA"(^) is zero, unless A' — A" = v in X. 23.1.3. U - b i m o d u l e s t r u c t u r e . Note that U is naturally a U-bimodule by the following rule: if if G U ( i / ) , s G U , i " G U(i/"), we have by definition, t'-KX>,\"{s)t" = irx>+u>,\"-v"{t'st") for all A', X" G X. For a l i i G / and A G X , we have the following identities in U: = l\+i>Ei, (EiFj-FjEi)lx

Fil\ = =

lx-i'Fi;

6itj[{i,\)]ilx;

and more generally, E^ U =

F^U

Ei°)F?hx = F V E f h

t> 0

t>o

=

x

U-ai'F^-,

if

a + b — (i, A) F> t

'1_A+(a+b-t)i'bi

a + b — (i, A) t

ip(a-t). I

1A-(a+b-t)i'^i

These identities follow from 3.1.9. 23.1.4. U n i t a l m o d u l e s . A U-module M is said to be unital if (a) for any ra G M we have 1 Am = 0 for all but finitely many A G X; (b) for any ra E M we have J^Aex

=

m

-

If M is a unital U-module, then we regard it as an object of C as follows. The decomposition M = ®M A is given by Mx = 1AM; the action of u G U is given by um = (ulx)m for any A G X and any ra G M A . Here ulx is regarded as an element of U as in 23.1.3. In this way, we see that to give a unital U-module is the same as to give an object of C.

23.1. Definition and First Properties of U

187

23.1.5. The comultiplication of U induces a similar structure on U. More precisely, for any A'1? A2, X'{, X2 G X, there is a well-defined linear map 1

Ai+A^Uy^// -> (a^UA^) (a^UA^)

such that AA^A'/.A^A-C^Ai+A^Ai'+A^W) = ( ^ . A ' / 0 7Ty2>y^)(A(x)) for all x G U. This collection of maps is called the comultiplication of U, and may be regarded as a single linear map from U to the direct product

AjjAj

II

(aiUA;')®(ASUa»).

23.1.6. The algebra automorphism u : U —> U induces, for each A7, A", a linear isomorphism A'UA" —• _A'U_A". Taking direct sums, we obtain an algebra automorphism u : U —• U such that u;(1a) = 1 - a for all A and u(uxx'u') = u(u)u(x)u(x')uj(u') for all u,u' G U and x, x' G U. It has square 1. The map a : U —• U induces, for each A', A", a linear isomorphism A'UA" —• _A"U_A'. Taking direct sums, we obtain a linear isomorphism a : U —• U such that -A"U_A'. Taking direct sums, we obtain a linear isomorphism S : U —• U (resp. S' : U —» U) such that S'(IA) = 1-A S(u')S(x')S(x)S(u) (resp. S'(1 A ) = 1-A) for all A e l , and S{uxx'u') = (resp. S'(uxx'u') = S'(u')S'{x')S'(x)S'(u)) for all u,u' G U and x,x' G U. These maps are related to o as follows. L e m m a 23.1.7. Let x G l A 'U(i/)l A ". We have S(x) = ( - 1 ) t r (">v p a(x) where v G Z[I] is such that u = X' - X" in and S'(x) = ( - 1 ) t r ^v~Pa(x), i/<(» • i/2) G Z. X and p= i/<(i, X' + A")(i • ij4) This follows easily from the definitions. 23.1.8. The Q-algebra homomorphism : U —> U induces, for any A, A' G X, a Q-linear map ~ : AUA' —» AUA'- Taking the direct sum of these maps, we obtain a Q-linear map : U —> U with square 1, which respects the multiplication of U, maps each IA into itself and satisfies tst' = tst' for M ' e U and s G U.

23. The Algebra U

188

2 3 . 2 . TRIANGULAR DECOMPOSITION, ,4-FORM FOR U

23.2.1. The U-bimodule structure on U gives us a f f o p p -module structure (x x') : u i—• x+ux'~ on U. Similarly, the U-bimodule structure on on U. From U gives us a f 0 f o p p -module structure (x x') : u h-> X~UX,+ the triangular decomposition for U, we see that U is a free f f opp -module with basis ( l A ) A e x for either of the two f f o p p -module structures. In other words, (a) the elements b+l\b'~ vector space U;

(b,b' G B,A G X) form a basis of the Q(v)-

(b) the elements b~l\b'+ vector space U.

(b,bf G B, A G X) form a basis of the Q(u)-

This is called the triangular decomposition

of U.

L e m m a 23.2.2. (a) The A-submodule of U spanned by the elements x+\\x'~ (with x,x' G a?) coincides with the A-submodule of U spanned by the elements x~l\x,+ (with x,x' G a W e denote it by ^ U . (b) The elements in 23.2.1(a) form an A-basis o / ^ U . The elements in 23.2.1 (b) form an A-basis 0/.4U. (c) .4U is an A-subalgebra of U . This algebra is generated by the elements for various i e I, n> 0 and A G X. (a) follows immediately from the commutation formulas 23.1.3. (b) follows from (a) and the fact that B is an .4-basis of ^ f . (c) follows from the commutation formulas 23.1.3. 23.2.3. From the integrality properties of the maps r, f , it follows easily that the maps AA^A'/.A^A? : l V i + A ^ U l A / ' + A - -> (1A>U1A>') ( l ^ U l ^ ) restrict to ^4-linear maps lAi+AiUU)lAi'+Ai' -

(lAiUU)lAi') <8U (lAiUU)lAi').

23.2.4. From the definitions we see that : U —• U leaves stable the ,4-subalgebra ^ U of U. The same holds for uj and cr. 23.2.5. Let {y' ,X',...) be the simply connected root datum of type (/, •) and let / : Y' —»• Y, g : X —> X' be the unique morphism of root data. We have an induced algebra homomorphism 0 : U ' —> U between the corresponding Drinfeld-Jimbo algebras (see 3.1.2). Assume that we are given ( e l ; let <' - g(Q G X'.

23.3. U and Tensor Products

189

Now 4> maps the left ideal U'(K^ of U ' into the left ideal X ^ e v ~ v ^ ' ^ ) of U; hence it induces a linear map on the quotients: Ulc. j>: U ' l c , Prom the triangular decomposition in U ' and U, it follows that the linear map (j) is an isomorphism. Using the definitions, we see that 0 restricts to an isomorphism of .4-modules Aj>:AV'lc*
2 3 . 3 . U AND T E N S O R PRODUCTS

23.3.1. Let A, A' G X. The U-module UMX <8> Mx> = f f belongs to C\ hence it is naturally a unital U-module. From 23.2.1, we see that the elements b+b'~lx» with b,b' G B and A" G X form a Q(v)-basis of U. Such an element (with X" = X' — X) acts o n l < g > l G f < g > f a s follows: (a) (6 + 6 /_ 1A'-A)(1 <8> 1) = 6+( 1 ®b') = b®b' + E ^ & i b'x where c ^ ^ G Q(t>), and the sum is taken over elements b\, b[ in B such that tr |&i| < tr |6|, tr \b[\ < tr |6'| and b[ belongs to the U-submodule of My = f generated by b'. Similarly, the elements b'~b+lx» with 6,6' G B and X" G X form a Q(t>)-basis of U. Such an element (with X" = X' - X) acts o n l < g > l e f ® f as follows: (b) (6 , -6+1 v _A)(1 ® 1) = 6 7 "(6 0 1) = 6 0 b' + £ c'bi b, 6X ® b[ where c'bi b, G Q ( f ) , and the sum is taken over elements 6i,6j in B such that tr |6I| < tr |6|, tr \b[\ < tr |6'| and 6i belongs to the U-submodule of wMA = f generated by 6. Using either (a) or (b), we see that (c) the Q(v)-linear map 7r' : U 1 A ' - A —'- U,MX®MX> given by u is an isomorphism.

U(

1(g) 1)

23.3.2. For any A G X+, we define a ^U-submodule .4 MA of the Verma module Mx as follows. As an ,4-module, it is the .A-submodule Af of f = Mx. This submodule is obviously stable under the action of the operators F ^ G U on MA; using repeatedly the commutation formulas 23.1.3, we see that it is also stable under the operators E\n>> G U on Mx; hence it is

190

23. The Algebra U

a ^U-submodule of M\. The same ,4-submodule is an ^U-submodule of "Mx, denoted The argument used to prove 23.3.1(c) can be repeated word for word for A instead of Q(u) and gives the following result. (a) Given A, A' € X, the map u H> u(l 0 1) defines an ^-linear isomorphism 7r7 of ^U1A'-A onto the ,4-submodule %Mx®a(aMy) oi"Mx®My. 23.3.3. Let <" e X and let a = that (i, C) — a[ — ai for all i € I. Let P(C,fl, a') be the left ideal U. Let A P ( C a , a ' ) be the left ideal of A i f .

G N[/],a' = J ^ a J i € N[J] be such n>a, n>a,

UF/n)lc +

n>ai

V E < n h c of

l C + Ei,n>a<

23.3.4. Let (Y',X',...) be the simply connected root datum of type (I,-) and let / : Y' —> Y, g : X —• X' be the unique morphism of root data. Let U' be the algebra defined like U, in terms of (Y'y X',...). Let £ € X, a, a' be as in 23.3.3, and let C' = g(0The natural isomorphism U'l^/ —> Ul^ (see 23.2.5) carries, for any i and any n > 0, the subspace XJ'F^lp isomorphically onto UF/ n ^l£ and the subspace XJ'E^l^* isomorphically onto U E ^ l ^ . Hence it carries the left ideal P(£',a, a') isomorphically onto the left ideal P(C,a, a'). The same argument shows that the isomorphism ^U'l^/ —> ^ U l ^ (see 23.2.5) carries the left ideal ^PfCjOja') isomorphically onto the left ideal 23.3.5. In the remainder of this section we assume that the root datum (y, X,...) is y-regular. Let A, A' € X+. We set (i, A) = a», (i, A') = a\ for all i and £ = A' — A. Let a = aii, a' = a[i. Let T (resp. T ' ) be the kernel of the canonical homomorphism of Umodules f = "Mx -> "Ax (resp. f = My Ay). By 14.4.11, T (resp. T ) is the subspace of f generated by a subset of the basis B, namely by D = Ui,n;n>aia ; n ). Taking tensor products, we obtain a surjective homomorphism of Umodules, or U-modules X'"Mx®

My -> "Ax <8> AA'

and we see that (a) the kernel of x IS the subspace T My + MA <S> T' of f ® f.

28.3. U and Tensor Products

191

Using 23.2.1(a),(b), and the description of T ' , T given above, we see that 7r' (see 23.3.1) maps (b) the subspace J2b£B,b'er>' Q(v)b+b'~l{ f

= ]Ci,n>(i,A')

v

of Ul<; onto the subspace YlbeB:b ei>' Q( )b ® b' = M\ T' of f ® f and (c) the subspace £ 6 G D ; b , e B Q(v)9/-b+lC

= Zi,n>(i,X)

of U l c onto the subspace £ 6 e D ; 6 / e B Q ( v ) b ®b' — T My of f f. Combining (a),(b),(c), we see that 7r' maps the subspace P ( £ , a , a ' ) of U l ^ onto the kernel of This fact, together with 23.3.1(c), implies the following result. P r o p o s i t i o n 23.3.6. In the setup above, the assignment u •—• U(£_A®T?A') defines a surjective linear map 7r:Ul^—» wAA <S> Ky with kernel equal to P(C,A,A').

23.3.7. Assume that A G X+. As in 19.3.1, we denote by the Asubmodule of A\ generated by its canonical basis. Using 19.3.2, we see that is an ^U-submodule of A\. The same .A-submodule is an ^ U submodule of ^AA, denoted by ^AA. From Theorem 14.3.2(b), we see that the kernel of the the .A-linear map ^ f —> .4 A a given by x »-> x~rj\ is the left ideal XT of ^ f generated by the elements for various i G / and n > (i, A). We can now repeat word for word the proof of Proposition 23.3.6 for A instead of Q(v) and we obtain the following result, where A, A' G X+. P r o p o s i t i o n 23.3.8. The assignment u u(€-\®r]y) defines a surjective A-linear map tt of aUl^ onto the A-submodule ^A\®a( A Ay) of"A\) is an ^U-submodule of the U-module "Ax®Ay. P r o p o s i t i o n 23.3.10. Let M G C' and let m G There exist A, A' G X+ such that X' — A = £ and a morphism f : u A\ Ay —> M in C' such that /K_A = mSince M is integrable, we can find integers ai, a[ G N such that E\a) m = 0 for all i and all a > ai and ^m = 0 for all i and all a' > a'{. Since the root datum is Y-regular, we can find X € X such that (i, A) > ai and (i, A + C) > Q-i for all i. Let A' = A + Then (i, X') > a'{ for all i. The existence of / follows now from 23.3.6.

CHAPTER

24

Canonical Bases in Certain Tensor Products 24.1.

INTEGRALITY PROPERTIES OF THE

QUASI-7£-MATRIX

24.1.1. Let M , M ' € C be such that either UM £ Chi or M' e Chi (see 3.4.7). We regard M ® Mf naturally as a U 0 U-module and we define M ® M' by 0 ( m m!) = ^ ©i/(m mf) a linear map © : M ® Mf (notation of 4.1.2.) This is well-defined since only finitely many terms of L e m m a 24.1.2. Let M, M ' be as above. We have (b) Assume that we are given Q-linear maps : M —> M and ~ : M ' —> M' such that urn — urn and um! — urn' for all u € U, m E M, m! G Mf. Let - = : M ® M ' -> M®M'. Then A ( u ) B ( m 0 m / ) = ©(A(t2)(m ® ra')) The set of u for which (a) holds is clearly a subalgebra of U containing all K^. Hence it suffices to check (a) in the special case where u is one of the algebra generators Ei, Fi. Applying both sides of the equalities 4.2.5(c),(d)

(Ei ® 1 + Ki ® Ei)Q(m ® m') = ©(£; 1 +

® ^ ) ( m ® ra')

This proves (a). Now (b) is a consequence of (a). The lemma follows. 24.1.3. The following property is just a reformulation of the property in Lemma 24.1.2(b). Let M, M ' be as in that lemma. Then for any ra 6

(a)

uQ(m ® ra') = Q(u(m ® ra'))

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010

192

24-1. Integrality Properties of the Quasi-H-Matrix

193

P r o p o s i t i o n 24.1.4. Let A, A' G X. Consider the Verma modules M\, Note that M = " Mx G C and M' = My G Chi; hence

My.

6 : M M ' -> M O M ' is well-defined. Then

6

leaves stable the A-submodule

A

M \

<8.4

(AM\>).

Since the ambient space of M and M' is f, we may regard ~ : f —• f as maps : M —• M, ~ : M' —• M'. Using the definition of Verma modules, we may identify M' = U / UEi + - v<"' V >l) as a U-module so that ~ : M' —> M' is induced by ~ : U —> U. It follows that um' = urn' for all u G U and m! G M'. Similarly, we have TZm = ufh for all u G U and m G M. As in Lemma 24.1.2, we set = ~ 0 ~ : M M ' and we have u 6 ( m <8> m') = Q(u(m m')) for all u G U, m G M, m' G M ' . In particular, taking m = l = l , r a ' = l = 1, we obtain (a) u(l ® 1) = 6(12(1 <8)1)) for all u G U , Then since 1 = 1,1 = 1, and 6 ( 1 1) = 1 1. Let x G ^ M A <8>A {AM\>). x = x' where x' G {AMy), since the involution ~ ~ : f —> f leaves ^ f ®A (A?) stable. With the notation of 23.3.2(a), we have x' = ir'(u') for some u' G .4Uly-A- Since ^U1A'-A is stable under the involution ~ : U —> U, we have u' = u for some u G ^U1A'-A- Hence x = x' = u( 1 1). Using (a), we have 9(x) = 6(12(1 <8> 1)) = u(l<8> 1) = TT'(U). Using again 23.3.2(a), T H E ®A U M X ' ) we have TT'(U) G ^ M a <8u U A f y ) ; hence G(x) G proposition is proved. C o r o l l a r y 24.1.5. Assume that the root datum is Y-regular. Let A, A' G X+. The map 6 : "A\ AA' —*• "Ax ® AA' leaves stable the A-submodule ®A

(AAy).

This follows immediately from the previous proposition since ^AA (^AAO is the image of ^ M A O ^ ( ^ M A ' ) under the natural map U w

A a ® Av.

C o r o l l a r y 24.1.6. Assume that ( / , •) is of finite type. Write e

=

E

E

Pwb~®b'+

(p^eQW).

M \ ® M Y

<8>A

—>

194

24- Canonical Bases in Certain Tensor Products

For any v° and any b°, b'° G B^o, we have Pbo,b'° € A. such that b° G B(-tu 0 (A)), b'° G B ^ o t A ' ) ) - By We can find A, A' G "AA ® Aa> maps the .A-submodule ®a UAA') 24.1.5, O : "AA ® AA' into itself. By 21.1.2, we may canonically identify as U-modules ^AA with A_u,0(A) and AA' with u;A_u,0(A') respecting the canonical bases. It follows A—woix)^ maps the .4-submodule that 6 : A_u;o(A)0U;A_u;o(A') ^A_U7o(A) <8>a (J4A_Wo(A')) into itself. In particular, this submodule contains the vector S(rj f ) = J2vYlb,b> Pb,b'b~rj ® b'+£. Here, rj =• r}_WoW and and b' runs over B ^ f l B ^ u ; o ( A ' ) ) . £ = fu>o(A'); b runs over BunB(-w0(X)) Since the elements b~r)<S> b'+£ (for all indices (f, 6, &') as in the sum) are a 6°, b'°) is an index part of an .A-basis of a^-w0(\) ®a (^A_ u;o (A')), and The corollary is proved. in the sum, it follows that Pb°,b'° € 2 4 . 2 . A LEMMA ON SYSTEMS OF SEMI-LINEAR EQUATIONS

L e m m a 24.2.1. Let H be a set with a partial order < such that for any h < h' in H, the set {h"\h < h" < h'} is finite. Assume that for each h < h' in H we are given an element rh,h' € A such that (a) rh,h = 1 for all h; (b) ^2h"-,hh,h' for all h
in H.

Then there is a unique family of elements Ph,h' £ h < h! in H such that

defined for all

(c) ph^h = 1 for all h G H; (d) Ph,h' ev~1Z[v~1]

for all h < h' in H;

(e) Ph,h' = Yjh"',hPh,h"rh",h' for

al1

h
in H.

For h < h' in i f , we denote by d(h, h') the maximum length of a chain h = ho < hi < h2 < • • • < hp = h! in H. Note that d(h, h') < oo by our assumption. For any n > 0, we consider the statement Pn which is the assertion of the lemma restricted to elements h < h! such that d(/i, h') < n. Note that property (e) is meaningful for this statement. We prove Pn by induction on n. The case n = 0 is trivial. Assume now that n > 1. Let h < h'. If d(h,hf) < n, then ph,h' is defined by Pn-\. If d(h,h') = n, we note that q = Y^h" h
24-3. The Canonical Basis ofuA\ Ay Indeed, using Pn-\ Q + Q=

=

and (a),(b), we have

h",h
Ph,h"fh",h' +

h1,h
PhMrhi,h"Th",h'

—

195

PhMrhuh'

+

PhM^uh'Th" ,h'

^ PhMrhuh' = 0» Ph,hirhi,h"Th",h' = h"M ;h

as required. Since g G ^ satisfies q + q = 0, there is a unique element q' G v~1Z[v~1] such that q' — q[ — q. We set ph,h' = q'> Then properties (c),(d),(e) are clearly satisfied as far as Pn is concerned. This proves the existence in Pn. The previous proof also shows uniqueness. The lemma is proved. 2 4 . 3 . T H E CANONICAL BASIS OF "A\

<s> Ay

In this section we assume that the root datum is F-regular. Let A, A7 G X+. We shall consider the following partial order on the set B x B: we say that (61,6'J < (62,62) if tr |6i| - tr |6;| = tr |62| - tr |6'2| and if we have either 24.3.1.

tr |6i| < tr 1621 and tr |6i| < tr |6'2|, or 61 — 62 and

= br2.

This induces, for given A, A' G X + , a partial order on the set B(A) x B(A'). As in 19.3.4, let ~ : Ay —> Ay be the unique Q-linear involution such that ufjy = ur)\' for all u G U; similarly, let ~~ : UA\ —• w Aa be the unique Q-linear involution such that = w£-a for all u G U. Let U ~ = ~ <8> ~ : wAA (8> Ay A\ Aa>. The elements 6+£_A ® b'~r)y with 6 G B(A) and b' G B(A') form a Q(v)-basis of UA\ Ay. They generate a Z[V -1 ]-submodule C — £A,A' and an ,4-submodule w Now © : W A A Ay A a ® Aa> is well-defined (see 24.1.1). w A a <8> Ay be given by = G(x). Since B Let ^ : Aa <8> Aa' stable (see 24.1.5), we have and ~ : " A \ AA' —• ^AA ® AA' leave = 1. We clearly ^ U £ ) C AC. Prom 24.1.2 and 4.1.3, it follows that

24.3.2.

w

24- Canonical Bases in Certain Tensor Products

196

have fx) = f^(x) for all / G A and all x. Prom the definition we have for all bx G B(A),6i G B(A'): ® b^-rjy) =

J2

PbuK-MbU-x

® b^rjy

6 2 GB(A),6^EB(A')

where pblyb>l]b2,b'2 G A, and pbl,b'l]b2,b'2 = 0 unless (bi,b\) > (b2,b2); hence the last sum is finite. Note also that pbl ^ ;bl ^ = 1 and Y Pbi ,b[ ;b2,b'2Pb2,b'2,b3,b'3 = b2€B(\),b'2eB(X)

,63^ »

for any 61, 63 G B(A), 6'1? 63 G B(A'); the last condition follows from \I>2 = 1. Applying Lemma 24.2.1 to the set H = B(A) x B(A'), we see that there is a unique family of elements nbl,b'1-,b2,v2 G Z[v _1 ] defined for 61,62 G B(A),6/1,62 G B(A') such that TTfei ,bi ;6x ,6'x = 1; "b^^b'

e t r ^ t r 1 ] if (biX) =

0

(b2,b'2);

unless (bub[) > (b2,b'2);

Kbub'i-tete =

,63 ^bi ,b[ ;b3 ,b'3 Pb$ ,b'3 ;b2 ,b2 for all (&l,&i) >

(b2,b'2).

We have the following result. T h e o r e m 24.3.3. (a) For any (bi,b[) G B(A) x B(A'), there is a unique element (&iO&i)A,A' € C such that and ( & I O & I ) A , A ' - b+Z-x ® b^rjy G v~lC. A ® b'^rjx' plus a (b) The element {b\§bi)x,x' in (a) is equal to linear combination of elements b2^-x®b2~r}x' with (b2, b2) G B(A) x B(A'), (62,^) < (61, b[) and with coefficients in v~1Z[v~1]. *((&IO&'I)A,A') -

(&IO&;)A,A'

(c) The elements (6i<^6J)a,a' with 61, b[ as above form a Q(v)-basis of "Ax <8> Aa', an A-basis of and a Z[v~1]-basis of C. (d) The natural homomorphism £fl\l>(£) —> C/v~lC

is an isomorphism.

The element (6IO&'I)A,A' = E b ^ ^ . & ' ^ ^ J f - A ® b'2~r)x> (see 24.3.2) satisfies the requirements of (a). This shows existence in (a). It is also clear that the elements (&iO&i)A,A' just defined satisfy the requirements of (b),(c),(d). It remains to show the uniqueness in (a). It is enough to show that an element x G such that x = x, is necessarily 0. But this follows from (d).

24.3. The Canonical Basis ofuA\

24.3.4. The basis W

(g> Ay

197

24.3.3(c) is called the canonical basis of

AA<8>AV.

24.3.5. Let A, A G X+. Let (Yf ,X',...) be the simply connected root datum and let / : Y' —> Y, g : X —• X ' be the unique morphism of root data. Let U ' be the algebra defined like U, in terms of ( y , X 7 , . . . ) . Let A', A' G be defined by A' = g(A), A' = g(A). Then w A a ® A x , defined in terms of U, has the same ambient space as wAA' <S> A^,, defined in terms of U 7 . We have a priori two definitions of the canonical basis of this space, one in terms of U, one in terms of U'. Prom the definitions, we easily see that these two bases coincide.

CHAPTER

25

The Canonical Basis B of U

25.1.1. In this section, the root datum is assumed to be F-regular. P r o p o s i t i o n 25.1.2. Let A, A' be dominant elements of X.

We write r\ =

(a) There is a unique homomorphism of U-modules x • AA+A' —• AA<8>AA'

sum over bi E B(A),6 2 G B(A'), with /(6;6I,6 2 ) G Z j v 1 ] . (c) If b~r)' + 0, then /(fe; 1, b) = 1 and f(b; 1, b2) = 0 for any b2 ± b. If The vector r) rfG A\ Aa' satisfies Ei(rj rj') = 0, K^(rj (g> 7/) = (»,\)+(»,x') ( b y 3.5.8). By the definition of comultiplicav u This implies s u m over tion in U, we can write x(6~r/") = b2 /(&; 6i, b2)b^r} ® b^rfi 6i G B(A),6 2 G B(A'), with f(b;bub2)'eQ{v) satisfying f(b;l,b2) = 1 if By 23.2.3, we have /(&;&i,&2) G A for all bi,6 2 . Hence to prove (b), it suffices to show that f(b; b2) G A for all 6i, b2. We have a commutative

where S is as in 18.1.4, the left vertical map is given by x »—• x~rj" and the right vertical map is given by x ® y —> (x~r]/) <8) y. The commutativity of the diagram follows from the definitions. Now our assertion on f(b\b\,b2)

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Media, LLC 2010 2011

198

25.1. Stability Properties

199

P r o p o s i t i o n 25.1.3. Let A, A' be dominant elements of X. We write f = A—A' • = = (a) There is a unique homomorphism of U-modules x' : "'AA+A' —* w AA' ® ^AA such that x'(F") = F ® F • (b) Le* 6 e B(A +A'). We have x'tt+t") = f, G Z[v - 1 ]. If 6 + f ^ 0, sum over G B(A),6 2 e B(A'), wtM f(b;bub2) Men /(6; 1,6) = 1 and /(6;1,6 2 ) = 0 for any b2 ± b. If 6 + f = 0, then / ( 6 ; 1,& 2 ) = 0 for any b2. We have a commutative diagram U —

U

U

u

*A ) u ® u

(*A as in 3.3.4.) Indeed, both compositions in the diagram are algebra homomorphisms; to check that they are equal, it suffices to check that they agree on the generators and that is immediate. Using this commutative diagram, we see immediately that the proposition follows from the previous proposition. P r o p o s i t i o n 25.1.4. Let 77 G Aa,£ G "'Aa be as above. (a) There is a unique homomorphism of U-modules 6\ : Aa —> Q(v), where Q(v) is a U-module via the co-unit U —> Q(v), such that = 1(b) Let 6, b' G B(A). Then 6x(b+C ® b'-j]) is equal to 1 if b = b' = 1 and otherwise. is in v~l7i[v~l\ The following statement is equivalent to (a). There is a unique bilinear pairing [, ] : AA x AA —> Q(^) such that [77,77] = 1 and [Fix, y]

-[K

Eiy], [x, Fiy] = -[.E{x, K-iy], [K-^x, K^y] = [x, y]

for all x, y G A, all« G I and all p G Y. We then have 6 \(x ® y) = [x, y]. This is also equivalent to the following statement. There is a unique bilinear pairing [, ] : AA X A A —> Q(V) such that [77,77] = 1

25. The Canonical Basis B o / U

200

and [ux,y] = [x,p{u)y] for all x, y £ A, and all u £ U, where p : U —• U o p p is the algebra isomorphism given by the composition Su (S is the antipode). This is proved exactly as in 19.1.2. It follows from the definition that [x,y] = 0 if x £ (A) v ,y £ (A)„/ and v ^ v'. Let (,) be as in 19.1.2, We show by induction on tr v > 0, where v £ N[/], that (C)

[«,»] =

(-l)tr"t;-H(«.»)

for all x, y £ (A)jy. This is obvious for v — 0. We assume that tr v > 1. We can assume that x = Fix' for some i such that vi > 0 and some a;' £ Then [x,y] = [Fix',y] = -[K^x^Eiy]

~[x\K^Eiy]

=

and (x,y) = (FiX,y) =

Vi(x\K-iEiy).

By the induction hypothesis, we have -\x\k.iEiy) = (-1)tr ^ . ^ ( x ^ K . i E i y ) hence [x,y] = (—1) tr uv_\u\(x,y), which completes the induction. Now let 6, h' be as in (b). We must show that [6—77, b'~rj\ is in v~1Z[v~1] unless b = h' = 1. We may assume that there exists v such that b~r},b'~rj both belong to (A)^. The result then follows from (c) since, by 19.3.3, we have (b~r],b'~r]) £ Z[v - 1 ]. 25.1.5. Let A, A7, A" be dominant elements of X. We define a homomorphism of U-modules t : "Ax+y ® AA'+A"

"AA' <8> AA"

as the composition of W

x' <8> X ' ^AA+A' <8> A V + A "

where x',X

a r e 95

1 0

6y

AA (8) "Ay

® Ax> ® A A " ,

25.1.3, 25.1.2, with 1 : " A A <8>

"Ay

<8>

Ay

A A " - >

U

AX

<8> A A " .

25.1. Stability Properties

201

L e m m a 25.1.6. (a) Let b E B(A), b" E B(A"). We have ® b"-rix> + \") = b + Z- X ® b"~rix»

mod

v~lCx,x"

(notation of 24.3.1). (b) Let b E B(A + A'), b" E B(A' + A"). Assume that either b <£ B(A), or b" ^ B(A"). We have t(b+^X-x> ® b"~rix'+X") = 0 m o d v " 1 ^ ' . (c) t is surjective. In this proof we shall use the symbol = to denote congruences modulo vtimes a Z[i; _1 ]-submodule spanned by the natural basis. Using 25.1.2, 25.1.3, 25.1.4, we see that if b,b" are as in (a), we have t(b+£-x-\> ®b"-r)y+x")

= (1 ®6\> ® 1 )(b+£-x ® =

® V\' ®

+

b £-x®b"-rix».

Using again 25.1.2, 25.1.3, 25.1.4, we see that if b, b" are as in (b), we have = 0. t{b+^x-x>®b"-r)x>+x») Now (c) follows from the fact that £_A r)x' is in the image of t and it generates the U-module "Ax ® Ay (see 23.3.6). Using the definition 24.3.3 of the elements (b^b')x,x', we can reformulate the previous lemma as follows. L e m m a 25.1.7. (a) Let b E B(A), b" E B(A"). We have t(b<>b")x+x>,\>+\" = (t>0b")x,x»

mod

v~lCx,x».

(b) Let b E B(A + X'), b" E B(A' -I- X"). Assume that either b (£ B(A), or b" £ B(A"). We have t(b<>b")x+x',x'+x" = 0

mod

v~1Cx,x"-

25.1.8. In the following result we show that the maps u

$ : "Ax+x' ® Ay+x"

Ax+x'

<8> A A ' + A "

and ^ -."AxteAx"

u

Ax 0 AA",

defined as in 24.3.2, are compatible with t.

202

25. The Canonical Basis B o / U

L e m m a 25.1.9. We have tV = Vt. We write £,77 instead of £_A-A', rjy+y. Since any element of wAA+A' 0 AA'+A" IS of the form «(£ 0 rj) for some u G U (see 23.3.6), it is enough to check that tG(u(£ 0 77)) = et(u(£ 0 77)) for all MGU, Using the definition of © and its property 24.1.3(a), we have tG(u(£ 0 77)) = t u ( G ( £ 0 77)) = tu(£ ® 77) = ut(£ 0 77) = u ( f _ A <8> and Gtiui^rj))

= Gu(t({, (8) 77)) = U 9 ( £ _ A 0 77A") = U ( £ - A 0 »?A")-

The lemma is proved. P r o p o s i t i o n 25.1.10. (a) Let b G B(A), b" G B(A"). We have t(b0b") x+x>,x'+x" = ( W ) A,A". (b) Lei 6 G B(A + A'), 6" G B(A' + A"). Assume that either b <£ B(A), orb" i B(A"). We have t(b0b") A+A',A'+A" = 0. The difference of the two sides of the equality in (a) is in V-1£A,A" (by 25.1.7) and is fixed by ^ : "AA 0 AA" —> "AA 0 AA", using the definitions and Lemma 25.1.9; hence that difference is zero, by 24.3.3(d). Thus the equality in (a) holds. Exactly the same proof shows (b). 2 5 . 2 . DEFINITION OF THE BASIS B OF U

T h e o r e m 25.2.1. Assume that the root datum is Y-regular. and let b, b" G B .

Let £ G X

(a) There is a unique element u = b<>c&" € ^ U l c such that u(Z-x®r)x")

= (bOb")x,x"

for any A, A" G X+ such that b G B(A), b" G B(A") and X" - A = (b) If A, X" G X+ b" <£ B(A"), then

are such that X" - X = ( and either b £ B(A) or 0 rjX") = 0 (u as in (a)).

25.2. Definition of the Basis B o / U

203

(c) The element u in (a) satisfies u = u. (d) The elements bQ^b", for various C,b,b" as above, form a Q(v)-basis of U and an A-basis of AU. Since the root datum is ^-regular, we can find A, A" G X+ such that b G B(A), 6" G B(A") and A" - A = C be the .A-submodule of ^ U For any integers N\,N2, let P(N\,N2) spanned by the elements b+b^ 1< where b\,b2 run through the set of pairs of elements of B such that tr |&i| < Nu tr |6 2 | < N2 and \bi\-\b2\ = |&|-| 6 "lBy arguments such as in 23.3.1 or 23.3.2, we see that any element of " M x ^ M y , or "AA^AA", of the form ®/?'~?7A"> with /?,/?' G B, is equal to wi(£_a<S>t?a") for some u\ G P( tr \/3\, tr |/?'|); moreover, u\ can be plus an element in P( tr \(3\ — 1, tr \(3'\ — 1). taken to be equal to /3+/3'~ From this we deduce that (e) (b§bn)A,A" £ "Ax <8> AA" is of the form u(£_A ® Vx") f° r some u G plus an P( tr |6|, tr |6"|); moreover, u can be taken to be equal to b+b"~ element in P( tr |6| - 1, tr |6"| - 1). Assume that u is such an element and that u' is another element with the same properties as u. Then (u — U')(£_A <8> rjy) = 0; hence, by 23.3.8, we have u-u'e

J2

AVFfn\+

i,n>(i,X")

^ i,n>(i,

n )

l

C

.

A)

Since u — u' G P( tr |6|, tr |6"|) we deduce that, if {i, A) and (i, X") are large enough (for alH), then we must have u = u'. Thus, for such A, X" the element u above is uniquely determined. We denote it by ux,x"Assume now that A, X" G X+ satisfy b G B(A), b" G B(A") and X" - X = Let X' G X+ be such that (i,X') is large enough for all i, so that v! = itA+A',A'+A" is defined. We show that x ® Vx") = (bW')x,x"Indeed, if t is as in 25.1.5, we have u'(€-x®r)x>>) = u'(t(£-x-x' = t((bOb")

<8>Vx'+x")) = A+A',A'+A") =

x-x

®VX>+X"))

(b0b")x,x»

where the last equality follows from 25.1.10. It follows that UA,A" IS independent of A, X", provided that (i, A) and (i, X") are large enough (for all i); hence we can denote it as u, without specifying A, A". It also follows that u satisfies the requirements of (a). This proves the existence part of (a). The previous proof shows also uniqueness. Thus (a) is proved.

25. The Canonical Basis B o / U

204

Now let A, A" be as in (b). Let A' G X+ be such that (i, X') is large enough for all i, so that WA+A',A'+A" IS defined (hence it is u). We have A ® F ] \ " ) = U ( T ( I - A - A ' 0 *?A'+A")) =

A - A ' <8>7?A'+A"))

= T ( ( & 0 & " W , A ' + A " ) = 0,

where the last equality follows from 25.1.10. This proves (b). We prove (c). Let u, A, X" be as in (a). We have U ( £ _ A 0 T?A") =

<8> F?A")

= O U ( ^ _ A
= ( W )

A,A-

Thus it satisfies the defining property of w. By uniqueness, we have u — u. This proves (c). We prove (d). From (e) we see that, for fixed we have bOcb" = b+b"~lc

mod P{ tr |6| - 1, tr \b"\ - 1).

Since the elements b+b"~l^ form an *4-basis of ^ U , we see that (d) follows. The theorem is proved. 25.2.2. We now drop the assumption that the root datum (F, X,...) is y-regular. Assume that we are given £ G X. Let ( Y ' , ^ ' , . . . ) be the simply connected root datum of type (/, •) and let / : Y' —• Y, g : X —> X' be the unique morphism of root data. Let U ' be the algebra, defined like U, in terms of (Y', X',...). Let C = g( C). By 23.2.5, we have a natural isomorphism (a) U ' l c , ^ U l c , > u+u'~ for all u, u! G f. For each b,b" G B, we defined by denote by b(}^b" the element of Ul^ corresponding under (a) to b(}^b" G U'l^/ (which is defined by the previous theorem.) Corollary 25.2.3. The elements b^^b" for various 6, b" G B and various £ G X form an A-basis o / ^ U and a Q(v)-basis of U . They are all fixed by the involution ~ : U —» U .

25.3. Example: Rank 1

205

25.2.4. R e m a r k . The basis of U just defined is called the canonical basis of U. We denote it by B. In the case where the root datum (Y, X , . . . ) is Y-regular, this canonical basis coincides with the one defined in 25.2.1. This follows immediately from definitions, using 23.2.5 and 23.3.4. From the proof we see that any element of B is contained in one of the summands in the direct sum decomposition 23.1.2 of U. T h e o r e m 25.2.5. Let ( e X and let a, a' be as in 23.3.3. Let P(C,a,a'), aP(Ci a') be as in 23.3.3. Then B n P ( £ , a, a') is an A-basis ofAF(C, a, a') and a Q(v)-basis of P(£,a, a'). Using the definitions and 23.3.4, we are reduced to the case where the root datum is simply connected. In that case, the result follows immediately from Theorem 25.2.1. We now show that B is a generalization of B. P r o p o s i t i o n 25.2.6. Let b G B and let ( e X . b+ l c G B.

Then b~ l c G B and

We can assume that the root datum is simply connected. Choose A, A' G such that A'—A = £ and such that b G B(A'). We have b~ lc(f_A77A') = £_a <8> b~7)\'. Using the definitions, we see that £_a <8> b~rjxf satisfies the defining properties of ( 1 < h e n c e it is equal to (106)A,A'- It follows = 6 ^ 1 . The that b~ 1<; = 10c b. A similar argument shows that proposition is proved. 2 5 . 3 . EXAMPLE: RANK 1

25.3.1. In this section we assume that I — {z} and X = Y = Z with i = leY,ir = 2eX. Consider the following elements of U: (a) ELA)L_NF^B)

(b) i ^

(6)

ln^

a)

( a , 6 , n G N , n > a + 6)

(a, 6, n G N, n > a + b).

Note that (c) E\a)l_nFlb)

= Flb)\nE\a)

iom = a + b.

P r o p o s i t i o n 25.3.2. The canonical basis of U consists of the elements 25.3.1(a) and (b), with the identification 25.3.1(c). More precisely, if n> a + b, we have Elah.nFlb) = ela)0-n+24b) and

25. The Canonical Basis B o / U

206

We compute the image of the elements 25.3.1(a),(b) under the map U —> "Ap® A g , with p,q> 0, given by u u(£-pr)q). The image of the element 25.3.1(a) is zero unless -n + 2b = q — p, in which case it is £ < 0 > i f >(£_p ® v<) = E(°}(fi-P ® i f >»,,) a'+a"=a

= =

E

a'+a"=a

J2<'a"-a"pEla'k-1

t>0

a" -b + q F t t i

( )

E f - \

\a" -b + q

E

a'+a"=a

=

s(a — s—p)

E

s- b+q

s>0;s
This element is fixed by the involution ^ of u A p A g , since the element 25.3.1(a) is fixed by ~ : U —> U. Using the definitions, we see that this element is Hence the element 25.3.1(a) is 0\a^ The image of the element 25.3.1(b) is zero unless n — 2a = q—p, in which case it is i f >i?< a) (£- p ® Vq ) = i f > ( i f ^ - p ® Vq ) b'+b"=b

= =

b'b"-b'q

E

5>J

b'+b"=b

t

t>0

E 8>0]8
s{b-s-q)

E t ^ F f - H - ^ F f ^

s — a + p s) Et t-P s

® F t ° \ -

As before, we see that this is equal to {e\a) 0e\b))Pyq. follows.

The proposition

2 5 . 4 . STRUCTURE CONSTANTS

For any triplet c G B, we define elements m £ b € A by ab = 1S Scma6c the product in U). We also define elements m" 6 G A by the following requirement: for any A^A^A^A^' G X and any c G B fl (A;+A^U a // +a ^) we have 25.4.1.

a,6

25.4• Structure Constants

207

in the last formula, A v ^ ^ ^ ^ ' is as in 23.1.5; a runs over B fl ( ^ U y ^ ) ; b runs over B fl (A(UA'2'). If a G B fl ^ B n (A^U a -) and c G an B fl d either A3 ± X[ + \'2 or A'3' ± A" + AJ, then mf is defined to be 0. The elements are called the structure constants of U. They satisfy the following identities, for all a, b, d, e G B and A G X:

(b) £ c m f m f = £ c m 2 c ™ c d ; (C) H c m C ab m e c d = £a',6',c',d' ^ a (d) m f j = 1 if a = 1 v , b = 1\», A' + A" = A and m f j = 0 otherwise. In each sum, all but finitely many terms are zero. The identity (a) expresses the associativity of multiplication in U; (b) is a consequence of the coassociativity of comultiplication in U; (c),(d) are consequences of the fact that the comultiplication A : U —• U ® U is an algebra homomorphism preserving 1. C o n j e c t u r e 25.4.2. If the Cartan datum is symmetric, then the structure constants are in N [ v , v - 1 ] . This would generalize the positivity theorem 14.4.13. For the proof an interpretation of (U, B) in terms of perverse sheaves, generalizing that of (f, B) will be required.

CHAPTER

26

Inner Product on U 26.1.

FIRST DEFINITION OF THE INNER

PRODUCT

26.1.1. In the following theorem, p : U —• U is as in 19.1.1, and p : U —> U T h e o r e m 26.1.2. There exists a unique Q(v)-bilinear pairing (,) : U x (a) (l^xlAa* I a ' ^ ' I a j ) is zero for all x , x ' G U, unless Ai = A'x and

(c) (X~1

X'~1

for all x , x ' G f and all A (here (x,x') is the

Let ( G I . If B is a basis of f consisting of homogeneous elements, the elements with i>, bf G B, form a basis of Ul^. (We use the triangular decomposition of U.) Hence there is a unique Q(t>)-linear map p : Ul^ —> Q(v) such that p(p(x~)x,~ 1^) = ( x , x ' ) for all x , x ' G f; here, (x,x') is as in 1.2.5. The properties of (x,x') imply that for homogeneous x,x', we have p(p(x~)x'~ 1^) = 0 unless x , x ' have the same homogeneity, = 1 ^p{x~)x'~. Thus, for ( ' G l different from in which case p{x~)xf~ we have = 0. It follows that, for any /i G Y, we have p((K^ — ^(m,C))ui c ) = 0. We now define a pairing : Ul^ x U l c —> Q(v) by — p(p(u)u'l{) where u,u' G U. To show independence of u, vl, we must check that p(p(u)u'l= 0 if either u or u' is in the left ideal of U generated by ( K ^ — v ^ ^ ) for some p, G Y; in the case of ti, this follows from the previous sentence, while in the case of u', this is obvious. Thus, fa is well-defined. We define the bilinear pairing (,) on U by (x, y) — /^(x, y) It is if x,y e U l c and by (x, y) = 0 if x G U l c and y G U1C/ with £ / clear that this pairing satisfies (a),(b),(c); the uniqueness is also clear from The pairing x, y (t/, x) satisfies the defining properties of (x, y) (since p2 = 1), and hence coincides with it. This proves (d).

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2010 2011

208

26.1. First Definition of the Inner Product

P r o p o s i t i o n 26.1.3. We have (xu,y) = (x,yp(u)) u G U.

209

for all x,y E U and

We may assume that u is one of the standard generators of U. Thus, we must verify that (.xEi,y) = (x,ViyFiK-i),(xFi,y)

= (x,ViyEiKi),{xK-^y)

=

(x,yK-^)

for all x, y G U and i G / , p G Y. We may assume that x = u' where u' G U and £ G X. Using 26.1.2(b), and setting p(u')y = y', we see that the previous equalities are consequences of v^Eiki) (1 tEirf) = (l£,ViyfFiK-i), (l{Fi,y') = (l c , and eY. for all y' G U and i G I,p We can assume that y' = where 2/1,2/2 £ f are homogeneous. Then the equalities to be proved can be rewritten as follows: (a) (yf £7
}

(yf l c , y ^ f i k ' + i ' )

(b) (yTFdc+i^y^U') = v l + { i * ' - i ' ) ( y i l < : , y 2 E i l C - i , ) (c) ( y i i c K - ^ i c ) = ( y T h i y 2 h ' K - n ) ' Now (c) is obvious and (b) follows from (a), using 26.1.2(d). It remains to prove (a). We may assume that f = £ — i'. We substitute 2/i Ei

=

, rj(yi) K-j - Kj(jr(yi) lc + zi Vi -

)

lc

and note that (Eiyl 1c,;i/2 1C/) = (^ l^^i^Fi^

lc>)

V,- — vvr<<,fV<(vi).») - ^ - " " ' ^ ( i K y i J . i t t ) Vi-Vi1

26. Inner Product on U

210

(see 26.1.2(b), (c)). We see that (a) is equivalent to:

Vi-Vi1

But this follows from the known equalities 1 V

1 - Vi

~i

(see 1.2.13(a)). The proposition is proved. P r o p o s i t i o n 26.1.4. We have (a(x),cr(y)) = (x,y) for all x,y e U . We must show that the pairing x,y (a(x),a(y)) on U satisfies the defining properties of (,). Property 26.1.2(a) is obvious and property 26.1.2(c) follows from 1.2.8(b). Since ap = pa : U —• U, we see that 26.1.2(b) for the pairing x,y i—• (a(x)i
t

p(<7(*+)))

= (1_a>
V~~NX~l-\)

The lemma is proved.

= (x'~l-\,x~l-x)

= (x',x) =

{x,x').

26.2. Definition of the Inner Product as Limit

211

P r o p o s i t i o n 26.1.6. We have (<jj(x),u(y)) = (x,y) for all x,y G U. We must show that the pairing x,y • (u(x)yw(y)) on U satisfies the defining properties of (,). Property 26.1.2(a) is obvious and property 26.1.2(c) follows from the previous lemma. Since up = pu : U —• U, we see that 26.1.2(b) for the pairing x,y •—• (u(x),u(y)) is equivalent to the corresponding identity for (x, y). The proposition follows. 2 6 . 2 . DEFINITION OF THE INNER PRODUCT AS A LIMIT

26.2.1 In this section we assume that the root datum is Y-regular. Let We consider the bilinear C G X and let A, A' G X+ be such that A' — A = = (x, y)\(x',y')\>. pairing (, )A,A' on "Ax ® Ay, defined by (x®x\y®tf) Here (, )y is the pairing on AA' defined in 19.1.2, and (, )A is the analogous pairing on A A which has the same ambient space as "A\. L e m m a 26.2.2. If x\,x2 G "A\ ® A\> and u € U, we have (uxi,x2)\,\>

= {xi,

p{u)x2)\,\>.

It is enough to check this in the case where u is one of the algebra generators Ei,Fi, K^ of U. The case where u = K^ is trivial. We now fix i and regard the U-module wAA ® Ay as an object of C[ (by restriction). It is enough to show that the form (,) on this object is admissible in the sense of 16.2.2. Using 17.1.3(b), we see that this would follow if we knew that the forms (, )A and (, )A' on wAA and A\> (regarded as objects of C-) are admissible. For (, )A' this follows from the definition. The same holds for (, )A on AA- One checks easily that applying u to an object of C[ with an admissible form gives a new object of C[ for which the same form is admissible. In particular, (, )A' is admissible for "A\. The lemma is proved. P r o p o s i t i o n 2 6 . 2 . 3 . Let x,y G Ul^. When the pair A, A' tends to oo (in the sense that (i, A) tends to oo for all i, or equivalently, (i, A7) tends to oo for all i, the difference A7 — A being fixed and equal to the inner product _1 (x(£_A <8>*7A')>2/(C-a ® V\'))\,\' E Q(v) converges in Q((v )) to (x,y). Assume first that x = have x{£-X®r)\')

—

where X\,y\ G f. In this case we = f-A ® XiVx'

and y{£-\ <S> 77a') = i-x

®yiV\r,

26. Inner Product on U

212

hence and, by 19.3.7, this converges to (xi,yi) when A —• oo. Since (x\,yi) = (a:]" the proposition holds in this case. Next, we prove the proposition in the case where x = and y is arbitrary. We may assume that y = p(x^)y^ where Xi,yi G f. Using the previous lemma, we have (1<(£-A ® ?7A')>2/(£-A ® 7?A'))A,A' =

terMf-A

® 7?A')>2/ric(f-A ® vx<))x,x>

and by the earlier part of the proof, this converges to (rr^l^, y f l ^ ) when A oo. Since yf 1^) = (1^,2/), the proposition holds in this case. We now consider the general case. We can write x = ul^ where u G U. Using the previous lemma, we have 0 Vx'),p{u)y{£-\

(U1C(£-A ® 77A'),2/(£-A <8>77A'))A,A' =

<8> Vx'))x,x>

and by the case previously considered, this converges to (1 £,p(u)y) A —• oo. Since (l^,p(u)y) = (ul^,y), the proposition is proved.

when

2 6 . 3 . A CHARACTERIZATION OF B L) ( - B )

In the following result there is no assumption on the root datum. T h e o r e m 2 6 . 3 . 1 . (a) Let 6,6',bi,b[ G B and let C,Ci € X.

We have

In particular, the canonical basis B o / U is almost orthonormal {or (,). (b) Let (3 G U. We have (3 G B U (—B) if and only if (3 satisfies the following three conditions: (3 G j3 = (3 and ({3,(3) G 1 + v_1A. Note that (a) is trivial when C Ci- Hence we may assume that £ = In this case, using the definitions we are immediately reduced to the case where the root datum is simply connected. Then 26.2.3 is applicable. Hence, using the definition, we see that it is enough to prove that (c) ((&0&')A,A', (6lO&i)A,A')A,A' = S B M ^ M for any A, A' G Since (b+€-x ® b'~r)\,

M O D

V

~'

A

7

such that A - A = £ and such that b G B(A), b' G B(A'). ® b[~r]x)x,x' = (b~r)x,b]~rix)x(b,~rix>,b,1-7)x')x>

G (fa,bi +v~ 1 A)(6 b ,y i + v~*A) =

6b,b16b>,b'l

+

26.3. A Characterization of B U ( - B )

213

we see that (c) follows from Theorem 24.3.3(b). We prove (b). If (3 G ± B then, by (a) and 25.2.1, it satisfies the three conditions listed. Conversely, if (3 G U satisfies the three conditions in (b) then, using (a) and Lemma 14.2.2(b), we see that there exists j3' G B such that f3 — (±/?') is a linear combination of elements in B with coefficients in v - 1 Z [ v - 1 ] . These coefficients are necessarily 0, since (3 — (±/3') is fixed by : U —> U. The theorem is proved. Corollary 26.3.2. If (3 G B, then a((3) G ± B and u((3) G ± B . a and u commute with ~ : U —> U, preserve the lattice ^ U and preserve the inner product (,) (see 26.1.4, 26.1.6). Hence the corollary follows from Theorem 26.3.1(b).

CHAPTER

27

Based Modules

27.1.1. In this chapter we assume that (/, •) is of finite type. Let M e C. We assume that M is finite dimensional over Q ( f ) . For any A £ we denote by M[A] the sum of simple subobjects of M that are isomorphic to A a . Then M = © ^ M [ A ] . We also define for any A £ X+:

27.1.2. A based module is an object M of C, of finite dimension over Q(v)

(b) the *4-submodule j^M generated by B is stable under ^ U ; (c) the Q-linear involution : M —> M defined by fb = fb for all / £ Q(v) and all b £ B is compatible with the U-module structure in the (d) the A-submodule L(M) generated by J5, together with the image of B in L(M)/v~~1L(M), forms a basis at oo for M (see 20.1.1). We say that ~ : M —> M in (c) is the associated involution of (M,B). The direct sum of two based modules (M, B) and (M', B') is again a based 27.1.3. The based modules form the objects of a category C; a morphism from the based module (M, B) to the based module (M', Bf) is by definition

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011

214

27.1. Isotypical Components

215

27.1.4. Let ( M , B ) be a based module and let M' be a U-submodule of M such that M' is spanned as a Q(t>)-subspace of M by a subset B' of B. Then (M', B') is a based module; moreover, M/M' together with the image of B — B' is a based module. For any A G X+, A A together with its canonical basis, is a based module. (See 19.3.4, 23.3.7, 20.1.4.) 27.1.5. Let (M, B) be a based module with associated involution ~ and let ra G M be an element such that ra = ra, ra G ^ M and m G B + v~1L(M) (resp. ra G v~1L(M)). Then we have ra G B (resp. ra = 0). Indeed, we can write m = JZbeB with CF, G A. By our assumption, we have Q> G A for all b. Hence Cb G Z[t>_1] for all b. We have Cb — Cb for all b. Hence c& G Z for all b, except for all b. Moreover, by our assumption, we have CF> G It follows possibly for a single b for which we have Cb = 0 or 1 mod that Cb = 0 for all 6, except possibly for a single b for which we have c& = 0 or 1. Our assertion follows. 27.1.6. Let (M, B) be a based module. Assume that M ± 0. Let Ai G X+ be such that MXl ^ 0 and such that Ai is maximal with this property. Let Bi = B n MXl. It is a non-empty set. Let M' = ©beBiAo^.b G C. Here A\ lt b is a copy of AXl corresponding to b; we denote its canonical generator by rjb. For any b G jE?i, we have Eib = 0 for all i G / by the maximality of Ai. Hence there is a unique homomorphism (j) : M' - > M o f objects in C whose restriction to any summand AXl,b carries 775 to b. Let B' be the basis of M' given by the union of the canonical bases of the various summands AXl,bProposition 27.1.7. In the setup above, B fl M[Ai] is a basis of M[\\] and defines an isomorphism M' = M[X\] carrying B' onto B fl M[Ai]. Thus M' is compatible under (j) with that of M. Indeed, both involutions are the identity on B\. (We regard B\ as a subset of M' by _ b 7]b.) Let b' <E B' fl AXlyb- We have b' = b'; hence 0(6') = (&') = (b'). Thus (f>(b') is fixed by ~ : M —> M. We know from 19.3.5 that there exists a sequence ii, •.. ,ip in I such that b' is equal to Fi1Fi2 • • • Fipr)b plus a v~lA-linear combination of elements of the same kind. Now the action of Fi on M' is compatible with

21. Based Modules

216

the action of Fi on M. Hence (f>(b') is equal to F^F^ ... Fipb plus a linear combination with coefficients in v - 1 A of elements of the same kind. By property 27.1.2(d) of B, we see that either (#) G B + v~lL(M) or 4>(V) G v~~lL(M). On the other hand, by the definition of the canonical basis of M', we have that b' belongs to the ^U-submodule of M' generated by 775; hence (p(b') belongs to the ^U-submodule of M generated by 6; by the property 27.1.2(b), we then have (j>{b') G ^ M . These properties of (br) G B or (f)(b') = 0 (see 27.1.5). The second alternative does not occur: indeed, the restriction of 4> to the summand Aa1;6 is injective since A\ lt b is simple. Thus we have <j)(b') G B. We see that (j> defines a bijection of the canonical basis of A\1,b with a subset B(b) of B. Next we consider an element b € B1 distinct from b. We show that B(b) is disjoint from B(b). Indeed, assume that 61 G B belongs to B(b) C\B(b). Then we have 61 = F^Fiz • ••Fiyb mod v~lL(M) and 61 = FjlFh • • • Fjgb

mod

v~1L(M)

for some sequences ii, «2> • - - , a n d j\,32, • — ijq m I27.1.2(d), we then have b = EjqEjq_l • • • Ej1Fi1Fi2 • • • Fipb

mod

By property

v~lL(M).

Hence b is equal to some element in B(b) plus an element of v~1L(M). It follows that be B{b). In particular, we have b = (6') for some b' G Since b ^ b, we have V / r]b\ hence b' G A ^ b with A7 < Ai. It follows that b G Mx> with A' < Ai. This contradicts the assumption that b G B\. We have proved therefore that B{b) is disjoint from B(b). Since B' is the disjoint union of the canonical bases of the various AAltb and these subsets are carried by <j> injectively onto disjoint subsets of B, it follows that (j) restricts to an injective map B' —> B. Since B' is a basis of Mit follows that 0 : M' —> M is injective. Thus we may identify M' with a U-submodule of M (via cj)) in such a way that B' becomes a subset of B. This submodule is clearly equal to M[Ai]. The proposition follows. Proposition 27.1.8. Let (M, B) be a based module and let A G X+. (a) B fl M[> A] is a basis of the vector space M[> A] and

Then

27.2. The Subsets B[X]

217

(b) B f l M [ > A] is a basis of the vector space M[> A]. First note that (b) follows from (a). Indeed, the vector space M[> A] is a sum of subspaces of form M[> A7] for various A' > A. To prove (a), we argue by induction on dim M. If dim M = 0, there is nothing to prove. Therefore we may assume that dim M > 1. For fixed M, we argue by descending induction on A. To begin the induction we note that if A) is sufficiently large, then M[> X] = 0 and there is nothing to prove. Assume that A is given. If M[X] = 0, then M[> A] is a sum of subspaces M[> A7] with A' > A; hence the desired result holds by the induction hypothesis (on A). Thus we may assume that M[A] ± 0. Then clearly Mx ± 0. We can find Ai G X+ such that Ai > A, M A l / 0 and Ai is maximal with these properties. Let M' = M[Ai] and let B' = B n M'. Then (M',B') G C by 27.1.7. Hence, by 27.1.4, M" = M/M 7 , together with the image B" of B-B', is an object of C. Since M' ^ 0, we have dimM 77 < d i m M ; hence the induction hypothesis (on M) is applicable to M". We see that 5 " n M " [ > A] is a basis of M"[> A]. Since M' = M'[Ax] and Ai > A, we see that M[> A] is just the inverse image of M"[> A] under the canonical map M —> M"\ moreover, a basis for this inverse image is given by the inverse image of B" D M"\> A] under the canonical map B —> B". The proposition is proved. 2 7 . 2 . T H E SUBSETS £ [ A ]

27.2.1. Let (M,B) be a based module. Let b G B. We can find A G X+ such that b G M[> X] and A is maximal with this property. Actually, A is unique. Indeed, assume that we also have b G M[> A;] and X' is maximal with this property. We note that M{> A] fl M[> A7] is a sum of subspaces M[> X"} for various A" such that A < X" and X' < X" and from 27.1.8 it follows that b G M[> X"] for some such X". If A ^ A7, then A77 satisfies A < A77 and A7 < A77, and we find a contradiction with the definition of A. Thus the uniqueness of A is proved. Let Z?[A] be the set of all b G B which give rise to A G X+ as above. These sets clearly form a partition of B. From 27.1.8, we see that, for any A G X+, the set Ux>€x+-,\'>\B[^'] is a basis of M[> X] and the set U A / g X + ; A / > a B[A 7 ] is a basis of M[> A]. P r o p o s i t i o n 27.2.2. Let f be a morphism in C from the based module (M,B) to the based module ( M 7 , B 7 ) (see 27.1.3). For any X G X+, we have /(B[A]) CB'[A]U{0}.

218

21. Based Modules

From the definitions, we see that f(M[> A]) C M'[> A] and f(M[> A]) C M'[> A]. Hence if b G B[A], then either f(b) 6 B'[\'] for some A' > A or f{b) = 0. Assume that f(b) $ B'[A]. Then f(b) G M'[> A]. Using the obvious inclusion f(M) D M'[> A] C / ( M [ > A]), we deduce that 6 G M[> A] -f k e r / . Since both M[> A] and k e r / are generated by their intersection with B, it follows that either b G M[> A] or b G k e r / . The first alternative contradicts b G 5[A]; hence the second alternative holds and we have f(b) = 0. The proposition follows. 27.2.3. Let ( M , £ ) be a based module. Let A G X+. We define B[X]hi to be the set of all b G B such that b G Mx and Eib G v~1L(M) for all i G I. We define B[X]l° to be the set of all b G B such that b G Mw°W and Fib G v _ 1 L ( M ) for all i G / . P r o p o s i t i o n 27.2.4. (a) We have B[X]hi C B[X] and B[X]l° C B[A]. (b) Let p : M[> X] —> M{> X]/M[> X] = M be the canonical map. Note that p defines a bijection of -B[A] with a basis B of M and that (M,B) belongs to C so that B[X]hl and B[X]l° are defined. Then p restricts to bijections B[X]hi B[X]hi and B[X]l° B[X]l°. We prove (a). Let b G B[X]hl. There is a unique A' G X+ such that b G B[X']. We must prove that A = X'. We have b G M[> A']. Replacing M with M[> A7], we may assume that M = M[> X']. Let ir be the canonical map of M onto M" = M/M[> A']. Then B[X'] is mapped by IT bijectively onto a basis B" of M" and we have 7r(b) G B". Moreover, ir(b) belongs to B"[X]ht and we are therefore reduced to the case where M = M". Thus we may assume that M — M[X']. Now 27.1.7 reduces us further to the case where (M, B) is k\> with its canonical basis. In this case, there are two possibilities for b: either b is in the A'-weight space or there exist i and b' G B such that F^b' -be v~1L(M). In the first case we have b G M A '; in the second case we have Eib — b' G v~lL(M)\ hence Eib £ v~lL(M), in contradiction with our assumption on b. Thus we have b G Mx , hence A = A;, as required. We have proved that B[X]hl C -B[A]. The proof of the inclusion B[X]la C B[A] is entirely similar. We prove (b). We assume that M = M[> A]. It is clear that p(B[X]hi) C B[X]hi and p(B[X]l°) C B[X]l°. Assume that b G £[A] satisfies b £ B[X]hi. We show that p(b) ^ B[X]hl. By our assumption, we have that either b e Mx> with Y ± X or that Eib £ v~lL(M) for some i. If be Mx' with X' / A, then p(b) G Mx' with A; ^ A; hence p(b) $ B[X]ht, as required. If Eib £ v~1L(M) for some i G / , then there exists b' e B such that Eib - b' G y-1L(M) and therefore Fib' -be v~lL(M).

27.3. Tensor Product of Based Modules

219

We consider two cases according to whether or not b' G M[> A]. In the first case (ib' G M[> A]), we have Fib' G M[> A] (since M[> A] is a subobject of M) hence b G M[> A]-\-v~ 1 L(M); this implies that b G M[> A] (using that 5 fl M[> A] is a basis of M[> A]). Then we have p(b) = 0 and, in particular, p(b) £ B[X]hl, as required. In the second case (b' £ M[> A]), we have b' e B[A]; hence n(b') G B. Let L(M) be the A-submodule of M generated by B. From Eib — b' G v~1L(M), we deduce Ei(Tr(b)) - 7r(b') G v~lL(M). In particular, we l ht have Ei(7r(b)) £ v~ L(M)\ hence p{b) B[X] , as required. Thus we have proved the equality p(B[X]ht) — B[X]hl. The proof of the equality p(B[X]l°) = B[X]l° is entirely similar. 27.2.5. C o i n v a r i a n t s . Let ( M , £ ) G C. Let M& 0] = Bx^0M[X]. The space of coinvariants of M is by definition the vector space M* = M / M [ / 0]. Clearly, 0] is equal to the sum of the subspaces M[> A'] for various + X' e X - {0}; hence, from 27.2.8, it follows that UAVO^[A7] is a basis of 0]. We deduce that under the canonical map 7r : M —• M* the subset JB[0] of B is mapped bijectively onto a basis B* of M*. Note that n is a morphism in C if we regard M* with the U-module structure such that M* = M* [0]. We see that (a) (M*, B*) is a based module with trivial action of U. P r o p o s i t i o n 27.2.6. We have £[0] = B[0]hi = B[0]l°. This set is mapped bijectively by n : M —> M* onto B*. To prove the first statement, we are reduced by 27.2.4(a),(b) to the case where M = M[0], where it is obvious. The second statement has already been noted. 2 7 . 3 . T E N S O R P R O D U C T OF BASED MODULES

27.3.1. Let (Af, B), (M 7 , B') be two based modules with associated involutions ~ : M —> M, ~ : M' -> M'. We will show that the U-module M <8> M' is in a natural way a based module. The obvious basis B M' into a based module, since the involution M' —• M ® M' given by m ® mf = m ® mf is not, in general, compatible with the U-module structure. We will define a new involution ^ : M M' —• M (8> M' by ^ ( x ) = O(x) for all x G MM'; here © : MM' M(g)M' is as in 24.1.1. Eventually, will be the associated involution of our based module.

21. Based Modules

220

Let C (resp. ^C) be the Z[v -1 ]-submodule (resp. ,4-submodule) of M M' generated by the basis B ® B'. From 24.1.6, we see that © leaves j^C stable and clearly ~ : M M' —> M M' leaves j^C stable; it follows that we have C From 24.1.2 and 4.1.3, it follows that = 1 and V(ux) = uV(x) for all u G U and all x G M M'. We shall regard B x B' as a partially ordered set with (bi,b[) > (62,63) if and only if 61 G M A L , 6 I G M / A ' I , 6 2 G Mx*,b'2 G M'x2 where AI > A 2 ,A'j < A2, Ai -I- Aj = A2 + A2. From the definition we have, for all 61 G B, b[ G B', b[) =

J2

Pbub'iMUh ® 63

b2£B,b'2£B'

where Pbub'i;b2,b'2 e A and also that

= 0 unless (61,6i) > (62,6'2). Note P6i,6'i;6i,6i = 1

and Pbi ,b[ ;b2,b'2Pb2,b'2;b3,bf3

=

M fy ,b'3

b2eB,b'2eB'

for any 61,63 G B and b[, 63 G the last condition follows from ty2 = 1. Applying 24.2.1 to the partially ordered set H — B x B', we see that there is a unique family of elements ^ Z[i>-1] defined for 61,62 G B and 61,63 G B', such that 7r

61,6i;bi,6/1 = Ii G v - 1 Z [ v - 1 ] if (61,6i) ^ (62,6'2);

Ttbub'ite^ = 0 unless (61, &i) > (62,6/2); = ]Cb3, 6^61,6i;63, b'3Pb3, b'3,b2 ,b'2 for all (61,6;) >(62,6^). We have the following result. T h e o r e m 27.3.2. (a) For any (bi,b[) G B x there is a unique element 6j<>6i G C such that ^ ( 6 i 0 6 ; ) = 6i<>6i and (6i6i) - 61 6^ G v~lC. (b) The element b\^b\ in (a) is equal to bi®b[ plus a linear combination of elements 62 <8> 62 with (62,63) ^ B x B', (62,62) < (6i,6'i) and with coefficients in v~1Z[v~1]. (c) The elements b\with 61, b\ as above, form a Q(v)-basis M 0 M', an A-basis of ^C and a 7*[v~l]-basis of C.

of

27.3. Tensor Product of Based Modules

221

b\O&i j u s t defined satisfy the requirements of (b),(c) and that (d) holds. It remains to show the uniqueness in (a). It is enough to show that an element x G v~1C such that x = x is necessarily 0. But this follows from (d). 27.3.3. The previous result, together with the known behaviour of bases at oo under tensor product, (see 20.2.2) shows that (M M',B<*>) is a based module with associated involution This is by definition the tensor product of the objects (M, 15), (M\ B'). 27.3.4. Let A, A' G X+. Applying the previous construction to M = and M' = Ay regarded as based modules (with respect to the canonical bases), we obtain a basis of uAA Cg> AAS which clearly is the same as that constructed in 24.3.3. Thus, U A\ Ay, together with its canonical basis in 24.3.3, is a based module. P r o p o s i t i o n 27.3.5. Let A, A7, A" G X+. (a) The U-modules M = ^AA+A' ® AA'+A" and M' = U A \ ® Ay with their canonical bases B,B' constructed in 24-3.3, are in C; moreover, t : M —> M' (see 25.1.5) is a morphism in C. (b) For any Ax G

we have t(B[Ai]) C B'[Ax] U {0}.

The fact that (M, B), (M', B') are objects of C has been pointed out in 27.3.4. The second assertion of (a) follows from Proposition 25.1.10. Now (b) follows from (a) and 27.2.2. 27.3.6. Associativity of t e n s o r p r o d u c t . Let (M, J3), (M', B'), and ( M " , B " ) be three based modules. On the U-module M M' M", we can introduce two structures of based module: one by applying the construction in 27.3.2 first to M M' and then to (M 0 M') ® M"\ the second one by applying the construction in 27.3.2 first to M' M" and then to M (M'®M"). Let Bi, B2 be the bases of MM'®M" obtained by these two constructions. We show that B\ = B2. By definition, the associated involutions to these two structures are given by ^

(A

^ L X E ^ E ^ R ® - & > - )

and J2 (1 ® A ) ( B ^ ) 0 ^ ( ~ ® _ ® " )

222

21. Based Modules

respectively. These coincide by 4.2.4. Next from the definitions, we see that the Z[v _1 ]-submodules of M M' ® M" generated by B\ or B2 coincide; they both coincide with the moreover, Z[v~1 ]-submodule C of M M' ® M" generated by B®B'®B"; if 7r : C —* C/v~lC is the canonical projection, then n(B\) = 7r(B2) = tt(B ® B' B"). To show that Bx = £ 2 , it suffices to show that (b<>b')<>b" = K>(&'0&") for any b G B,b' G B',b" G B". Let bx = (b<>b')Ob" G Bx and b2 = b0(b'0>b") G B2. From the definitions, we have that n(bi) = 7r(6 6' 6") and 7t(62) = n(b <8> b' 6"). Hence 7r(&i) = 7r(fe2). Then b\ — b2 G and 6i — b2 is fixed by the associated involution. This forces &i = 6 2 , as required. Thus we may omit brackets and write b()b'()b" instead of ( b § b ' ) W or b§(b'§b"). This implies automatically that the analogous associativity result is also true for more than three factors. 27.3.7. C o i n v a r i a n t s in a t e n s o r p r o d u c t . Let (M, B),(M', B') be The two based modules. We form their tensor product (M M',B$). following result describes the subset [0] of B^. P r o p o s i t i o n 27.3.8. Let be B,b' G B'. We have Bol0] = Uy e X +{bOb'\b e B{-w0(X')}lo,b'

G

B'[\']hi}.

Let b e B,b' e B' be two elements such that b G M A , b' G M ' A \ According to 27.2.6, the condition that b$b' belongs to £<>[0] is that A + A' = 0 and Fi(b^}bf) G v~1L(M b') G ® M'). By 20.2.4, our condition is equivalent to the following one: A + A' = 0, Fi(b) G v~1L(M) and Ei(b') G v~1L(M') for all i G I. The proposition follows. 27.3.9. We consider a sequence Ai, A 2 ,. •., An of elements of X+. According to 27.3.6, the tensor product AA: <8> AA2 • • • ® AA„ is in a natural way a based module (hence has a distinguished basis) and according to 27.2.5, the space of coinvariants (AAx <8> AA2 • • • ® AA„)* inherits a natural based module structure (hence has a distinguished basis). 27.3.10. Let us assume, for example, that the root datum is simply connected of type £) m , that n = 2n' and that Ai = A2 = • • • = An = A is such that AA is the standard (2m)-dimensional module. Then we may identify the space of coinvariants (AAx ® AA2 • • • ® AA„)* naturally with the dual space of E n d u ( A f n ). Hence, from 27.3.9, we obtain a distinguished basis

27.3. Tensor Product of Based Modules

223

for the algebra Endu(A® n ), the quantum analogue of the Brauer centralizer algebra. This basis is of the same nature as the basis of the Hecke algebra of type A defined in [3].

CHAPTER

28

Bases for Coinvariants and Cyclic Permutations

28.1.1. In this chapter we assume that (/, •) is of finite type. Let A G m • • • > ^n) I such that S i S i • • • S i is a X+. For any sequence i = ( i i , reduced expression of an element w £ W, we consider the element 0(i, A) = 1

2

N

P r o p o s i t i o n 28.1.2. The element 0(i, A) depends only on w and not on Assume first that w is the longest element in the subgroup of W generated by two distinct elements i,j of I . In that case the assertion of the lemma is the quantum analogue of an identity of Verma, whose proof will be given in 39.3. We shall assume that this special case is known. We now consider the general case. Let i' = (ji, ? • • • ,3N) be another sequence like i (for the same w). To prove that 6{i, A) = #(i',A), we may assume, by 2.1.2, that i' is obtained from i by replacing a subsequence hj, h h • • • ( m consecutive terms) of i by j, z, j, i , . . . , (m consecutive terms), where i,j are as above and m is the order of S{Sj. But this follows imme-

28.1.3. By Proposition 28.1.2, we may use the notation 9(w1 A) instead of P r o p o s i t i o n 28.1.4. The element 0(w,X)~rj\ € AA is the unique element of the canonical basis of A\ which lies in the w(X)-weight space. We prove this by induction on iV, the length of w. If N = 0, there is nothing to prove. Assume now that N > 1. Let ( n , . . . ,ijv) and

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Science+Business Science+Business Media, LLC 2010 2011

224

28.1. Monomials

225

si2si3 * • • siN so that w = Siiiv'. Let bf (resp. b) be the unique element in the canonical basis of A A which lies in the tz/(A)-weight space (resp. the w( A)weight space). Using the induction hypothesis, we see that it is enough to prove that b = F^b'. We have w{A) - s^w^X) = w'(X) - {ii,w'(X))i\ = w'(A) — a\i[ so that f / " 1 V is a non-zero vector in the same weight space as 6; thus, F^l]b' = fb for some / G A - {0}. Next we note that Ei x b' — 0, since the (w'(A) + i'J-weight space is zero. Otherwise, the w/~1(w'(X) + ii )-weight space would be non-zero, hence the (A -f u; /-1 (z / 1 ))-weight space would be non-zero, contradicting the fact that A is the highest weight, since ty / - 1 (i / 1 ) > 0. From the definition of F{ lt it then follows that = F ^ b ' . By the properties of the basis at oo of AA, the previous equality implies that f = c mod v~*A where c is 0 or 1. Hence we have / = c mod v~l7*[v~l] where c is as above and / ^ 0. The involution

> AA keeps b, b' fixed and we have F£x))V

: AA

=

pMy fb = Jb = fb. It follows that / = / ; hence f = c. = FMh>. hence Since / ^ 0 and c is 0 or 1, it follows that / = 1. The proposition is proved. P r o p o s i t i o n 28.1.5. We have

A)) =

A(9(W0,

9(WQ,

—

WQ(X)).

Let i = (ii,«2» • • • i ^ n ) be a sequence in W such that SilSi2 • • • S{N is a reduced expression of Define a 1,02,...,ajv as in 28.1.1. Then i' = • • • s^ is a reduced expression of WQ. (^tsMat-i, .. - , i i ) is such that S{NSIN_1 Let 61,62,..., 6AT be defined by bi = {sh • • •

(iN), -iu 0 (A)), • • • , bN-1 = (sh (i 2 ),

-w0(X)},

bN - (ii, -^o(A)); then 61 = ajv,&2 = g n - i , . . .

= a\. Using 28.1.2, we have

a(e(w0, A)) =
• • • *£">)

ViN-l =

€

L

}

€ T

•••

E

U

N )

=

-^o(A))

=

0(WO,-WO(X)).

The proposition is proved. 28.1.6. We have (a)

S'(e(w0,X)-)

=

(-l){2pA)v-n^^Kua(0(wo,X))~

226

28. Bases for Covariants and Cyclic Permutations

where 2p and n are as in 2.3.1, i with p(i) G Y as in 2.3.2, and

This follows from 2.3.2(a) and 3.3.1(d), applied to x = 0{wOlX). Note that = x G fjy, where i/ is as above and tr v = J2p=i(Sis ''' siP+1(v)> (2p> A). 2 8 . 2 . T H E ISOMORPHISM

P

28.2.1. C o i n v a r i a n t s a n d a n t i p o d e . Let M, M ' be two objects of C and let u E U. For x G M, x' G M', we have (a) ux x' = x ® S(u)x' in the coinvariants (M ® M')*. We may assume that x G M A , x' G M / A . First note that x ® x' = 0 (in the coinvariants) unless A + A7 = 0. We show (a) for u = Ei. Both sides are zero unless A + A' + i' = 0, when we have EiX x' = -

v

f

®

Eix' = —x<8> K-iEix'

= x0

S(Ei)x'

(in the coinvariants). We show (a) for u = Fi. Both sides are zero unless A + A' — i' = 0, when we have Fix <8> x' = - v j ' ! X ) x ® Fix' = - x ® P ^ x ' = x ® S(PI)x'

(in the coinvariants). We show (a) for u = KWe i ^ x x' = v{^x)x

have x', X ® SiK^x'

= v~^x>)x

® x'.

But we can assume that = . Now if (a) holds for u, u', then it also holds for linear combinations of u,u' and for uu'. Indeed uu'x ® x' = w'x 5(w)x / = x 5(w , )5'(w)x / = x 0 S(uu')x' (in the coinvariants). Thus, (a) is proved. An equivalent form of (a) is: (b) S'(u)x ®x' = x® ux' in the coinvariants (M ® M')*.

28.2. The Isomorphism P

227

28.2.2. Let A = AA where A G X+, and let (M,B) be a based module. Let rj = r)\ and let £ be the unique element in the canonical basis of A in the wo(A)-weight space. We define an isomorphism of vector spaces P:AM-*MAby P(x ®y) = (_1)<2M> v -n(C) y 0 ^ for x G Ac and y G M. Here 2p G Y, n : X —> Z are as in 2.3.1. We show that P maps E{( A M ) _ i ' into Fi(M (8) A); F{( A Mf' into Fi(M A); and (A ® M) c into (M A)c for any ( e l . Indeed, if then x G A c , y G M^, and £ + C + = P(Ei{x ® y)) = P(EiX y + = (_l)<2M) v -n(C+i') y 0 ^

Fi?/) +

= ( - 1 )(2p,C)^i
If x G A c , ?/ G M c ', and < + <' ~

(_1)<2p,Cv<-<<<,0/2~n{0E.y
^ + vi-i(iX')/2y

(_l)<2P.O, ; -n(C) F . y 2

0 x +

^

= 0, then

P(Fi(x (8) 2/)) = P(a; ® FiV + v'^^^FiX =

0

0 2/)

(_1)2p,Cv-<-i«,C')/2-n(C-<')y ®

/

= (-l)< P.Oi;-i-<<<,C >/2-n(C-<')(y 0 = (_l)<2M)v-i-i(i,C'>/2-n(C-i')jp.(2/

FiX

+ v-i-W,C)/2F.y

0

x)

0

It follows that P induces an isomorphism of vector spaces P : (A ® M)m —> (M A)*. 28.2.3. Let ~ : M —• M be the associated involution of the based module (M,B). Recall that on A we also have an involution ~ associated with its natural structure of based module. Then A M and M A are naturally based modules with associated involution ©~ (see 27.3.3) and the spaces of coinvariants (A <8>M)* and (M A)* inherit from them structures of based modules (see 27.2.5) with trivial action of U. P r o p o s i t i o n 28.2.4. P : (A M)* = (M A)* is an isomorphism of based modules. The proof will be given in 28.2.8. It will be based on a number of lemmas.

228

28. Bases for Covariants and Cyclic Permutations

L e m m a 28.2.5. For any y G M~w°(x\ P{Z®y)

we have

= 6{w0,-w0(\))

y®r]

(equality in (M <S> A)+). Using 28.1.4, 28.2.1(b), 28.1.6(a), 28.1.5, we have P ( £ ® y) = ( - 1 )<2p,A> v -n(«,o(A)) y

= v -n(«;o(A)) i; -n(A)+c 1 ^ j/ ^( ti;0j = u -n( Wo (A))-n(A)+c 1 +c a ^( 1i;0j

0

£

=

(_ 1 )<2p,A> t; -n(«,o(A)) J/

0

^ ^

x

y ^

—Xf70(A)) —J/ T)

- ^ ( A ) ) " ^ ® T)

(equalities in coinvariants) where c2 = — ' A)/2 = —c\. Note also that -n(w 0 (A) - n(A) = 0 by 2.3.1(b). The lemma is proved. L e m m a 28.2.6. Let be B.

Then

£<8>6 = £ O 6 G A M

and b®ri = b<^rj G M A. From the definitions we see that £ ® b (resp. b ® rj) is fixed by the involution 0 ~ of A (g> M (resp. M ® A). Hence the result follows from the definition of and b< In the following result, B[X]hi, B[X]l° are defined in terms of ( M , B ) as in 27.2.3. L e m m a 28.2.7. There is a unique bijection i?[A] 1 <-» B[\]l° such that the following two conditions for b G B[X]hl,b' G B[X]l° are equivalent: b <-> b'; 0(wo,\)-b-b' G M [ > A]. Replacing M by M[> A], we are reduced to the case where M = M[> A] (see 27.2.4(a)). Then replacing M by M/M[> A], we are reduced to the case where M = M[A] (see 27.2.4(b)). Using 27.1.7, we are reduced to the case where (M, B) is A with its canonical basis. In this case, we have B[X]hi = {rj} and B[X]l° = {£} and the result follows from 28.1.4.

28.2. The Isomorphism P

229

28.2.8. P r o o f of P r o p o s i t i o n 28.2.4. Let B'^ (resp. ) be the basis of A ® Af (resp. M A)) defined as in 27.3.3, in terms of the based modules (A, P i ) and (M, P ) . Here P i is the canonical basis of A. Let 7r : A ® M — (A ® M)* and 7r' : M A —• (M ® A)* be the canonical maps. We know that TT defines a bijection of P^[0] onto a basis (P^)* of (A M)* and TT' defines a bijection of Pj>[0] onto a basis (P A)*. Let b G (P^)*. Let 6 be the unique element of 0] such that 7T(6) = b. + By 27.3.8, there exists A' G X and elements bx G Pi[-iuo(A')] i o , b2 G B[X']ht such that b — b\^b2. In A, we have that Pi[—wo(A')] is empty unless — wo(X') = A and Bi[X]l° = {£}. Thus we have b = £0^2 where b2 G P[—iwo(A)]fcl and by Lemma 28.2.6, we have b = £ b2. By Lemma 28.2.5, we then have P(b) — Q(WO, — M0(X))~b2 ®R] modulo the kernel of TT'. By Lemma 28.2.7, we can find an element b'2 G B[—wo(A)]io such that 0(wOt —wq(X))~b2 - b'2

G M[>

-w0(X)].

Then we have P(b) = b'2r} modulo the kernel of 7r'. Note that M[> -iw0(A)l (8> A is contained in the kernel of TT' . By Lemma 28.2.6, we have b'2 rj = b'2<}r]. By 27.3.8, we have that B2<>V € 0]. It follows that Tr'(P(S)) belongs to 7r(££[0]) - (££)*. We have therefore proved that P maps (P^)* into (P^)*. The proposition is proved. 28.2.9. Let Ai, A2,..., An be a sequence of elements of X+. As in 27.3.9, the space of coinvariants (AAl ® AA2 * • -<8> A An )* has a natural based module structure (hence has a distinguished basis). This last based module has the following property of invariance by a cyclic permutation: there is a natural isomorphism ( A A L <8> A A 2 • • • <8> A A J *

( A A 2 <S> A A s • • • A\n

A A L ) *

induced by the map xi x2 • • • xn ^ (~l)(2pxi)v-n(Ci)X2

0

X3

... 0

Xn


where xp G A a ^. This isomorphism maps the distinguished basis onto the distinguished basis (see 28.2.4). If we compose the n iterates of this isomorphism, we get the identity map of (AAl ® AAz • • • ® A An )*, since we may assume that Ci + C2 + • • • + Cn = 0.

A Refinement of the Peter-Weyl Theorem

29.1.1. In this chapter we assume that (/, •) is of finite type. Let (3 be an element in the canonical basis B of U. We associate to f3 an element Ai G as follows. We have /3 G U l ^ for a unique £ G X. Choose A, A" G X + such that A" — A = £ and such that (i, A) is large enough for all i. Then /?(£-A ® is in the canonical basis B of ^AA ® AA", and by 27.2.1, it belongs to B[Ai] for a unique Ai G X+. We want to show that Ai depends only on /3, and not on the choice of A, X". It is enough to show that, if A, A" are replaced by A + A', A' + A", then the procedure above leads again to Ai. This follows from 27.3.5. Thus we have a well-defined map B —> X+ (/? • Ai). We shall write B[Ai] for the fibre of this map at Ai. 29.1.2. For any Ai G we denote by U[> Ai] (resp. U[> Ai]) the Q(t>)-subspace of U spanned by U ^ ^ A x B ^ ] (resp. by • 1B[A2]). L e m m a 29.1.3. The following conditions for an element u G U are equiv-

(c) for any object M G C of finite dimension over Q(f) and any vector (d) if X2 G X + and u acts on A\2 by a non-zero linear map, then X2 > The equivalence of (a) and (b) is clear from the definition. The equivalence of (c) and (d) follows by expressing M in (c) as a direct sum of simple objects. Clearly, if u satisfies (c), then it satisfies (b). Conversely, assume that u satisfies (b); we show that it satisfies (c). We may assume that m is in a weight space of M. By 23.3.10, we can find A, A" G and obviously have / ( w A a ® A A ")[> ^i] C M[> Ai]. Since (b) holds for u, it follows that um = uf{^-X ® ?7 ") = / ( ^ ( ^ - ® V\")) ^ M\> Ai]. Thus the equivalence of (b),(c) is established. The lemma is proved.

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_15, © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010

230

29.2. The Finite Dimensional Algebras U / U [P]

231

L e m m a 2 9 . 1 . 4 . The following conditions for an element u G U are equivalent: (a) u G U [ > AX]; (b) for any A , A " e I + we have X®

€ ( W A A ® A A " ) [ > AI].

(c) for any object M G C of finite dimension over Q(v) and any vector m G M, we have um G M[> AI]; (d) if

G X+ and u acts on A\2 by a non-zero linear map, then A2 >

AI.

This follows from the previous lemma or can be proved in the same way. L e m m a 2 9 . 1 . 5 . Let Ai G X+. The subspaces U[> Ai] and U[> Ai] of U are two-sided ideals. Hence U[> Ai]/U[> Ai] is naturally a U-bimodule. This follows from the descriptions 29.1.3(c), 29.1.4(c) of U[> AI] and U [ > AI].

29.1.6. The U-module structure on AAl gives us a homomorphism of algebras U —> End(A Al ). This restricts to a homomorphism of algebras (without 1) U[> Ai] —• End(A Al ) whose kernel is, by Lemma 29.1.4, exactly U[> Ai]; hence we have an induced homomorphism of algebras (a) U[> Ai]/U[> Ai]

End(A Al ) which is injective.

In particular, we have (b) dim(U[> Ai]/U[> Ax]) < 00, or equivalently, (c) B[Ai] is a finite set for any Ai G X+. 2 9 . 2 . T H E F I N I T E DIMENSIONAL ALGEBRAS U / U [ P ]

29.2.1. Let P be a subset of X+ with the following two properties: (a) if A G P and A' G X+ satisfies A' > A, then A' G P; (b) the complement of P in X + is finite. Note that such P exist in abundance. We denote by U[P] the subspace of U generated by UA6pB[A]. From Lemma 29.1.5, we see that U[P] is a

232

29. A Refinement of the Peter- Weyl Theorem

two-sided ideal of U, and from 29.1.6(c), we see that the algebra U/U[P] is finite dimensional. Note that this algebra has a unit element (unlike U). Indeed, since l c G B for all ( 6 I , we have that l c G U[P] for all but finitely many Then which is not meaningful in U, is meaningful in U / U [ P ] and is the unit element there. Let A G X + — P . We show that U[P] acts as zero on the U-module AA- Indeed, let /3 be an element of B fl P (these elements span P.) We have /3 G B[A'] for some A' G P . If the action of (5 on A A were non-zero, then from Lemma 29.1.3, it would follow that A > A'; using the definition of P , it would follow that A G P , a contradiction. We have proved that U[P] acts as zero on AA, hence AA may be regarded as a U/U[P]-module. This module is simple. Indeed, even as a U-module it has no proper submodules. It is clear that for A ^ A' in X+ — P , the U/U[P]-modules AA,AA' are not isomorphic (they are not isomorphic as U-modules). By the standard theory of finite dimensional algebras, it follows that dim(U/U[P]) > ^ ( d i m A A ) 2 A

(sum over all A G X+ — P). On the other hand, by 29.1.6(a), we have dim(U/U[P]) = ] T u [ > A]/U[> A] < ^ ( d i m A A ) 2 A

A

(both sums over all A G — P). Comparing with the previous inequality, we see that dim(U/U[P]) = ^ ( d i m A A ) 2 A

and dimU[> A]/U[> A] = (dimA A ) 2 for any A G X+ — P. This implies the following result. P r o p o s i t i o n 29.2.2. (a) The algebra (with 1) U / U [ P ] is semisimple and a complete set of simple modules for it is given by AA with A G X+ — P. (b) For any A G X+, the homomorphism U[> A]/U[> A] —• End(A\) (see 29.1.6(a)) is an isomorphism. Actually, we get (b) for A G X+ — P; but for any A G X+ we can find P as above, not containing A.

29.3. The Refined Peter-Weyl

Theorem

233

29.2.3. From the definition, we see that the finite dimensional semisimple algebra U / U [ P ] inherits from U a canonical basis, formed by the non-zero elements in the image of B. 2 9 . 3 . T H E REFINED P E T E R - W E Y L T H E O R E M

L e m m a 29.3.1. Let A G X+. (a) The anti-automorphisms U [ > -wo( )}.

S, 5',cr : U —• U carry U[> A] onto

(b) The automorphism u : U —> U carries U[> A] onto U[> — wo(A)]. Ay are such that Let u U[> A]. Assume that A' X+ and m S(u)m ^ 0. The map 6y : "Ay ® Ay -> Q(v) (see 25.1.4) may be considered as a non-degenerate pairing; hence there exists m! "Ay such that 8y(m' ® S(u)m) ± 0. Using 28.2.1, we see that Sy(m' ® S(u)m) = 6y(um' m); hence um' / 0. Since "Ay = A_ Wo (a'), we see using 29.1.3, that — wo(X') > A, or equivalently, that A' > — ^o(A). Using again 29.1.3, we deduce that S(u) U[> — wo(A)]. An entirely similar proof shows that S'(u) U[> —i«o(A)]. Thus ( U [ > A]) U[> -w0(\)] for all A and S'(;U[> -^o(A)]) C U[> A] for all A. Since SS' = S'S = 1, the assertions about S and S' in (a) are proved. The assertion about a follows from the assertion for 5, using 23.1.7 and the fact that U[> A] is generated by elements in B which are contained in the summands of the decomposition 23.1.2 of U. We prove (b). Let u G U[> A]. Assume that G X+ and ra G Aa' are such that a ) ( u ) m ^ 0. Then um ^ 0 in "'Aa' which is isomorphic to A_u,0(a')5 hence, by 29.1.3, we have WQ > A or equivalently, Using again 29.1.3, we deduce that u(u) G U[> — woA]. Thus, U( [> A]) C U [ > -w Similarly, w(U[> - ^ o ( A ) ] ) C U [ > A]. The lemma follows. 29.3.2. We shall use the following terminology: an element (3 G B is said to be involutive if au;(/3) = ±{3. (Recall that era; = ua maps B to ± B . ) The following theorem is, in part, a summary of the results above. T h e o r e m 29.3.3. Given X G X+, we define U[> A] (resp. U[> A]j as the set of all u G U with the following property: if X' G X+ and u acts on Ay by a non-zero linear map, then X' > X (resp. X' > X). (a) U[> A] and U[> A] are two-sided ideals of U, which are generated as vector spaces by their intersections with B . The quotient algebra

234

29. A Refinement of the Peter- Weyl Theorem

U[> A]/U[> A] is isomorphic (via the action of U on A\) to the algebra End(A\); in particular, it is finite dimensional and has a unit element, denoted by 1A- Let n : U[> A] —• U[> A]/U[> A] be the natural projection. (b) There is a unique direct sum decomposition of U[> A]/U[> A] into a direct sum of simple left U-modules such that each summand is generated by its intersection with the basis 7r(B[A]) of U[> A]/U[> A]. (c) There is a unique direct sum decomposition of\J[> A]/U[> A] into a direct sum of simple right U-modules such that each summand is generated by its intersection with the basis 7r(B[A]) of U[> A]/U[> A]. (d) Any summand in the decomposition (b) and any summand in the decomposition (c) have an intersection equal to a line consisting of all multiples of some element in the basis 7r(B[A]). This gives a map from the set of all pairs consisting of a summand in the decomposition (b) and one in the decomposition (c), to the set 7r(B[A]). This map is a bijection. (e) Each summand in the decomposition (b) and each summand in the decomposition (c) contains a unique element of the form 7t(/3) where (3 G B[A] is involutive. (f) Let b,bf G B[A]. There exists b" G B[A] and Cb,b',b" £ A such that bb' = c6f6',b"&" mod U[> A]. (a) has already been proved, (b) follows from the definitions, using 27.1.7, 27.1.8 with M = (UAX, ® A A ")[> A]/(WAa' ® A a «)[> A] and with Ai = A (for various V, A" G X+). (c) follows from (b) using the anti-automorphism a u = uj<j of U which maps B into itself, up to signs, (see 26.3.2) and maps U[> A] and U[> A] into themselves (see 29.3.1). We prove (d). The two subspaces considered in the first sentence of (d) are a minimal left ideal and a minimal right ideal in the algebra U[> A]/U[> A] which has 1, and is finite dimensional and simple (by (a)). Their intersection is therefore a line. Since both these subspaces are spanned by a subset of the basis 7r(B[A]), the same is true about their intersection, and the first assertion of (d) follows. The map in the second sentence of (d) is obviously surjective. It is a map between two finite sets of the same cardinality (dim AA)2 (see 29.2.2); hence it is a bijection. We prove (e). Let G b e a summand in the decomposition (b). The map aw : U —• U induces an involution t of the vector space U[> A]/U[> A]. The image of G under i is a summand in the decomposition (c), which by (d) intersects G in a line spanned by a vector in 7r(B[A]). This line is necessarily stable under L (since t is an involution); hence our vector in this

29.4-

Cells

235

line is preserved up to a sign by i. This proves (e) as far as G is concerned. The same proof applies to summands in the decomposition (c). We prove (f). The product 7r(6)7r(6/) is in the intersection of the left ideal of U[> A]/U[> A] generated by ir{b') with the right ideal generated by 7r(b), hence, by (d), is of the form ,b"7r(b") for some b" G B[A] and some Cb,v,b" £ Q(v), which is necessarily in A since the structure constants of the algebra U with respect to B are in A. The theorem is proved. 29.4.

CELLS

2 9 . 4 . 1 . The subsets B[A] (for various A G are called two-sided cells; they form a partition of B. For each A, the two-sided cell B[A] is further partitioned into subsets corresponding to the bases of the various summands in the decomposition 29.3.3(b) (these are called left cells) and it is also partitioned into subsets corresponding to the bases of the various summands in the decomposition 29.3.3(c) (these are called right cells). Then 29.3.3(d) asserts that any left cell in B[A] and any right cell in B[A] have exactly one element in common; 29.3.3(e) asserts that any left cell and any right cell contain exactly one involutive element. Since the number of left cells (or right cells) in B[A] is dimA A , it follows that the number of involutive elements in B[A] is also dimA A .

29.4.2. Let A be an associative algebra over a field K with a given basis B as a K-vector space. We do not assume that A has 1. The structure constants Cb,b',b" £ K oi A (where b, b', b" G B) are defined by bb' = c b,b',b"b". Generalizing the definition of cells in Weyl groups (which goes back to A. Joseph), we will define certain preorders on B as follows. If 6, b' G B, we say that b' {,a,6a+1 ^ 0 (resp. Cbaipatba+1 0) for s = 1 , 2 , . . . , n— 1. We say that b'
236

29. A Refinement

of the Peter- Weyl

Theorem

coincides with that given 29.4.1. (This can be easily checked.) The involutive elements in Theorem 29.3.3 are the analogues of the Duflo involutions from the theory of cells in Weyl groups. 2 9 . 4 . 3 . We will describe explicitly the two-sided cells of B in the simplest case where I = {i} and X = Y = Z with i = 1 G Y, i' = 2 G X. (See 25.3.) For each n > 0, we consider the subset

6 ( n ) = { ^ a ) l _ n J F ; ( 6 ) ; n > a + b} U {F^ lnE$a) ;n > a + 6} of the canonical basis B (with the identification 25.3.1(c)). Note that 6(N) consists of (n + 1 ) 2 elements. The product of two elements of 6 ( n ) is given by the following equalities (modulo a linear combination of elements in 6(n + l ) U 6 ( n + 2 ) U - ) : E\a)\.nFld) E\ah^nFiu>Eri-nFr

M) = <

iib = c,n>a

Fln~a} 1nE\n~d)

+d

\ib = c,n
+d

I\ 0 if b ' ^ c E\a)l^nFln-d)

if b + c =

F^n-a)lnE^d)

n,d>a

if b + c =

n,d
0 if b + c ^ n F(d)lnE(a)

F ^ l ^ F ^ U E ^ =<

H

b = :

n b E[n-d)\.nF^n-a) 0 if b / c (a)UE(n-d)

(d)

F ^ L N E ^ E F H - N F ^

=

E

,(n—a)

1 -nFj;d)

^

n

>

a

+

d

iib = c,n
if6 + c =

n j d

+d

>fl

ii b + c = n, d < a '

0 if b + c ± n Hence 6 ( n ) are the two-sided cells. The involutive elements in 6 ( n ) are (o) 1 rp(a) E\a)\^nFla} Aa) with a > 0,2a < n and with a > 0,2a < n, with (a) _ r W a) a ) if % _ , the identification = F l \ n E \ if 2a = n.

29.5. The Quantum Coordinate 29.5.

Algebra

237

T H E QUANTUM COORDINATE ALGEBRA

2 9 . 5 . 1 . Let O be the vector space of all Q(v)-linear forms / : U —> Q(V)

with the following property: / vanishes on U[> A] for some A G X+. If all a G B, then the linear form a : U —• Q(t>) given by a(a') = 8a,afor a G B, belongs to O and {a|a G B} is a basis of O. This follows from 29.3.3. We define an algebra structure on O by the rule ab = Xlc7™?6^, where c runs over B and rh"b are as in 25.4.1. The previous sum is well-defined: all but finitely many terms are zero. This product is associative by 25.4.1(b). c The linear map A : O —» O ® O given by A(c) = b m aba ® b (with c m ab as in 25.4.1) is well-defined. All but finitely many terms in the sum are zero. This map is called comultiplication. It is coassociative by 25.4.1(a) and it is an algebra homomorphism by 25.4.1(c). The element lo is a unit element for this algebra. Consider the linear function O —• Q ( f ) which takes a to 1 if a = 1a for some A G X, and otherwise, to zero. This is an algebra homomorphism. Thus O becomes a Hopf algebra called the quantum coordinate algebra. It is easy to see that this definition is the same as the usual one. We call {a|a G B} the canonical basis of O. 29.5.2. Let ^ O be the ,4-submodule of O generated by the basis (a). Since the structure constants are in A, it follows that ^ O inherits from O a structure of Hopf algebra over A. If now R is any commutative ^-algebra with 1, we can define a Hopf algebra over R by ^ O = R®a (AO).

The Canonical Topological Basis

30.1.

T H E DEFINITION OF THE CANONICAL TOPOLOGICAL

BASIS

In this chapter we assume that (/, •) is of finite type. We denote by (U~ ® U+)* the closure of U~ (8) U+ in (U U)f (see 4.1.1). The elements of (U~ ® U + )f are possibly infinite sums of the form Ylb b'eB cb,b'b~ ® with c^b' £ I n this chapter we shall construct + a canonical topological basis of (U~ (g>U )T which gives rise simultaneously to the canonical bases of all tensor products of type A a <8> w Av. Let ~ : (U~ & > (U~~ ® U + )f be the ring involution defined as the unique continuous extension of ~ ® ~~ : U~ U + —> U~ (g> U + . Note that we have G E (U~ U + J (see 4.1.2). By 24.1.6, we can write uniquely 30.1.1.

30.1.2.

From 4.1.2, 4.1.3, it follows that the Q-linear map

given by >P(x) = Ox (product in (U~ U + J) satisfies \I>2 = 1. We clearly have ^ ( / x ) = f*f?(x) for all / E A and all x. Hence if we set

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © © Springer Springer Science+Business Science+Business Media, LLC LLC 2010 2011

238

30.1. Definition

of the Canonical

Topological

Basis

239

The last sum is finite by the previous statements. Applying 24.2.1 to the set H = B x B we see that there is a unique family of elements PbiMfaM £ Z[F -1 ] defined for bi,b'1,b2,b'2 G B such that Pbifi'^biM — 1; PbxKM

€ v~1Z[v~1] if (61,14) +

(hMY,

Pbub'i;b2,b'2 = 0 unless ( 6 1 , b [ ) < (b2,b'2); PbuKfaM = 52b3,b'3Pbi,b>1\b3,b'3rb3,b'3]b2,b!2 for all (bijb'i) < (b2,b2). Thus we have the following result. P r o p o s i t i o n 30.1.3. For any G B x B, there is a unique element + G (U~U y such that Qf3bl,b\ = Pbxft and such that fo^b^-hi ®b[+ is an (infinite) linear combination of elements b2 b2+ with (b2,b2) > (bi,b\) and with coefficients in v~1Z[v~1]. We have /?6l)6> = E b ^ P & i . & i ®

b +

2 -

30.1.4. The elements /?6l>6/ G (U~ 0 U+)T, for various (61, b[) G B x B, are said to form the canonical topological basis of (U~ U + )f. This is not a basis in the strict sense. = where Taking 61 = b'x = 1, we obtain an element T = Tu G U~ U+ for all v and (a)

r 0 = 1 ® 1.

Hence T is an invertible element of (U~ U + )f. By definition, we have GT = T; hence (b)

e = rf~1.

Note also, that if v ^ 0, then Tu is a linear combination of elements b~ bf+ (b,b' G B„) with coefficients in v - 1 Z[u - 1 ]. This property, together with (a),(b), characterizes T. By the general construction in 27.3.2, the 30.1.5. Let A, A' G X+. U-module A\ has a canonical basis B I t consists of elements (b-rj\)0(V+Z-\>) for various 6 G B(A) and b' G B(A'). Note that um is a well-defined element of A A "Ay, for any u G (U~ U + J and any m G A\ "Ay, by regarding the last space as a U
30. The Canonical Topological Basis of (U

240

U + )

P r o p o s i t i o n 30.1.6. Let b,b' G B. (a) Ifb G B(A) andb' G B(A'), then pb,b>(rix®Z-x>) = (b) If either b £ B(A) or b' <£ B(A'), then (3b,b'(r)\ <8>

(b~Vx)0(b,+^-y). = 0.

This follows immediately from the definitions and from 27.3.2. 30.1.7. Example.. Assume I = {i} and X = Y = Z with i — 1 G Y, i' — 2 G X. The canonical topological basis of (U~~ U + )f consists of the elements {c>d>

0)

and

with the identification xCfd = yc^d for c — d. 30.2.

O N THE COEFFICIENTS

P^^-M^

30.2.1. The canonical topological basis in the previous section is completely determined by the set of coefficients P b i ^ M ^ £ Z[i? -1 ] defined for all 6i, 62, b'2 in B. In this section we make a proposal for a possible topological interpretation of these coefficients, assuming that the Cartan datum is simply laced (of finite type). We shall assume that (61, b[) < (b2,b'2); otherwise the coefficient is zero. 30.2.2. Let (I, H,...) be the graph of (/, •) (see 14.1.3); note that 1 = 1. Assume that we have chosen an orientation for this graph. According to 14.5.1, to give bi,b' 1 ,b2 J b 2 in B is the same as to give four objects of GVl,GV;,GV2,on V i , V i , V 2 , V^ of V and orbits 01,0'1,02,0,2 E v 1 , E v ' 1 , E v 2 , E v ^ , respectively (notation of 9.1.2). Hence we may write Vox ,o; ; o 2 ,o'2 instead of pbl >6/ ;i)2 ^ . Let V = V 2 © V2 G V and let x G E v be an element such that V2 and V2 are x-stable and the restriction of x to V2 (resp. V2) is in 02 (resp. in Ol2). Let J be the stabilizer of x in G v and let Z be the J-orbit of V 2 in the variety of all I-graded subspaces of V. Note that V 2 is a point of Z and that any W G Z is x-stable. Let Z' be the subvariety of Z consisting of all subspaces W G Z such that (a) W fl V 2 = V i (in V); (b) V / ( W + V 2 ) = V^ (in V);

30.2. On the Coefficients

Pbi^bi^

241

(c) the element of Ewnv 2 defined by x corresponds under an isomorphism as in (a) to an element of Oi C Ev^; (d) the element of Ev/(w+v 2 ) defined by x corresponds under an isomorphism as in (b) to an element of 0'x C E v ; . One can hope that the coefficients of the various powers of v in P0i,0i;0 2 ,0^ a r e equal to the dimensions of the stalks of the intersection cohomology complex of the closure of Z' in Z at the point V2 E Z, and that they are zero if V2 is not in that closure.

Notes on Part IV 1. The algebra U has appeared in [1], in a geometric setting (in type An), but its definition in the general case is the same as that in type An. One of the main results of [1] was a topological definition of a canonical basis of U (in type An), generalizing the author's definition of the canonical basis of f. The method of [1] works in almost the same way for affine type An, but the extension to other types remains to be done. 2. In [2], Kashiwara conjectured the existence of a canonical basis of U , for Cartan data of finite type, and constructed a basis of the quantum coordinate algebra O in which the structure constants were in Q [v^v"1]; this is presumably the same as the basis in 29.5.1, in which the structure constants are in Z [ v ^ 1 ] . 3. The definition of the canonical basis B of U , in the general case was given in [6]. Most results in Chapters 24 and 25 appeared in [6]. 4. Something close to Lemma 24.2.1 has been used in [3] to attach a polynomial to two elements of a Coxeter group. 5. Expressions like those in 25.3.1 appeared in [2], and are implicit in [1]. 6. I do not know what is the relation, if any, between the form ( , ) on U , in 26.1.2, and the form on U defined in [7]. 7. Propositions 27.1.7, 27.1.8 (and also results similar to 27.2.4) appear in [2]. 8. Theorem 27.3.2 is similar to results in [6]; the analogous result for more than two factors (see 27.3.6) is new. 9. The existence of a canonical basis on the space of coinvariants {A\l A\2 ... ® Aa„)* (see 27.3.9) is new; it was known earlier for n = 3, see [5]. It implies that the corresponding space of coinvariants over A is a free ^4-module; this answers a question of D. Kazhdan. (There is a somewhat analogous result about the space of "coinvariants in the tensor product" (see [4]) of several Weyl modules with the same negative central charge over an affine Lie algebra: this space has dimension independent of the central charge. There are other analogies between the two theories, for example the invariance property under cyclic permutations (28.2.9) has a counterpart in the theory over affine Lie algebras.) 10. The results in Chapters 28, 29 and 30 are new. 11. The positivity conjecture in 25.4.2 is made plausible by the results in [1].

References

243

REFERENCES 1. A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of GLn, Duke Math. J. 61 (1990), 655-677. 2. M. Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), 455-487. 3. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. 4. , Tensor structures arising from affine Lie algebras, parts I and II, J. Amer. Math. Soc. 6 (1993), 905-947, 94&-1011. , Tensor structures arising from affine Lie algebras, parts III and IV, J. Amer. Math. Soc. 7 (1994). 5. G. Lusztig, Canonical bases arising from quantized enveloping algebras, II, Common trends in mathematics and quantum field theories, (T. Eguchi et. al., eds.), Progr. Theor. Phys. Suppl., vol. 102, 1990, pp. 175-201. 6. , Canonical bases in tensor products, Proc. Nat. Acad. Sci. 89 (1992), 81778179. 7. M. Rosso, Analogues de la forme de Killing et du theoreme d'Harish-Chandra pour les groupes quantiques, Ann. Sci. E. N. S. 23 (1990), 445-467.

Part V CHANGE OF RINGS

Let R be a commutative .A-algebra with 1. The main topic of Part V is the i?-algebra RU, obtained from ^ U by tensoring with R over A, and its modules. Chapter 31 contains a general discussion of and its module category. In Chapter 32, assuming that the Cartan datum is of finite type, and that a certain root of v is given in R, we show that the integrable modules of ftU form a braided tensor category. In Chapter 33 we consider the specialization v = 1 and we establish the connection with Kac-Moody Lie algebras. Chapters 34, 35, and 36 are concerned with the case where v is a root of unity in R. In Chapter 34 we establish various properties of Gaussian binomial coefficients at roots of 1. In Chapter 35 we construct a quantum analogue of the Frobenius homomorphism (under some rather mild assumptions). This includes as a special case the classical Frobenius homomorphism over fields of positive characteristic and also the exceptional isogenies (in small characteristic) defined by Chevalley [1]. In Chapter 36 we study the Hopf algebra which in some sense, is the kernel of the Frobenius homomorphism. This algebra is finite dimensional if R is a field and the Cartan datum is of finite type.

CHAPTER

31

The Algebra ^U

31.1.1. Prom now on, R will be a fixed commutative ring with 1, with a given invertible element v. We shall regard R as an ^4-algebra via the ring

We have a direct sum decomposition R{ = ®„(i*fi/) where v runs over N[J] and RIv = R (gu CA^). The canonical bases B and B of ^ f , ^ U give rise to R-bases of , consisting of elements 1 (g> 6 where b is in B or B; we write b instead of 1 0 b. In particular the elements 1a G RU are well-defined for all A G X. They satisfy as in U, 1a 1A' = 5A,A'1AThe structure constants rncah^m^b of U (see 25.4.1) can be regarded as R. The identities elements of R via the ring homomorphism (j> : A The comultiplication of ^ U (a collection of maps as in 23.1.5) is defined ( -4 f opp )-module structure (x ® xf) : u x+ux'~ on 31.1.2. The A f ^ U , by change of scalars, induces a ^f <S>R (#f opp )-module structure on denoted in the same way. Similarly, the ^ f f opp )-module structure (x<8>xf) : u 1—• x~ux,Jt on ^ U , by change of scalars, induces a R{^R(R{° module structure on R\J denoted in the same way.

PP

(a) the elements b+l\b'~

(b,b' G B,A G X) form a basis of the R-

(b) the elements b~l\b'+

(6,6' G B,A G X) form a basis of the R-

(c) the i?-algebra R\J is generated by the elements E^LX,

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © Springer Science+Business Science+Business Media, LLC 2010 2011

for

245

)-

246

31. The Algebra R\J

31.1.3. We will give an alternative construction of in terms of Let RA be the algebra generated by the symbols ,x~L^x'+ with x £ i?f/y, x' G flfi/', for various i/, i/, and £ G X; these symbols are subject to the following relations: (fl{'>)+lc(tfW)- = (0f>)-lc+oiW(0,
(0<°>)+l_ c (0f>)-= £ > ( t> o

0" i

a + b- (z,C) t

t>0

x+l^ =

~a + b-(i, t

= l£_„x~ for x G f„;

(x+l c )(l c >x'~) = <5C)C/X+1CX'-, ( x ~ l c ) ( l c / x ' + ) =

lcx'+;

(x+l c )(l c >x'+) = <5 C)C ,l c+l/ (xx') + , ( x - l c X l c ^ 7 - ) =

if

x G fift,; (rx + r'x')+l^ if x, x 7 G

= r x

+

+ r ' x / + l < ( r x + r ' x ' ) ~ l £ = rx~l<; -I- r ' x / _ l £

and r, r' G i?.

If x or x' in x + l ^ x ' - or x ~ l ^ x / + is 1, we omit writing it. We have an obvious surjective i2-algebra homomorphism RA —-> Using the relations of RA, we easily see that the symbols x + l ^ x / - generate RA as an .R-module. In other words, the elements b+l^bf~, with b,b' G B and ( E l , generate RA as an -R-module. Since they form an R-basis of RXJ, they must also form an R-basis of RA and we deduce that: (a) the natural algebra homomorphism RA —> RU is an isomorphism. 31.1.4. There is a natural i?-linear involution cr : R{ —> it is given by a change of rings from the analogous involution for R = A, which is the restriction of a : f —> f. The automorphism u : U —> U restricts to an automorphism u : ^ U —> ^ U ; tensoring with R, we obtain an i?-algebra automorphism u : RU —> KU.

31.1.5. As in 23.1.4, we say that a ^U-module M is unital if (a) for any m G M we have \\m = 0 for all but finitely many A G X; (b) for any m G M we have XIagx

=

m

-

31.1. Definition of R\J

247

We then have a direct sum decomposition (as an abelian group) M = where Mx — 1\M; we can regard M as an i?-module by rm = ®\exMx ^ A ( r l A ) ( m ) for r G R,m G M. Then the decomposition above is as an /2-module. The unital # U-modules are the objects of an abelian category RC with the morphisms being homomorphisms of ^ U-modules. 31.1.6. Let M G RC, let i G I and let n € Z. We define iMinear : M ^ M by E\n)m = E A C ^ 1 * ) ™ and maps E\n) : M 1 F^m = E A C ^ * ) ™ for all m G M. (Recall that E\n) 1 a and i^ ( n ) l A are elements of B C U, hence are well-defined in /?U.) It follows immediately : M —> M) and 0?n) ^ (*\ : M from the definitions that D\N) H-> i > x+m and M) define two Rf-module structures on M, denoted by x,m — x, m H-> x~m respectively. We have (a) E\n)Mx

C Mx+ni\

F^n)Mx

C Mx~ni'

for any i G I,n G Z and

x e x. Moreover, for any £ G X and any m G (b) E^F^m

= F^E^m

(c) E ^ F ^ m =

E

t

we have

if i ± j; >

0

^ r ^ ^ F ^ E ^ m ,

(d) F ^ E ^ m = E t > o H [ ~ a + b 7 { i , ° ] i ) E i a ~ t ) F ? ~ t ) m ' 31.1.7. Conversely, let M be an i2-module with a given direct sum decomand given /^-linear maps E\N),FLN) (for position M = (B^ZXM^ i G J , n G Z) satisfying 31.1.6(a)-(d) and such that E\N) = F^N) = 0 for {E\U) : M - M) and 0 a?+m and x, 771 '—• x~m, respectively. Then this structure comes from a well-defined structure of unital rU-module on M. Indeed, it is clear that this structure gives an / ^ - m o d u l e structure on M hence a ^U-module structure (see 31.1.3). 31.1.8. Let M, M' G RC. The tensor product M ® R M ' (as iZ-modules) will be regarded as a RU-module by the rule c(x {'^lcb)ax ® bx'. (All but finitely many terms in the last sum are zero.) The fact that the rule above defines an # U-module structure follows from the identity 25.4.1(c). This U-module is unital, by the identity 25.4.1(d). Thus M ®R M' is naturally an object of RC. Now let M, M', M" be three objects of RC. By the previous construction, the H-module M <S>R M' ®R M" can be regarded as an object of RC in two

248

31. The Algebra R\J

ways, (M ®R M') ®R M" and M (Mf ®R M"). In fact these two ways coincide; this follows from the identity 25.4.1(b). 31.1.9. From the definitions it is clear that, in the case where R = Q(v),v = v, we have RC = C and the tensor product just defined coincides with the one introduced earlier for C. 31.1.10. To any object M of RC, we associate (as in 3.4.4) a new object of RC as follows. u M has the same underlying jR-module as M. By definition, for any u G /{U, the operator m on W M coincides with the operator aJ(U) on M. 31.1.11. If R' —R is a homomorphism of commutative *4-algebras with 1, we have ^ U = R R> ( k ' U ) and for any object M G R>C, we may regard R ®R> M naturally as an object in RC with the induced # U-module structure. This gives a functor R>C —> RC called change of rings, or change of scalars. It commutes with tensor products (as in 31.1.8) and with the operation u in 31.1.10. 31.1.12. Let (Y',X',...) be another root datum of type (/, •) and let / : Y' —> y, g : X —> X' be a morphism of root data. This induces a homomorphism : U 7 —• U between the corresponding Drinfeld-Jimbo algebras (see 3.1.2). For each <' G X' and £ € X such that g(() = f let A<j) : ^U'l^/ = ^ U l ^ be as in 23.2.5. By tensoring with R this gives rise to R<j>: R U'1 C , = flUl^- Let M be a unital R U-module. We can regard M as a unital /JU'-module by the following rule: if m G Mt and u G FLU'L^' then um is defined to be ( R 4>(u))m if £ = g(C), and 0, otherwise. This gives a functor from unital R U-modules to unital ^U'-modules. 31.1.13. Let A G X. The ,4-submodule j^M\ of the Verma module M\ is a unital ^U-submodule (see 23.3.2); by change of scalars, it gives rise to an object RM\ of RC, called an R- Verma module. We have an exact sequence in RC: ei,n>oUUlA+ni') ->

flUlA

MX

0,

where the first map has components given by right multiplication by 1 x + n i ' E ^ and the second map is given by u — i > ul (1 is the canonical generator of M\). This is deduced by tensoring with R from the analogous exact sequence over

31.2. Integrable RU -modules

249

Let M e RC and let M e Mx be such that E\n)m = 0 for a l M e l and all n > 0. From the previous exact sequence, we see that there is a unique morphism t : M\ —• M such that t(l) = ra. 31.2.

INTEGRABLE

/JU-MODULES

31.2.1. In this section we assume that the root datum is y-regular (except of A v is a unital ^ U in 31.2.4). Let A, A' e X+. The ,4-submodule submodule (see 23.3.7); by change of scalars, it gives rise to an object RAX>

— R <8>A U A

v

)

of

RC.

Similarly, the .4-submodule ^AA <8a (^A^O of "Ax <S> AA' is a unital .4U-submodule (see 23.3.9), in fact a tensor product in ^C; by change of scalars, it gives rise to the object ^AA ®R (HAA') of RC. Let ( = X' — X e X. Consider the following morphisms of RU-modules -ni')) © (®i,n>(i,A)

4 -I

R^X®R(RAV)

1 0 where / has components given by right multiplication by l x - n i ' F ^ (in the first group of summands), l x + n i ' E ^ (in the second group of summands) and 7r(u) = w(£_A <8> Vx') • We write £_A instead of 1 £-A and similarly for r]x>. P r o p o s i t i o n 31.2.2. The sequence above is exact. When R is A and v = v, this is a restatement of 23.3.8. The general case follows from this by taking the tensor product with R, by the right exactness of tensor products. We can state the previous proposition in the following equivalent form. C o r o l l a r y 31.2.3. n is surjective and its kernel is the left ideal £

i,n>(i,A')

HUFF>l{

+

FIU£lc

i,n> (i, A)

of

RU.

250

31. The Algebra R\J

31.2.4. In this subsection, the root datum is arbitrary. An object M G RC is said to be integrable if for any m G M and any i G / there exists n 0 > 1 such that E^m = F^m = 0 for all n > In the case where R = Q(v), v = V, this coincides with the earlier definition of an integrable object of C. Prom the definitions we see immediately that: (a) if M, R

M'

G RC are integrable, then

M

<S>R M'

6

RC

is integrable;

(b) if R! —> R is as in 31.1.11, and if M' G R>C is integrable, then M' G RC is integrable.

Let RC' be the the category of integrable unital # U-modules, regarded as a full subcategory of RC. 31.2.5.

Returning to the assumptions of 31.2.1, we note that RA\ and are integrable. Indeed, this is already known over Q(v); from this, the result over A follows, since our objects over A are imbedded in the corresponding objects over Q(t>) and finally, this implies the general case, by 31.2.4(b) with R' = A. ^AA

®R

(RA\')

Let M be the RP r o p o s i t i o n 31.2.6. Let M G RC; let A, A' G X+. submodule of Mx'~x consisting of all m such that E^m = 0 for all i and all n > (i, A) and such that F^m = 0 for all i and all n > (i, A'). Then ®R (RA\>),M) M given by f ^ /(£_A ® RJY) IS the map HomRlJ(RA\ an isomorphism. This follows immediately from Corollary 31.2.3. P r o p o s i t i o n 31.2.7. Let M G RC. Then M is integrable if and only if it satisfies the following condition: (a) M is a sum of subobjects each isomorphic to a quotient object of some ^AA <8>R (RA\>) with A, A' G X+. We know already that any object of the form ^AA ®R (RAAO with A, A' G X+ is integrable. It follows immediately that, if M is as in (a), then M is integrable. We now prove the converse. Assume that M is integrable and that m G M£ where £ G X. We can find integers ai,a[ G N such that E\a) m — 0 for all i and all a > a, and F^a ^m = 0 for all i and all a' > a[. Since the root datum is Irregular, we can find A G X such that (i, A) > ai and {i, A + C) > for all i. Let A' = A + Then (i, X') > a[ for all i. By the previous proposition, there exists a morphism / : ^AA <8>R (RA\>) —• M in RC such

31.3. Highest Weight Modules

251

that F(£-\<8>R)\>) = ra. The image of / is a quotient of ^A* <8>R { R A X ) and it contains ra. Hence M satisfies (a). 31.3.

HIGHEST WEIGHT

MODULES

31.3.1. In this section, we assume that the root datum is X-regular. Let M be an object of RC. We say that M is a highest weight module, with highest weight A G X, if there exists a vector ra G Mx such that (a) E\n)m

= 0 for all i and all n > 0;

(b) M = {x~m\x G i*f}; and (c) Mx is a free .R-module of rank one with generator ra. In this case, we have

M

=

2A'
X

' .

P r o p o s i t i o n 31.3.2. Assume that R is a field. (a) For any A G X, there exists a simple object (unique up to isomorphism) RL\ of RC which is a highest weight module with highest weight A. (b) If A / A' then RL\ is not isomorphic to RL\> . (c) If M is a highest weight module in RC with highest weight X, then M has a unique maximal subobject; the corresponding quotient object is isomorphic to RL\. Let M be as in (c). A subobject M' of M is distinct from M if and only if M' C £ v < a MX . This shows that the sum of all subobjects of M distinct from M is a subobject distinct from M. Thus, M has a unique maximal subobject, hence a unique simple quotient object, which is clearly a highest weight module with highest weight A. Applying this to the Verma module RM\, which is a highest weight module with highest weight A, we obtain a simple quotient RL\ of this Verma module; this proves the existence part of (a). If L' is a simple object of RC which is a highest weight module with highest weight A, then, by 31.1.13, we can find a non-zero morphism RM\ —• V. This is necessarily surjective. Since RM\ has a unique simple quotient, we must have that L' is isomorphic to RL\. Thus (a) and (c) are proved, (b) is now obvious.

32.1.1.

In this chapter we assume that the Cartan datum is of finite type.

(a) M is integrable if and only if it is a sum of subobjects which are (b) If M is integrable, then for any m G M there exists a number N > 0 Assume first that M is integrable. By 31.2.7, M is a sum of subobjects which are quotients of objects of the form ^AA <8># ( R ^ X ' ) with A, A' G these objects are finitely generated (free) ii-modules. It remains to show that an object M G RC which is finitely generated as an it-module, is integrable and satisfies (b). This follows from the fact that there are only finitely many A G X such that Mx ^ 0, together with the fact that the root datum is X-regular. If x G R{u and ra G M A , then x+ra G Mx+U and

Such / exists: for example, we can choose a set of representatives H for the cosets X/Z[I] and an arbitrary function c : H x H ^ Q, and set for

This function satisfies (a) and conversely, any function satisfying (a), is of this form for a unique function c for fixed H. A function / satisfying (a)

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © Springer Science+Business Media, LLC 2010 2011

252

32.1. The Isomorphism

/7£m,M'

253

C)« • V2>

(b) / « , C + i') ~ /(C» CO =

/(C + < / ,C / )-/(C,C , ) = -(i,C / >i-i/2, / ( C - < / , C / ) - / ( f , C ' ) = (i,C , ><-i/2, f(CC-i,)-f(CX,) = for all C, C' € ^ and i G / .

{iX)i-i/2,

32.1.4. We fix an integer d > 1 and a function / : I x I - > Q a s i n 32.1.3, such that the values of / are contained in ^Z. Such / exists even with integer values: it suffices to take the function c in 32.1.3 with integer values. Assume that we are given an element v G R such that vd = v. For any rational number q G ^Z, we will write v 9 instead of v d g . This is the usual power of v, when q is an integer. Given two objects M, M' in RC' , we define an (invertible) linear operator 11/ : M ® M ' -» M®M' by II/(m®m') = v-f(A,A')ram/ for M G M X , M ' G /A M . Let s : M'<8>M —• MM' be the isomorphism of /2-modules given by s(m'®m) = m®m'. We define the /^-linear map 0 : M % M ' —• by f M'eB„ is 35 i n 4 1 2 2 4 1 6 where 9 = EM'GB, ® --» ( w i t h PM' G A ) and (j): A —• R is as in 31.1.1. By 32.1.2 applied to Monly finitely many terms in the sum are non-zero for any given m,m'. Similarly, the R-linear map © : M (&r M' —• M <&r M' given by

4>(Pby)b~rnb'+rn'

©(ra ra') = v

b,b'e

for any ra G M, ra/ G M', is well-defined. From 4.1,3, we see that ©, © : M ®R M' —> M ®R M' are inverse to each other. T h e o r e m 32.1.5. Let fKM,M' = ®n/s is an isomorphism in RC.

: M>

®M

M

®

• Then

Let ( / ' , d') be another pair like (/, d), but with d' = 1; thus / ' has values in Z. Assume that the theorem holds for / replaced by / ' ; we show that it holds for / . Since /(A, A') — /'(A, A') is constant when A, X' run through M' M is an fixed cosets of Z[/] in X, the operator s l l j / l l / s : M' M isomorphism in Since / 7 Z M , M ' = /'T^A^Af'sIly/lI/s, our claim follows. Thus, in the rest of the proof we shall assume that d = 1 so that / takes values in Z; we then have v = v.

254

32. Commutativity

Isomorphism

IS a n so Prom the remark preceding the theorem, we see that i ~ 1 morphism of R-modules; its inverse is s ^ I I ^ © : M M' —*• M' ® M. We must show that u©Il/(m<8>ra/) = GHfu(m' ®m) for all homogeneous ra, m' and all u € R\J. Using the characterization 31.2.7 of integrable objects, we <%>R (RA\>); axe reduced to the case where both M, M' are of the form since such objects are obtained by change of rings from the analogous objects over A, we may assume that R = A. This can obviously be reduced to the case where R = Q(v). We may assume therefore that R = Q(v). It suffices to show that

A ( u ) e i l / ( m ® m ; ) = 0(II/ t A(u)(m U—>U®Ube the algebra automorphism given on the generators by a(Ei ®l) = Ei® a(Fi 1) = Fi ® a ( l Ei) = K - i Ei, a{l®Fi)

=

Ki®Fi,

We have the identity A ( u ) = a( t A(w)) for all u 6 U . Indeed, both sides can be regarded as algebra homomorphisms U —* U <S> U; hence it suffices to check that they agree on the generators E i , F i , K w h i c h is immediate. Therefore, the identity in 24.1.2(a) can be rewritten as follows: A(w)©(ra 0 ra') = ©(<*('A(u))(m m')). We will show that (a)

= n/AMnj

1

'.M®M'->M®M',

for all « G U . Therefore we obtain A(u)©(ra ra') = © ( n / A M I T ^ r a ra')), for all ra,ra'. This implies A ^ e n ^ m ^ m ' ) = ©(Il/A(u)(ram')),

32.2.

The Hexagon

Property

255

for all ra,ra', as required. It remains to show (a). Clearly, if (a) holds for u, v! then it holds for W , and for linear combinations of u, u'. Hence it suffices to verify (a) in the case where u is one of the generators Ei,F{, K^. The verification for K^ is trivial. We apply both sides of (a) (with u = Ei, resp. Fi) t o m ® m ' where m G G Mn» . The left hand side is Eim

K-ifn ® Eim!

(resp. ra i^ra' + i^ra ® Kim'). The right hand side is v~f(C.C')(t;/(C,C

/

+i,)m 0 E . M '

+

V/(C+I',C')£.M

0

(resp. 0771' + v^^'^K-im <8> i^ra')). It remains to use the identities 32.1.3(b) for / . The theorem is proved. In the following corollary, we do not assume the existence of roots of v. Corollary 32.1.6. If M, M' are in RC', then M ® M' and M' M are isomorphic objects of RC'. Indeed, / can be chosen with integer values. 3 2 . 2 . T H E HEXAGON PROPERTY

32.2.1. Let M, M', M" be three objects of RC . We define a linear isomorphism fW : M <8> M' ® M" M <8> M' <8> A/" by /II'(ra ® ra' ® m") = v /( A ^ A + A V/(A'',A')-/(A'',A) m ® ra' ® ra" for all ra G M A ,ra / G M A ',ra" G M A ". We define a linear isomorphism /II" : M" ® M ® M' —• M " M ® M' by /II"(ra" ® ra ra') = v ^ ^ ' ' ) - ' ^ " ) " ' ^ ' ' > r a " ® ra m' for all ra G M A ,ra' G M A ',ra" G M A ". Proposition 32.2.2. (a) The map

256

32. Commutativity

Isomorphism

coincides with the composition 1

1

—> M (8> M <8> M

M®M ®M

• M M ® M .

(b) The map

coincides with the composition M " ® M ® M '

M ® M " ® M '

' " P ' * " ' - " " ,

M ® M ' ® M " .

Let ( f ' , d f ) be another pair like (/, d), but with d! = 1; thus / ' has values in Z. Assume that the proposition holds for / replaced by / ' ; as in the proof of Theorem 32.1.5, we see that it also holds for / ' . Thus, in the rest of the proof, we shall assume that d = 1 so that / takes values in Z; we then have v = v. Using the characterization of integrable objects given in 31.2.7, we are reduced to the case where each of M, M ' , M" is of the form ^A A ®R (RA\>)-, since such objects are obtained by change of rings from the analogous objects over A, we may assume that R = A; this case can be obviously reduced to the case where R = Q(f). We may assume therefore that R = Q(v). Let me Mx,m' e M'x\m" e M"x". Using the definitions and 4.2.2(b), we have fI^M",M®M'{m 0 m ' ® m") = t/( a "> a + a ') ^

©„(ra" (ra ra7)) V

and (/7£m",M ® 1 M')(!M ® fI^M",M')(m = vfix">x,) ^2(fnM",M

® m ' ® m")

® 1 M>)&l3(rn m" ra')

On the other hand, using the definitions and 4.2.2(a), we have fft<M®M>,M"(m" ®m®rn') v'y

= vf(x+x^x")

y^ Q„((m ® ra') ® ra")

32.2.

The Hexagon

Property

257

and ( 1 M ® j1Z>M',M")(f'R>MYM"

f{x x,,)

= v ' (iM

®

1

M'){M"

m

ra')



The proposition follows since /(A" - i/", A) - /(A", A) = /(0, A) - /(*/', A) and / ( V , A" + v") - /(A', A") = /(A', i/") - /(A', 0) for i/" G Z [ / ] . L e m m a 32.2.3. Let d > 1 be the order of the torsion subgroup of X/Z[I]. There exists a symmetric 7i-bilinear pairing f : X x X —> ^Z which satisfies 32.1.3(a). We can find a direct sum decomposition X = X\ © X2 such that Z [I] is contained in X\ as a subgroup of index d. There is a unique symmetric bilinear pairing fi : X\ xXi —> ^Z such that fi(i', j') = -i-j for all i,j G I. fi(xi,x[). For xi,x[ G Xi and x2,x'2 G X2, we set f(xi 4- x2,x[ -f x2) — This has the required properties. P r o p o s i t i o n 32.2.4 ( H e x a g o n p r o p e r t y ) . Let M , M ' , M " G RC. Let ( f , d ) be as in the previous lemma. Assume that we are given an element v G R such that vd = v. Then the map /7£M",MM' : M <S> M' ® M" —• M " M ' coincides with the composition M ® M ' ® M "

I

M

%

,

1

I

M

"

M

\ M ® M " ® M '

M " ® M ® M '

and the map /^MM',M"

: M"

® M ® M '

M ® M '

®M"

coincides with the composition

This follows from Proposition 32.2.2, since in our case, /II 7 , / I I " are the identity maps.

CHAPTER

33

Relation with Kac-Moody Lie Algebras 33.1.

T H E SPECIALIZATION v =

1

Let j / f be the free associative algebra over R with generators OI (i E I). As for 'f, which corresponds to the case R = Q(^), we have a natural direct sum decomposition = S i / ^ ' f u ) where v runs over N[/]; each % is a free /2-module of finite rank. 33.1.1.

Let be the quotient of the algebra R{ by the two-sided ideal of R{ generated by the elements .

E

(-1)p>(

P + P .

P

for various i ^ j in I. Recall that : A —> R is given. Let nf u be the image of R i u under the natural map —> It is clear that we have a direct sum decomposition = @ u (Rf l/ ). From the definition, we have, for any z/, an exact sequence of i?-modules (K%") —•

v

fifv

—+ 0

where the indices satisfy i / , i / ' G N[J] and z/ + z/' + (l — (z, j'))i+j = v\ the first map has components x,x' x$ijx'. If we take this exact sequence for R = Q[v, v" 1 ] and we tensor it over Q[i>,i;-1] with Q(v) or with RQ (a field of characteristic zero, regarded as an *4-algebra or Q[i7, -algebra via v i—• 1), we obtain again exact sequences, by the right exactness of tensor product. We deduce that RQf„ — RQ <8> (Q[v,t;-i]f|/) and Q(v)lv

= Q ( V ) ® (Q[V,V-I]FI/).

Since Q[V)V-i]f„ is a finitely generated Q[t;,t>_1]-module and Q(v) is the quotient field of Q[v, v - 1 ], we deduce that (a) d i m Q W ( Q W f t , ) < dim

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010 DOI 10.1007/978-0-8176-4717-9_30, © Springer

258

33.1. The Specialization v =1

259

By 1.4.3, there is a unique (surjective) algebra homomorphism Q(v)f —• f which takes Qi to Qi for all i (and preserves 1). It is clear that this homomorphism maps Q( v )f u onto f„, hence (b) dimQ(v) f„ < dimQ( v )( Q ( v )f I/ ) for any v G N[/]. Note that ^ f is the ifo-algebra defined by the generators Qi (i G I) and the Serre relations p+p' = l-(i,j')

for various i ^ j in I. Thus it is the enveloping algebra of (the upper triangular part of) the corresponding Kac-Moody Lie algebra over Ro. 33.1.2. Assume now that the root datum is y-regular and X-regular. Let A G X+ and let M — R^A* G R^C'. The linear maps EiyFi : M —• M satisfy in our case: (a) EiM< C FiM< C for any i G I and ( G l ; (b) (EiFj - FjEi)m = Sij(i, £)m for any i, j G I and m G M c ; (c) T,p+p>=i-(i,j>)(-1)p'(Ei/Pl)Ej(Ef in / ;

/P'1) = 0 : M

M for any i ^ j

(d) = 0 : M - M for any i ? j in I. This shows that M is an integrable highest weight module of the KacMoody Lie algebra attached to the root datum. By results in [3], namely, the complete reducibility theorem of Weyl-Kac and the Gabber-Kac theorem, M is simple as a module of that Lie algebra and the Ro-\meoi map i

given by KK--K

^FilFi2'--Fivr)x

is an isomorphism. It follows that (e) RqA\ is a simple object of R0C and (f) for given v G N [/], we have dim/?0(/i0fly) = dim R o ( f l o A A -") provided that (i, A) are large enough for all i.

260

33. Relation with Kac-Moody

Lie Algebras

From the definition of A\ and R0A\, it is clear that (g) dim f l 0 (fl 0 Aj" , / ) = dim Q ( v ) A ^ - " for any A, v and (h) for given v £ N[/], we have dimq( v )(f I/ ) = dimq( v ) A* - " provided that (i, A) are large enough for all i. Prom (f),(g),(h) we deduce that dimQ( v )(fj,) = dim/^tfofj,) for all v. Combining this with the inequalities 33.1.1(a),(b), we see that those inequalities are in fact equalities. In particular, the natural surjective f must be an isomorphism. Similarly, the natural homomorphism Q(v)f surjective homomorphism n 0 f —• # 0 f is an isomorphism since dimHoUo^) > dim flo (f I/ ) = dim Q ( t; )(f I/ ) = d i m ^ ^ f , , ) for all v. Thus we have the following result. T h e o r e m 33.1.3. (a) The natural algebra homomorphism Q(„)f —> f is an isomorphism. (b) We have diniQ^) iu = dim^ 0 (fl 0 f I/ ) for any v. (c) The natural algebra homomorphism

Roi

—•

is an isomorphism.

(d) If A £ X+, then the dimension of the weight spaces of A\ are the same as those of the simple integrable highest weight representation of the corresponding Kac-Moody Lie algebra. 33.1.4. R e m a r k . Parts (a), (b) and (c) of the theorem hold for arbitrary root data, since only the Cartan datum is used in their statement. Corollary 33.1.5. The algebra U can be defined by the generators Ei (i £ I), Fi (i £ / ) , KM (p £ Y) and the relations 3.1.1(a)-(d), together with the quantum Serre relations for the Ei's and for the Fi's. 3 3 . 2 . T H E QUASI-CLASSICAL CASE

33.2.1. In this section we assume that the ^4-algebra R is a field of characteristic zero and {yi) — ± 1 in R for all i £ I. We then say that we are in the quasi-classical case; this is justified by the results in this section. We also assume that (/, •) is without odd cycles (see 2.1.3). Then, by 2.1.3, we can find a function

33.2. The Quasi-Classical

Case

261

(a) i t—• ai from I to {0,1} such that ai + a j = 1 whenever (i, j') < 0. Let Ro be the ring R with a new ,4-algebra structure in which v £ A is mapped to 1 € R. We want to relate the algebras and R0f. We cannot do this directly, but must enlarge them first as follows. Let A be the i?-algebra defined by the generators Qi, Ki (i £ I) subject to the following relations: the Qi satisfy the relations of the Ki commute among themselves and KiQj = ^ [ v i ^ ^ O j k u for all i,j £ I. Let Ao be the i?-algebra defined by the generators Qi, ki (i £ I) subject to the following relations: the Qi satisfy the relations of j ^ f , the k i commute among themselves and kiQj = (vi)(l,:> )Qjki, for all i,j £ I. It is clear that, as an /^-vector space, A (resp. ylo) is the tensor product of ftf (resp. # 0 f) with the group algebra of Z1 over R, with basis given by the monomials in k i . P r o p o s i t i o n 33.2.2. (a) The assignment Ei E[ = Eikand for all i, defines an isomorphism of R-algebras AQ —• A.

ki

> ki

(b) dim f l ( f l f I / ) = dimfl(/ Jo f |/ ) for all v. (c) The natural algebra homomorphism

—> r{ is an isomorphism.

Let i, j £ / be distinct. A simple computation shows that we have in A: =

E'fE'jE'f

The factor ^(VJ^CP+P'XP+P'-1) is 1 since (vi) = ± 1 and the exponent is even. By the definition of ai, we have

and this equals < j > ( i i f p + p' = 1 — {i,j'}- Note also that 4>([n]\) = since (vi) = ±1. Hence, if p + p' = 1 - (i,j;), then

=

<j)(vi)(p+p'Kp+p'-1)/2(j>(vi rpp'ci>(vi)p^-p)p\p,}-

It follows that E

(-ir'(E^/p\)E'J(El"'/p'\)

p+p' =

l~{i,j')

=

ttepSM-VJ'))/'

262

33. Relation with Kac-Moody Lie Algebras

This shows that the assignment in (a) preserves relations, hence defines an algebra homomorphism A0 —• A. The same proof shows that the asKi defines an algebra homomorphism signment Ei »-> EiK~ a i and Ki A — i t is clear that this is the inverse of the previous homomorphism. This proves (a). For any v, the isomorphism in (a) maps the subspace f u isomorphically onto the subspace (NFU)K where A' is a monomial in the KI depending only on v. This proves (b). The homomorphism in (c) maps onto Riu for any V. Using (b), it is an isomorphism. This implies (c). follows that this restriction RIv —> The proposition is proved. The next result compares the algebras # 0 U and Proposition 33.2.3. There is a unique isomorphism of R-algebras f : H0U —> such that f(Eil c) = cf>(vi)a^0Eilof(Filc)

=

= lc

for alii € I, C G X. Here ai is as in 33.2.1(a). To construct / , we take advantage of the fact that for the algebra # 0 U, we know a simple presentation by generators and relations, while for we do not. It will be easier to first construct a functor from RC to RqC and then to show that it comes from an algebra homomorphism. Let M be an object of RC. The linear maps EI, FI : M —*• M (i G I) satisfy (a) EiM x C Mx+i',

FiM x C M x - { ' for any iel

(b) (EiFj - FjEi)m = M;

and A G X;

A)m for any i j G I and m €

x

= 0 :M

(c) any i ± j in I;

M for

(d) =0:M M for any i ^ j in I. For any a G Z, we define linear maps Pi,a ' M —• M by Pi>a(m<) = (j>(vi)a(l'x)m for m G Mx. We define linear maps E[, F[ : M —> M in terms of the function 33.2.1(a) by E\ = EiP^ai,F( = <J>(vi)FiPiti_a.. As in the proof of the previous proposition, we can check that: (al) E[MX C M A + i \

F(MX C MA-*' for any iel

and A G X;

(bl) (E{Fj - F<E%)m = 6ij(i, A )m for any i j el and m G M A ;

33.2. The Quasi-Classical

Case

263

(cl) E p + ^ i - ^ ^ t - i F ' f ^ / p O ^ t ^ ' / p ' ! ) = 0 : M i ^ j in / ;

M for any

(dl) = 0 : M - M for any i j in / . Since /^f = # 0 f (see 33.1.3) has the presentation given by the Serre relations, it follows that M with its weight space decomposition and with the linear maps E'I} F( is an object of R0C. Thus, we have defined a functor T : RC —• R0C. This functor has the following obvious property: it associates to M an object T(M) with the same underlying R-module as M and any endomorphism of M in RC is at the same time an endomorphism of F(M) in RqC. Applying the functor T to flU, regarded as a left module over itself, we obtain a structure of a unital left U-module on Thus, we have an i?-bilinear pairing ^ U x —• ftU, denoted by a, b a * 6; this has the properties (aa') *b = a* (a' *b) and (a * b)b' = a * (bb'). The last property holds since right multiplication by b is an endomorphism of in RC, hence also in R0C. We define a map / • R 0 U —• flU by f(a) = E c g x a * ^C' o n ^y finitely many terms in the sum are non-zero. We have /(a)/(a') = $ > *

lC)(a'*lC')

= Ea*E1c(«,*1c) c c * (a' *

= c

=

E

a a

' *

=

f(aa')-

c

It is clear that / has the specified values on the algebra generators ^ l c , F a c , l c °f u. We show that / is an isomorphism. The elements are algebra generators of ^ U , since 4>([N]\) = ± n ! is invertible in R for any i £ I and any n > 0. It follows that / is surjective. As in 31.1.2, (resp. RQIJ) is a free RF ^f o p p -module (resp. /j 0 f ® R0fopp-module) with generators (under (x x') : u »-*• x+ux'~ ), and / carries the subspace of ^ U spanned by x+l^x'~ with X £ © t r V
£ © tr

u
and fixed £ onto the analogous subspace of Since these subspaces are both of the same (finite) dimension over R, the restriction of / must be an isomorphism between them. This implies that / is an isomorphism. The uniqueness of / is obvious. The proposition is proved.

264

33. Relation with Kac-Moody Lie Algebras

Corollary 33.2.4. Assume that the root datum is Y-regular and X-regular and that A £ X+. Then RA\ is a simple object of RC. By the proof of 33.2.3, this is equivalent to the statement that R0A\ is a simple object of R0C. (See 33.1.2(e).)

CHAPTER

34

Gaussian Binomial Coefficients at R o o t s of 1

34.1.1. Let I be an integer > 1 . In this chapter we assume that the AR and with v = <j)(v), is such that R is an integral algebra R, with (j) : A domain and the following hold vzl = 1 and wlt ^ 1 for all 0 < t < Z. All the identities proved in 1.3 for Gaussian binomial coefficients imply, after applying , corresponding identities in R. However, certain identities will be satisfied only in R. We shall now give some examples of such identities. L e m m a 34.1.2. (a) If t > 1 is not divisible by I, and a G Z is divisible by I, then 0([®]) = 0. (b) If a\ G Z and t\ G N, then we have

(c) Let a G Z and t G N. Write a = ao + la\ with ao> a i G Z such that 0 < ao < I ~ 1 and t = + Hi with to^ti £ N such that 0 < to < I — 1. We have 2 a a ^ ^ = v (aoti-aito)«+(ai + l)tii 0 ^ ° ^ Here ("*) G Z is an ordinary binomial coefficient. We prove (a) for a > 0 by induction on a. If a — 0, we have trivially ["] = 0; if a = Z, the equality (f>{ [*]) = 0 follows directly from the definitions. Assume now that a > 21 and that (a) holds for a — I instead of a. By 1.3.1(e), we have

\L J /

\L

J/

\ L J /

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © © Springer Springer Science+Business Science+Business Media, LLC 2010 2011

265

266

34-1- Gaussian Binomial Coefficients at Roots of 1 266

For each term in the sum, we have that either t' or t" is not divisible by /; hence the sum is zero by the induction hypothesis. This proves (a) for a > 0. We now prove (a), assuming that a < 0. Write t = to + lt\ with 0
-

J-a+lti)t"-(to-l)t'

£



t'+t"=t

—a + lt\ t'

0

t0 - 1 t"

Consider a term in the sum corresponding to ( t t " ) . Since — a+lt\ > 0 is divisible by we see from the earlier part of the proof that <j){ [""J'* 1 ]) = 0 unless t' is divisible by I. But then t" is congruent to t modulo I. Hence t" is congruent to to modulo I. It follows that t" > to, hence P 0 " 1 ] = 0. Hence our sum is zero and (a) is proved. We prove (c), assuming (b). In the setup of (c) we have 4>

=

aof'-lmt' £ f+t"=t

4>

By (a), the sum may be restricted to indices such that t" = It" for some t" G N and such that ^ < 0 0 - Then t' is congruent to t modulo /, hence t' is congruent to t0 modulo I. Since both t', t0 are in [0, / — 1], we must have t' = to and therefore t" = ItThus, (d)

_ ylaoti-laito

0



4>

Idi Iti

This shows that (c) is a consequence of (b). We prove (b) for a\ > 0 by induction on a\. The case where a\ = 0 or 1 is trivial. Assume now that a\ > 0. By 1.3.1(e), we have 4

la\ Iti

=t'+t"=iti E *l(a\ — \)t" — It'4

la\ — t'

Mi

By (a), we may assume that the sum is restricted to indices divisible by namely t' = It^^t" = lt'[ with t[ +t'2 =t\. Using the induction hypothesis, we get 4>

la\ It 1

=t'l+ti'^i E

=

vi

2

(*i+i)ti(ai

267

34-1- Gaussian Binomial Coefficients at Roots of 1

We have used the identity 1.3.1(e), specialized for v = 1. This proves (b) for ai > 0. We now prove (b) assuming that a\ < 0. We have

la\ lt\

—la\ + lt\ — 1 Iti (I - 1) + l(-ai + ti - 1) Iti

= (-1

1)

= (-1

The last equality follows from (d). By the part of (b) that is already proved, we have l{-ai -[-tiltx

d>

1)

=

=

^(-ax+tot!

f-ai+h-1 tl ^ ( - a a + t O t i ^ j ^ t i ( a1 tl

It follows that la i 4> Iti

It remains to observe that (e) vl*+l =

{-l)l+1.

Indeed, if I is even, we have v' = — 1 and both sides of (e) are —1; if / is odd, then v* = ± 1 and both sides of (e) are 1. The lemma is proved. 34.1.3. Let p > 0. We have MP\-/(MP)=P]-vl2p{p-1)/2>

(a)

We prove (a) by induction on p. If p = 0 or 1, then (a) is trivial. Assume that p > 2. We have 4>m'/(mp) = m i p - ^ / ( w r v

Q'f]) •

Using 34.1.2 and the induction hypothesis, we see that

This proves (a).

268

34-1- Gaussian Binomial Coefficients at Roots of 1 268

L e m m a 34.1.4. Assume that 0 < r < a < 1. We have (a) E U " ' 1 ( - 1 ) ' " ' + 1 + < ' v " ( ' " r ) ( a " ' + 1 + < ' > + ^ ( [ i 7 ] ) =

v (

' °~rV(["])-

In the left hand side of (a) we may replace v i2 ~ z by (—l) z+1 . Note also that I — r > 1; hence l-r i)v<1-,+r> q=0

l-r .

Q .

= 0

(see 1.3.4). Hence the left hand side of (a) equals (-ly+lv^a+l-O-'a

l-r

l-r

( - 1 )9V9(1~Z+rV

q—l — a

.

Q .

q' = 0

\l

Q

J/

For any q' in the last sum, we have by 34.1.2(c), {[l~,r]) = v' 9 V([~/ r ]) • Thus the left hand side of (a) is a-r ^ ^^a-r-q'^Iq'+

(_iy+l-r+lvr(a+l-l)-la

(l-r-q')(l-l+r)^

q'= 0 =

=

(_l}a+l-r+lwl-(l-r)l-la

^

1

,r

r

^L9 ^-q'+r(a-r-<7/)^

q> =0 - r - 1 (_1 \a+l-r+l l-(l-r)l-la a—r a (^y+lvl-d-rV-la^ a—r

We have used vl = v

/



. The lemma is proved.

CHAPTER

35

The Quantum Frobenius Homomorphism

35.1.1. In this chapter we fix an integer I > 1. Then the integers h > 1, the new Cartan datum (/, o) and the new root datum (y*, X * , . . . ) of type (/, o) are defined in terms of Z, (/, •), (y, X , . . . ) as in 2.2.4, 2.2.5. 35.1.2. The assumptions (a),(b) below will be in force in this chapter, (a) for any i ^ j in I such that lj > 2 , we have li > — (z, j'} + 1; Note that (a) is automatically satisfied in the simply laced case: in that case, we have li = Z for all in the general case, the assumption (a) can be violated only by finitely many Z. Note also that (b) is automatically

35.1.3. Let Z' be one of the integers Z, 2Z, if Z is odd, and let I' be equal to 2Z, if Z is even. Let A' be the quotient of A by the two-sided ideal generated

In this chapter, we assume that the given ring homomorphism <j>: A —» R factors through a ring homomorphism A! —* or that R is an *4'-algebra 35.1.4. When (/, •) is replaced by (/, o), the element Vi G A, whose definition depends on the Cartan datum, becomes v* = v l ° 1 / 2 = v1^. For any P G A, we denote by P* the element obtained from P by substituting v by v*. For each i G / , we set v* — (vi) and v* — (f>(v*) =

L e m m a 35.1.5. (a) Letae

ZfZ andtG Z*N. We have ([?];) =

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2010 2011

[?/£]*)

269

270

35. The Quantum Frobenius

Homomorphism

(b) Let a G UZ and lett G N be non-divisible by U. We have (f>( [®]i) = 0 (equality in R). It suffices to prove this for R = A'. This is an integral domain in which v f < = 1 and v f ^ 1 for all 0 < t < U. We prove (a). Applying 34.1.2(b) to Vi and li instead of v and I, we see that ^(["j^) = vj0"*"'*^

Applying

the same result to v* and 1 instead of v and /, we see that tf>( [^J*] ) = v *(o+ii)t/i?

(ajhy

It remains to use the

equality v* = v f . The proof of (b)

is entirely similar; it uses 34.1.2(a). 35.1.6. Let f* (resp. U*) be the Q(v)-algebra defined like f (resp. like U), in terms of the Cartan matrix (/, o) (resp. in terms of (F*, X*, ...))• Then the .R-algebras r{, Rf*, r U , are well-defined. We state the main results of this chapter. T h e o r e m 35.1.7. Recall that R is an A'-algebra. There is a unique Ralgebra homomorphism Fr : ^f —> rV such that for all i e I and n G Z, Fr(e\n)) equals 6^ n//i) if n G UTi, and equals 0, otherwise. T h e o r e m 35.1.8. There is a unique R-algebra homomorphism n) = e\nli) for all i € I and all n G Z. Rf* -> Ri such that Fr'{0l )

Fr' :

T h e o r e m 35.1.9. There is a unique R-algebra homomorphism Fr : RU —> ftU* such that for all i G / , n G Z and £ G X, we have: F r ( ^ n ) l c ) equals E\n/li)lc wise;

if n G UZ and < G X*, and equals 0, other-

Fr(F^l^) wise.

if n G kZ and £ G X*, and equals 0, other-

equals

We give a proof of the last theorem, assuming that theorem 35.1.7 is known. Using the presentation of the algebra RU in terms of RF given in 31.1.3, and the analogous presentation of the algebra in terms of Rf*, we see that it is enough to prove that the assignment rc+l^x- ' • Fr(x+)l<^Fr(x~),x~l^x+ for x,x' G R f ,

Fr(x~)l^Fr(x+)

CeX*, x + \ ^ x ~ i—•

— ' •0

for x, x' G Rf, C £ X — X* respects the relations described in 31.1.3. (Here, Fr is the homomorphism given by Theorem 35.1.7.) This is immediate, using Lemma 35.1.5.

35.2. Proof of Theorem

35.1.8

271

35.1.10. R e m a r k . The homomorphism Fr constructed in Theorem 35.1.9 is called the quantum Frobenius homomorphism. It is compatible with the comultiplication on and RXJ* (proof by verification on the generators ^ n ) l c and i ^ ( n ) l c ) . 35.1.11. The uniqueness part of Theorems 35.1.7 and 35.1.8 is clear. To prove the existence part of these theorems, we note that the general case follows by change of scalars from the case where R = A'. Since A! is an integral domain, it is contained in its quotient field K and the algebras .4'f, ^4'f* are naturally imbedded in the corresponding algebras over K. Thus, if the theorems are known over K then, by restriction, we see that they hold over A'. We are thus reduced to proving the theorems assuming that R is the quotient field of A!. The proof in this case will be given in 35.2, 35.5. 3 5 . 2 PROOF OF THEOREM 3 5 . 1 . 8

35.2.1. In the rest of this chapter (except in 35.5.2, 35.5.3), we assume that R is the quotient field of A'. Note that R is a field of characteristic zero and that the order of v 2 = <j>(v2) in the multiplicative group of R is /. Thus, we have v 2 i = 1 and v 2 t ^ 1 for all 0 < t < I. By the definition of k, we have v 2Zi = 1 and v?* ± 1 for all 0 < t < U. In particular, ([n]\) is invertible 212 in R, if 0 < n < For any i we have v* = ± 1 since v i 1 = 1 ; hence, when dealing with the algebras RU*, we are in the quasi-classical case (see 33.2). L e m m a 35.2.2. The R-algebra Rf is generated by the elements of^ I) and by the elements 9{ for i G I such that U>2. Recall that the i?-algebra R{ is generated by the elements i and n > 0. Writing n = a + lib with 0 < a < k and be ^ s i n g 34 1 2 ) . On the other hand, e(n) = vabiie(a)e(hb) (a) e\ a ) = ^ ( H i ) - 1 * ? (see 35.2.1) and (b)

= (6!)"ivf

The equality (b) follows from the equalities (0<'
in

= w ^ f

(t l>/2

f

and M M / A U M

(see 34.1.3). The lemma is proved.

-

)

(i G

for various N, we have

272

35. The Quantum Frobenius

Homomorphism

35.2.3. We now give the proof of Theorem 35.1.8. As observed in 35.1.11, it is enough to prove the existence statement in 35.1.8, assuming that R is as in 35.2.1. We will show that there exists an algebra homomorphism Rf* —> such that Oi H-> for all i. Since the assumption 35.1.2(b) is in force, the algebra r{* has a presentation given by the generators 6i and the Serretype relations; this follows from 33.2.2(c) . Since <£(([«]')*) = v f n ( n _ 1 ) / 2 n ! (see the proof of Lemma 35.2.2), we see that it suffices to prove that, for any i ^ j, we have the following identity in Rf: l

W lsp+p' = l-(i,j')lj/li\ li) b

)

v

v

i

i

l Kb 1)/2

ib)

(equality in Using (0\ ) = b\v ^ ~ ef rewrite (a) in the following equivalent form:

p! Vj R

p'\

f , see 34.1.3), we can

0>) = 0It remains to prove (b). Let a = —(i,j'). For any q G [0, U — 1] we set gq =

£

(-i)S(''-1-,)«{r)ej,,)«<w e

^

r+s=alj+li — q

This is fij-ijfaij-Mi-g;-i m the notation of 7.1.1. Let g = Y!qZo{-l)qvfiq+liq~qgq0\q). By the higher order quantum Serre relations (see 7.1.5(b)), we have gq = 0 for all q € [0, li — 1]; hence g = 0. On the other hand, setting s' = s + q, we have g = E r ^ r + y ^ - N , CrA^'W* ^ li-1 Y^(-iy+*vri(li-1-qHotljq+liq-q s Crs, = L9J

9=0

Taking the image of g = 0 under the obvious map ^f —> Rf, we obtain r,s' \r+s'=alj +li

For fixed s', we write s' = a + lib where 0 < a < k — 1. We have = v r ^ ' ^ C l i ) ( s e e 34.1.2); hence

ftcrs) = (-l)rv-(,i_1> E ( " l ) < v f < p q=0

q=0

^ L9-1 »'

35.3. Highest

Weight Modules of RU

273

We have used the identity v | i ( / < - 1 ) = see 34.1.2(e). (b) follows. Thus, we have an algebra homomorphism Fr' : Rf* —• Rf such that Fr'(6i) = o\li) for all i. To complete the proof we must compute Fr'(9^) for n > 0. We have Fr'(9^n)) = mrfYir'Fr'iOir

= vtrI?n(n-1)/2(n!)-1(^,<))n =

o\nli).

Theorem 35.1.8 is proved. 3 5 . 3 . STRUCTURE OF CERTAIN HIGHEST W E I G H T MODULES OF RU

P r o p o s i t i o n 35.3.1. Assume that i ^ j in I satisfy li > ~{i,j') + 1. The following identity holds in Ri:

r=0

^L

r

-!«'

We set a = ~(i,j'). Using Corollary 7.1.7 with m = li,n = 1, we see that we are reduced to checking the identity h-a-l

E

for any r G [0, a]. This follows from Lemma 34.1.4. P r o p o s i t i o n 35.3.2. Assume that the root datum is X-regular. Let A G X be such that (i, A) G UZ for all i G I. Let M be a simple highest weight module with highest weight A in RC and let rj be a generator of the R-vector space Mx. (a) / / ( G X satisfies M^ ^ 0, then £ = A — particular, (i,Q G UZ for all i G I.

knii', where ni G N . In

(b) If i G / is such that li > 2, then Ei, Fi act as zero on M. (c) For any r > 0, let M'r be the subspace of M spanned by the vectors F^^F^1^ • • • F^^rj for various sequences 2i,«2, ... ,ir in I. Let M' — M

'v

Then M' = M.

Clearly, M'r is spanned by vectors in M^ where C is of the form £ = A — ^iUnii', with ni G N. Such £ satisfies (i,() G UZ for all i G I. We use the fact that, for j G I, the integer lj(i,j') is divisible by UWe show by induction on r > 0, that

274

35. The Quantum Frobenius

Homomorphism

(d) EiM'r = 0, FiM'r = 0 for any i e I such that U > 2. Assume first that r = 0. Then E{MQ = 0 is obvious. Assume that for some iel such that li > 2, we have x = Fit] ^ 0. For any j e / , we have EjX = EjFirj = FiEjf] + 6id ([<», X)]i) v = Sid ( [ ^ j ) ry. thus If Since (i, A) G hZ, and k > 2, we have ^ ( [ ^ L ) = > Ejx = n > 2, then = E^Firj is an iMinear combination of FiEj^rj and of

Ejn~^rj, hence is again zero. Thus, E^x = 0 for all j e I and all n > 0; since x e Mx~l , there exists a unique morphism in RC from the Verma module M\-i> into M which takes the canonical generator to x; its image is a subobject of M containing x but not rj. Since M is simple, we must have x = 0. Thus (d) holds for r = 0. Assume now that r > 1 and that (d) holds for r — 1. To show that it holds for r, it suffices to show that E i F ^ m = 0, F i F ^ m = 0 for any i,j in I such that k > 2 and any m e If lj > 2, then E i F ^ m is an R-linear combination of F^Eim and of Fj3~1m, hence is zero since Eim = 0, Fjm = 0, by the induction hypothesis. If lj = 1, then EiFjm = FjEim + Sijfi)^'®].)™, where (i,f) e hZ, hence ^ ( [ ^ L ) = 0, as above; since ^ m = 0, by the induction hypothesis, we have again EiFjm = 0. If i ^ j , then from the identity in 35.3.1, we deduce by interchanging i,j and applying cr, that FiF^m is an .R-linear combination of FiF^m for various r with 0 < r < —(j,if) < lj. For such r we have FiF$ r ) m = 0. (Indeed, if lj > 2, then F i F ^ m = 0, by the induction hypothesis; if lj = 1, then r = 0 and F i F ^ m = Fim = 0, again by the induction hypothesis.) Thus, we have FiF^m = 0. If i = j, then F i F f j ) m = F f j ) F i m = 0, by the induction hypothesis. This completes the inductive proof of (d). Next we show by induction on r > 0 that (e) E\li)M'r C M;_ for any i e / , where, by convention, M'_x = 0. This is clear for r = 0. Assume now that r > 1. We must show that

Fj1^m' e M'r_l for any j and any m e

is an iMinear combination of F^E^m! (which M'r_x. Now E^F^m! is in M'r_x by the induction hypothesis) and of elements E^^m' with t > 0 such that t
35.3. Highest Weight Modules of

RU

275

Prom (d), (e), 35.2.2, and the obvious inclusion F ^ M ' r C we see that Yhr ^ r 1S a n ^U-submodule of M . (It is certainly equal to the sum of its intersections with the weight spaces of M since it is spanned by homogeneous elements.) Since M is simple, we must have M = Thus (c) is proved. Now (b) follows from (d) and (c). We prove (a). Let C G X be such that M^ ± 0. By (c), we have M ' c ^ 0. Then, as we have seen at the beginning of the proof, £ is of the required form. The proposition is proved. C o r o l l a r y 35.3.3. There is a unique unital RXJ *-module structure on M in which the weight space is the same as that in the RU-module M, for any £ G X* C X, and such that Eu Fi G Rf* act as E\lF^li) G R f . Moreover, this is a simple highest weight module for R\J* with highest weight A G X*. We define operators ei, fi : M —> M for i G I by ei = E\li\ fi — Using Theorem 35.1.8, we see that the e^ satisfy the Serre-type relations of Rf* and that the fi satisfy the Serre-type relations of Rf*. If C G X - X* we have = 0, by 35.3.2(a). If £ G X* and m G M^, then, by 31.1.6(c), we have that ( e i f j — f j e i ) ( m ) is equal to ]i)m plus an i?-linear combination of elements of the form F i I i - t ^ i - t ( m ) with 0 < t < k which are zero by 35.3.2(b). Since (i, 0 G kZ, we see from 35.1.5 that

Therefore, ( a f j - f j e i ) ( m ) = 6ij<j>([ {iX l /li ]*)m. It is clear that E I ( M C ) C M<+lii' and fi(M^) C M^~lii>. Thus, we have a unital #U*-module structure on M . By 35.3.2(c), this is a highest weight module of R \J* with highest weight A. This ftU*-module is simple. Indeed, assume that M" is a non-zero ^U*-submodule of M. Then M" is the sum of its intersections with the various (with ( 6 X ) and is stable under all : M —• M (in the ^U-module structure). Now M" is automatically stable under E f , F? (in the /^U-module structure) for any i and a such that 0 < a
276

35. The Quantum Frobenius

Indeed, the module RA\ of we may apply 33.2.4. 35.4.

Homomorphism

i s simple. By the assumption 35.1.2(b),

A T E N S O R P R O D U C T DECOMPOSITION OF

Rf

35.4.1. Definition. Let f be the .R-subalgebra of generated by the elements Oi for various i such that h >2. (Note that without the assumption 35.1.2(a), the definition of f should be more complicated.) We have f = ©i/fiv where f„ = R i v D f. T h e o r e m 35.4.2. (a) If i £ I and y G f„, the difference of^y — (b) The R-linear map x ' R,f* ®R f an isomorphism of vector spaces.

given by x 0 y •—> Fr'(x)y

We prove (a). If (a) holds for y and y', then it also holds for yy'. Hence it suffices to prove (a) when y is one of the algebra generators of f. Thus, we may assume that y — Oj where j satisfies lj > 2. By our assumption, we then have li > — {i,j') + 1. Therefore, we may use the identity in )OjO^ is an i?-linear Proposition 35.3.1, and we see that Of^Oj — combination of products O ^ O j O f ^ ^ with 0 < r < < h; these products are contained in f, by the definition of f. This proves (a). We prove (b). We first show that \ is surjective. Using Lemma 35.2.2, we see that is spanned as an R-vector space by products X\X2"-XP where the factors are either in for some v (factors of the first kind) or of the form o f ^ (factors of the second kind). By (a), any product xsxs+i with xs (resp. xs+i) a factor of the first kind (resp. of the second kind) is equal to v™x s+ ix s plus an element of fv< for some n and some v'. Applying this fact repeatedly, we see that x\x(f>)yb = 0 in ^f. We must prove that 2/6 = 0 for all b. We may assume that each yt, belongs to for some v. Assume that 0 for some b. Then we may consider the largest integer N such that there exists b with yb ^ 0 and tr |6| = N.

is

35.4- Tensor Product Decomposition of R{

277

In this proof we shall assume, as we may, that (Y, X,...) is both Irregular and X-regular. Let A G X+ and A' G X; assume that {i, A) G kZ for all i (i.e., that A G X*). We consider the objects M = RL\, M' = RM\> of RC (see 31.3.2, 31.1.13); let 77,77' be generators of the .R-vector spaces M \ M,X'. Then M' ® M G In M' M we have Fr'(b)~y^(r/ 0 77) = 0. We have 3/^(77' 0 77) = v^^VbW) ® ^ f° r some integer n(6), since any element of f„ with v ^ 0 annihilates 77 (see 35.3.2(b)). Hence we have (c) Ebvn(b)Fr'(b)~(yb(v') <8> 17) = 0 in M' M. Al C M where the sum is taken over all Ai G X of the Let Mi = ©M form A — ^2iliPii' with J^iPi = N. Let n : M —> M\ be the obvious projection. We apply 1 7r : M' M —> M' ® M\ to the equality (c). We obtain =0 (d) Zbvn(b)y;(n')®Fr'(b)-(r,) where the sum is taken over b subject to tr \b\ = N. By 35.3.3, we may regard M as a #U*-module; this is a simple highest weight module of RU* which is just RA\ (see 35.3.4). Note also that Fr'(b)~r) in the RU-module structure is the same as b~r) in the ^U*-module structure. We shall assume, as we may, that (i, A) are not only divisible by li, but are also large for all i, so that the vectors b~rj G M are linearly independent when b is subject to tr \b\ = N (a finite set of b's). Here we use that M = RA\ as a /?U*-module. Then from (d) we deduce that Y^(RJ') = 0, hence 2/6 = 0 for all b such that tr |6| = N. (We use the fact that M' is a Verma module.) This is a contradiction. The theorem is proved. 35.4.3. We assume that the root datum is simply connected. Then there is a unique A G X+ such that (i, A) = li — 1 for all i. Let 77 be the canonical generator of RA\. P r o p o s i t i o n 35.4.4. The map x

x~rj is an R-linear isomorphism f —>

Let J = Ei ! n>/ i (H f 6 , i U ) )- lt suffices to show that J © f = R f . An equivalent statement is that a( J) © f = since f is cr-stable. We have a are suc n ( J ) = Y2in>i ^i^flf- If h — then we can write n = a + kb with 0 < a < li and b > 1 and we use the formulas in the proof of Lemma 35.2.2. We see that 0\n) C 0\li)Rf. It follows that cr (J) = S i ^i'^fif- The fact that and f are complementary subspaces of R{ follows easily from Theorem 35.4.2(b). The proposition follows.

278

35. The Quantum Frobenius

Homomorphism

3 5 . 5 . PROOF OF THEOREM 3 5 . 1 . 7

35.5.1. As we have seen in 35.1.11, we only have to prove existence in 35.1.7, assuming that R is as in 35.2.1. By Theorem 35.4.2, there exists a unique i^-linear map P : Rf —• such that P(0^l) • • • 9?*n)9jl • • • 9jr) is equal to 9h • • • 0in if r = 0 and to 0 if r > 0. (Here z'i,..., zn is any sequence in I and j i , . . . , j r is any sequence 2.) in I such that lh >2,...,ljr> We show that P is an algebra homomorphism. It suffices to show that (a) P(x9i) — P(x)P(9i) for any x E R{ and any i such that h>2

and

(b) P(x9^li)) = P(x)P(9\li)) for any x e Rf and any i. (a) is obvious. We prove (b) for x = of*1^ • • • • • • 9jr by induction on r > 0. The case where r = 0 is trivial. Assume that r > 1. We have x = x'Oj where j = jr and x' = e[[11} ••• 9?*n)9jl • • • 9ir_x. If i ^ j then, using the identity in 35.3.1 (after applying
= P{x'9fi))P(9j)

=0

and F ( x , ^ r ) ^ ( 9 f i - r ) ) = P ( x , ^ r ) ( 9 j ) P ( 6 l f _ r ) ) = 0. It follows that P(x/6>J6'f<)) = 0. If i = j, then x'OjO^ = x'O^Oj and the same proof as the one above shows that P(x'9j9f = 0. On the other hand, we have from the definition P{x) = P(xf9j) = 0. Hence (b) holds in this case; both sides are zero. To complete the proof we must compute P(fi\n^) for n > 0. Writing n = a + lib with 0 < a < k and b € N, we have 9\n) = vfli9\a)9fib) as in n) W4 o) <6) Lemma 35.2.2, hence P ( ^ ) = v? P(0 t ! )P(0? ). This is zero if a > 0, i.e., if n is not divisible by U. If a = 0, then P(*4<»>) = p(e\w)

=

(wr'vf1

= (W)-1(v*)6(f,"1)/2«,-

The theorem is proved.

35.5. Proof of Theorem

35.1.7

279

35.5.2. We now discuss to what extent the assumptions 35.1.2(a),(b) are necessary. Theorem 35.1.8 depends only on the assumption 35.1.2(b). This assumption can be replaced by the assumption that I is odd; then essentially the same proof will work (using the results in 33.1, instead of those in 33.2). If the Cartan datum is of finite type, then as pointed out in 35.1.2, the assumption 35.1.2(b) is automatically satisfied; hence 35.1.8 holds in this case. We now discuss Theorem 35.1.7. Here we may again substitute the assumption 35.1.2(b) by the assumption that I is odd. If the Cartan datum is irreducible, of finite type, 35.1.2(b) is automatically satisfied, but 35.1.2(a) can fail; more precisely, if 35.1.2(a) is not satisfied, then we may assume that (a) we have {i,j') = —2 for some i,j G I and I = 2 or (b) we have I = {i,j} with (i,jf) = - 3 , ( j , i ' ) = —1 and I = 2 or 3. In case (a), the algebra is known in terms of explicit generators and relations (see [6]) and the statement of 35.1.7 can be verified by checking that these relations are satisfied in In case (b), the explicit presentation of the algebra is not known for general I; however, for small I (for example I = 2 or 3), it is possible to again write generators and relations, using the formulas in [6], and with their help to verify 35.1.7. We omit further details. We see that 35.1.8, 35.1.7 (hence also 35.1.9) hold unconditionally in the case where the Cartan datum is of finite type. It is likely that they hold without any restriction whatsoever. 35.5.3. Frobenius h o m o m o r p h i s m in t h e classical case. We now assume that I is a prime number and that the „4-algebra R is such that v = 1 and I = 0 in R. (For example, R could be the finite field with I elements.) Let V = I if I is odd and let I' = 4 if / = 2. Then the value of the /'-th cyclotomic polynomial at v = 1 is divisible by /; hence if we define A' as in 35.1.3 (with the present choice of /'), we have that R is an ^'-algebra. Hence Theorems 35.1.7, 35.1.8, 35.1.9 hold in this case. (For Cartan data of finite type, the assumptions in 35.1.2 are not needed; for infinite types, 35.1.2(a) is needed, and 35.1.2(b) is needed only if / = 2.) For Cartan data of finite type, we thus obtain the transpose of the classical Frobenius map or of an exceptional isogeny in the sense of Chevalley.

36.1.1. In this chapter we assume that the Cartan datum is simply laced. As in the previous chapter, we fix an integer / > 1. We preserve the assumptions of 35.1.1- 35.1.3. Note that in this case, 35.1.2(a) is automatically satisfied. Note also that in this case, we have li = I and v^ = v for all i. 36.1.2. We define an /?-algebra as follows. If R = A\ then is the i?-subalgebra of ^f generated by the elements 0^ for various i, n such that 0 < n < I. In the general case, we define Rf = (*4'f)- We have a direct sum decomposition = 0i,(#fi/) indexed by v E N[/]; for R — Ait is induced by the analogous decomposition of and, in general, is obtained by extension of scalars from the special case R — Af. Prom the definitions we see that in the case where R is the quotient field of A', is the ii-subalgebra of R{ generated by the elements 0^ for various such that 0 < n < Z, or equivalently, by the elements Oi for various i (if I > 2) and by 1. (We use the fact that ([n]!) is non-zero in this field for 0 < n < I.) It follows that in this case, Rf is the same as the

L e m m a 36.1.3. Let (a) 0^0^

£ I be such that (i,j')

0. Let ra, n E N be such

G .4/f is an A1 -linear combination of elements uso\3^ where

(b) O^Oj171^ E .4'f is an A'-linear combination of elements d\8^uf3 where We have (i, f ) = —1, since the Cartan datum is simply laced. We prove (a). We may assume that m > 0. Then m > Z, hence m > n + 1 and 7.1.7 is applicable. Thus we can express 0 ^ 0 ^ as an ^-linear combination of terms 0^0^0\s ^ where r, s' E N, r + s' = ra, m — n < s' < ra. For such a term, we have r < n < Z, hence 0^0^ E ^/f and 0\3 ^ is either 0^

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © Springer Science+Business Science+Business Media, 2011 Media, LLC 2010

280

36.1. The Algebra

Rj

281

or, if s' < m, a power of v times 0\s ~ m + l ) g(m~l) ^ s e e 34 \ 2). In the last expression we have ra — I G IZ and 0 < s' —ra+1 < I; (a) follows. Now (b) follows from (a) by using the involution a. T h e o r e m 36.1.4. The R-module

is free for any v G N[/].

L e m m a 36.1.5. . Assume that the root datum is simply connected. Let A G X+ be defined by (i, A) = U — 1 for all i. Let RF be the canonical generator of RA\. The map x 1—• x~rj is an isomorphism —> RA\. It suffices to prove this in the case where R = A'; the general case follows by change of rings. The fact that this map is injective follows from Proposition 35.4.4 (over the quotient field). We prove surjectivity. The argument is similar to the one in the proof of 35.4.2. Note that Rf is spanned as an R-module by products X\X2' — xp where the factors are either of the form 9^ with 0 < n < I (factors of the first kind) or of the form 9 w i t h ra G /N (factors of the second kind). By Lemma 36.1.3, any product xtxt+i with xt (resp. xt+i) a factor of the second kind (resp. of the first kind) is an ^'-linear combination of products of the form where x\,x'2,... ,2^.-1 a r e factors of the first kind and x'T is a factor of the second kind. Applying this fact repeatedly, we see that X\X2 - • -xp is a linear combination of analogous words in which any factor of the first kind appears to the left of any factor of the second kind. Since the .R-module RA\ is generated by elements x~t] with x G we see from the previous argument that RA\ is generated by elements x'~x\x2 • • • x~T) where x' G and xi, X2,..., xp G Rf are factors of the second kind. Since (9\m^)~ri — 0, for any m such that m > (i, A) = k — 1, we have ~r) = 0 for any m G IN such that ra ^ 0. It follows that the .R-module A\ is generated by elements x'~r] with x' G /jf. The lemma follows. R 36.1.6. P r o o f of T h e o r e m 36.1.4. We may assume that the root datum is simply connected. Hence Lemma 36.1.5 is applicable. The isomorphism in that lemma is compatible with the direct sum decompositions according to 1/; it remains to observe that the canonical basis of RA\ provides a basis for the summand corresponding to v. The following result is an integral version of Theorem 35.4.2(b); here R is not assumed to be a field.

282

36. The Algebras

RU

T h e o r e m 36.1.7. The R-linear map x Rf* ®R (/if) x y Fr'(x)y is an isomorphism of R-modules.

• R? given by

It is enough to prove this in the case where R = A'; the general case follows by change of rings. The fact that x is surjective has already been proved in the course of proving Lemma 36.1.5 (actually the products in that proof are in the opposite order of what we need now, so we must apply a to them). Next we note that x is a homomorphism between two free A'modules (the freeness of R{* and of R{ is already known; the freeness of rtf follows from 36.1.4). Hence to prove that x 1S injective over A!, it is enough to prove the corresponding statement for the quotient field of A'. That statement is already known (see 35.4.2(b)). The theorem is proved. 3 6 . 1 . 8 . Let Rj —• flf be the .R-algebra homomorphism induced by change of scalars from the analogous homomorphism for R = A', which is the obvious imbedding.

Corollary 36.1.9. The natural algebra homomorphism ^f —• R{ is an imbedding; its image is the R-subalgebra of R{ generated by the elements 0^ for various i, n such that 0 < n < I. We shall identify

with a subalgebra of Rf, as above.

3 6 . 2 . T H E ALGEBRAS RU,

RU

3 6 . 2 . 1 . Let RU be the P-subalgebra of RU generated by the elements -E^l^, for various z, n such that 0 < n < I and various £ G X. Note that RU is the free ®R (flf opp )-submodule of /jU with basis (1^) (the module structure being (x x') :MH x+ux'~); the same statement holds for the module structure ( x ^ i ' ) : u •—• x~uxf+.

L e m m a 36.2.2. RU is closed under comultiplication. This is easily proved by checking on the algebra generators of RU. 3 6 . 2 . 3 . In the rest of this chapter we assume that 1 = 1' is odd. Then vz = 1. We introduce a certain completion RU of RU as follows. Note that any element RU can be written uniquely as a sum (A) EC,C'€X

X

C,C

where x G are zero except for finitely many pairs (CCOWe now relax the last condition and we consider infinite formal sums (a) in which the only requirement is that there exists a finite subset F C X

283

36.2. The Algebras RU, RU

such that G are zero unless ( - (' G F . The set of all such formal sums is denoted by (Note that the set F varies from element to element of #u.) The /^-algebra structure of RU extends in an obvious way to an /^-algebra structure on /^u; this algebra has a unit element 1^. opp <S>R (/?f )-module structures on RU extend in Note that that the two an obvious way to two Rf (^f o p p )-module structures on ^ Let J For any X*-coset c in X, we define l c = ECGC (resp. J') be the iZ-submodule of RU generated by the elements x+lcx'~ for various c G X/X* and x, x' G Rf. (resp. x~lcx'+) L e m m a 36.2.4. (a) F^u C J for any u G J and any i,b such that 0 < b < I. (b) J is an R-subalgebra of RU and J = J'. To prove (a), we may assume that u = E ^ • • • e\^\cX'~ where a i , . . . , ap G [0, Z — 1], c G X/X* and x' G j?f. We argue by induction on p. ••• If p = 0, the result is trivial. Assume that p > 1. Let x\ = We have u = \dE\*x)xtx'for some c' G X/X*. If i ^ ii, the desired result follows immediately. Assume that i = i\. We have if>u =

E

E

*

a\ + b — (i, C)" t

L

C-(ai+*>-t)i'*i

X

l

X

For each £ in the sum we have 0 • We have ai + b - (i, Co) ai+6-(t,C>' = t t (see 34.1.2); here we use the hypothesis that Z is odd. Hence we have j f >« =

E * t>0;t
ai + 6 - (i, Co)' t

'

lc-

tion hypothesis, we see that F^u G J ; (a) is proved. Using repeatedly (a) and the identities l c l c / = £ CjC /l c , for c, c' G X/X*, we see that J is a subalgebra of RU. Again, using (a) repeatedly, starting with lcx'+ G J for c G X/X*,x' G R{, we see that J ' C J . By symmetry, we have J C J ' , hence J — J'. The lemma is proved.

284

36. The Algebras

R

f , RU

36.2.5. Definition. is the JR-subalgebra J = J' of Note that the algebra Ru has a unit element l c ; here c runs through the set X/X*. The set X/X* is finite, since by the definition of X*, the map C —• (i,C) m ° d ' defines an injective map X/X* —> (Z/IZ)1. Prom the definition, we see that is the free <&R (/jf opp )-submodule of R U with basis { L | C G X/X*} (the module structure being ( X ® X ' ) : u x+ux'~)\ the same statement holds for the module structure (x®x') : u — i> x~ux'+. C

Notes on Part V 1. T h e results in Chapter 32 are due t o to Drinfeld, for R = Q ( f ) . T h e extension to the case where R is a field and v is a root of 1 in R is new; it answers a question that Drinfeld asked me in January 1990. 2. T h e fact that the simple integrable modules of a K a c - M o o d y Lie algebra admit a q u a n t u m deformation (Chapter 33) was proved in [4]; for Cartan d a t a of finite t y p e this was also stated in [8], but the proof there has a serious gap. (It appears [2] that, for Cartan data of finite type, this result was known t o Drinfeld.) T h e results in 33.2 are new. 3. T h e results in Chapter 34 have appeared (for I odd) in [5]. 4. T h e quantum Frobenius homomorphism, for Cartan d a t a of finite type and w i t h some restrictions on I , was implicit in [5] and explicit in [7]; its generalization given in Chapter 35 is new. 5. In the case where R is a field of characteristic zero, v is a root of 1 in R, and the Cartan d a t u m is of finite type, is the finite dimensional Hopf algebra defined in [6], [7] (with some restrictions on the order of v). T h e extension t o infinite t y p e s is new.

REFERENCES

1. C. Chevalley, Seminaire sur la classification des groupes de Lie algebriques, Ecole Norm. Sup. Paris, 1956-58. 2. V. G. Drinfeld, On almost cocommutative Hopf algebras, (Russian), Algebra i Analiz 1 (1989), 30-46. 3. V. G. Kac, Infinite dimensional Lie algebras, Birkhauser, Boston, 1983. 4. G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras,, Adv. Math. 70 (1988), 237-249. 5. , Modular representations and quantum groups, Contemp. Math. 82 (1989), 59-77, Amer. Math. Soc., Providence, R. I.. 6. , Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257-296. 7. , Quantum groups at roots of 1, Geom. Dedicata 35 (1990), 89-114. 8. M. Rosso, Finite dimensional representations of the quantum analog of a complex semisimple Lie algebra, Comm. Math. Phys. 1 1 7 (1988), 581-593.

Part VI BRAID GROUP ACTION

In the classical theory of semisimple Lie algebras, the Weyl group plays an important role. Now the Weyl group is not quite a subgroup of the corresponding simply connected Lie group; only a finite covering of it is. As Tits has shown [9], one can choose such a covering which is naturally a quotient of the braid group. In particular, there is a small obstruction to making the Weyl group act on a simple integrable module for the Lie algebra; what acts naturally is a quotient of the braid group, which is a finite covering of the Weyl group. Since in this case, the obstruction involves only signs, it is almost invisible. In the quantum case, the obstruction becomes quite serious, and in this case, not even a finite covering of the Weyl group can be made to act; the braid group still acts, but in general not through a finite quotient. In Part VI we explain how the braid group acts on integrable U-modules and on U itself. (In fact there are several braid group actions, but they are related to each other in a simple way.) The symmetries T'i e, T"e of an integrable U-module have already been introduced in Chapter 5. In Chapter 39 it is shown that these symmetries satisfy the braid group relations, hence they define braid group actions. These symmetries are studied simultaneously with the analogous symmetries of U (see Chapter 37) which also satisfy the braid group relations. In Chapters 38 and 40 we study the connection between the symmetries of U and the inner product (,) on f. In Chapter 41 we define a braid group action on RU and on its integrable modules for any R. In Chapter 42 we assume that the Cartan datum is simply laced and of finite type and we use the braid group actions to give a purely combinatorial parametrization of the canonical basis B in terms of reduced expressions for the longest element of W.

CHAPTER

37

T h e Symmetries T-,e , T"e of U

37.1.1. In this section we fix i E I and e = ±1. Recall that in 5.2.1 we have defined some symmetries T[ , T"e : M —• M for any integrable module M of U. In the following proposition we define analogous symmetries P r o p o s i t i o n 37.1.2. (a) For any u E U, there is a unique element u" £ U such that for any integrable U-module M and any z E M, we have (b) The map u

u" is an automorphism of the algebra U, denoted by

(c) For any u £ U , there is a unique element v! £ U such that for any integrable U-module M and any z £ M, we have T'i e[u'z) — uT(e(z). (d) The map u

u' is an automorphism of the algebra XJ, denoted by

Thus, for any integrable U-module M, any z £ M, and any u £ U, we

37.1.3. The proof will be given in 37.2.3. The proof will give at the same time the following formulas for the values of the automorphisms T"_ e , T[ e :

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, Media, LLC 2010 DOI 10.1007/978-0-8176-4717-9_30, © Springer Science+Business Science+Business Media, 2011

287

288

31. The Symmetries

Tl[_c{Fi) =

T'ie

T^c(Ei)

= -FiK-ei,

K-eiEj)

= £

TlUFj)

= Zr+s=..{i,r)(-iyvrrFt)FJFt)

f

, T(e

-KeiEi;

O

T

3 * ii for j ? i;

More generally, we have, for any n > 0:

T

U

^

)

f O T

=

I J t e ( f f >) = £ r + s = _ ( . ,

Ty_e(Ffn>)

,

)

f

n

o

r

3 *

^

j / i;

= (-l)"t,-en("~1)tfeni£f); for

=

i * i;

=

for j * i.

3 7 . 2 . CALCULATIONS IN RANK 2

37.2.1. Given i j in / , we set a = ~(i,j') set (compare 7.1.1) r -—r^n,m;e -— /f+ 2-1,.7;n,m;e n ,m;e ~~

r/

( V

v

A

G N. For any m , n G Z, we

/

^ U

r+s=m

_ x

x,j;n,m;e

/ x

_

n,m;e

r

/ v ^ r+s=m

= lfc.,mi. =

yi,j;n,m;e

2/n,m;e

=

f-DVr(an'ra+1)£;!,)E!n)£;!r)eu+' A

/

^ ^

£

'

g U";

/n,m;—e

V ( _ l/ ) r% v / y \ r+s=m

e r

(

a n

-

m + 1 %

With this notation, we have the following result.

) J| r ( %r )

G

U~.

,

37.2. Calculations

in Rank 2

289

L e m m a 37.2.2. Let M be an object of C' and let z 6 M. We have (a) T['_e{xn
=

E^T^e(z);

=

Ef>rtie(z); Fln)Tl[_e(z);

= =

F^TlJz).

We may assume that z e M \ We have i „ i m ; e z € M x + a n i ' + n i ' . Let g = {i, \),g'

= (i, A + a n i ' + ra/) = g + an.

We have T t W n ^ z ) =

Y.

(-l)

b

vr(-

a

^

b )

E^F^Efx

n

^z.

a,b,c£N; — a+b—c=g' By 7.1.3(a) and 7.1.5(a), we have /r( c ) r

_ ..eacn

17(c)

Hence

By 7.1.3(b) and (a), this can be written: y^

b y^

^b+b'v~e(-ac+b)+eacn-e(bb'-b')

x E ^ K ^ . x ^ ^ F ^ E ^ z

x

v

i

eacn-eCftft'-bO-efc^^A+cz'-fcz'+am'+nj') 77.(0)

j?(b-b') r-.(c)

&i

^n,an-6';e"j

v E E l ( ' r «! a,6,c€N;-a+6-c=j' 6'=0 a'=0

cm — 6' + a' a'f

^

-2;

290

37. The Symmetries

T'i e , T(e

where (e) t = e(ac- b + acn + bb' + b' - b'g - 2b'c- ab'n + aan + aa' -a' - 2ab'). We make a change of variable: a" = a — a' ,b" = b — b'. The exponent (e) then becomes t = e(a'(—1 + b" — a" — c — g) + {a"c - b") + {a" + c ) ( - a " + 6" - c - g)) We use the condition — a' — a" -f b' 4- b" — c = g + an which is equivalent to — a + b — c = g'. We have x n y a n -b'+ a ',e = 0 unless 0 < an — b' + a' < an (see 7.1.5), or equivalently, 0 < —a" + b" — c — g < an; hence we may add the condition 0 < — a" 4- b" — c — g < an in the summation without changing the sum. In the presence of the equation — a' — a" 4- b' + b" — c = g + an , the inequality a' > 0 implies b' > 0 since a" — b" + c + g -f an > 0. The sum becomes ^_1y"ve((a"c-b")+(a"+c)(-a"+b"-c-g)) a",b",c€H;0<-a"+b"-c-g
£

{_ir'vea>(-l-a"+b"-c-9)

-a" + b"

-c-g]

%n, — a"+b"—c—g;e

a'GN

x E ^ F ^ E ^ Z . The sum over a' is zero unless — a" + b" — c — g = 0 (see 1.3.4). Hence the sum becomes

a",6",ceN;-a"+6"-c=s

which equals E^ n ) Tl f _ e (z). This proves (a). Using 5.2.3(b), we have Tt[_ e z = ( - 1 ) « - » v r < i , * > 7 l , - e z and TH- e (x n , a „;ez) = ( - 1 ) < i ^ + a " v - e ( i - X > ~ e a n T l _ e ( X n t a n : e z ) . Hence (a) implies Tl_e(xn,an;ez) = ( - 1 r « v r n E j n ) T l _ e ( z ) .

37.2. Calculations in Rank 2

291

Substituting here x

—/ 1\an-ean x n,Qn;-e — V L) vi n,an;e)

we obtain Replacing e by — e, we obtain (b). We now apply (a),(b) to z and W M instead of M. The action of T"_e (resp. T'i e) in ^ M is the same as the action of T[ _ e (resp. T£'e) in M; thus, (a) and (b) for UM imply Tl_^(Xn,«n-,e)z)

Fjn)Tl_e(z)

=

and = We have Cj(x n?an; e) = Tf

y'n,an-e

^

and

( /

x

^( 'n,Qn-,e)

_

i,-ewn,an;-e

/

—

j

M.

2M,an;-e. z

=

Hence

i.-C

and T!'te(yn,an:-ez)

=

F^T^z.

Replacing e by —e, we obtain (d) and (c). 37.2.3. P r o o f of P r o p o s i t i o n 37.1.2. We show the uniqueness of u" in 37.1.2(a). It suffices to show that, if u € U satisfies T"_e(uz) = 0 for any integrable U-module M and any z G M, then u = 0. Since T"_e is invertible on M, we have that u annihilates any integrable U-module. But this implies u = 0 (see 3.5.4). Thus the uniqueness in 37.1.2(a) is proved. To prove existence, we observe that, if (a) holds for two elements U\,U2, then it also holds for U\U2 and au\ + bu2 for any a, b 6 Q(v); indeed, it is clear that (U\U2)" = u'lu'i and (au\ + 61*2)" = o,u'{ + bu Hence it suffices to prove the existence of u" in the case where u is one of the generators E j , Fj, K^ of U. For K^, this follows from 5.2.6. For Ei,Fi this follows from 5.2.4. For Ej,Fj with j ^ i, this follows from 37.2.2. This proves (a). The argument above shows that u >—• u" is an algebra homomorphism. In an entirely similar way we see that (c) holds and that u' is an algebra homomorphism. It remains to show that (u")' = u and (u')" = u for all u. For any M, 2 as above, we have, using 5.2.3(a): (u'Y'z = T[ieTi:_e(u')"z

=

T ^ T ^ Z )

=

<

e

l?> =

UZ.

Thus (u')" — u annihilates any integrable U-module, hence (u')" = u. The identity (u")' = u is proved in the same way. The proposition is proved.

292

37. The Symmetries

T'i e

,T{e

37.2.4. We have UT^LJ

= T[[e : U

U and aT[ea = T['_e : U

U.

The symmetries T/ e , T"e are related to the involution ~ : U —> U as follows: KM Let

= ir,-.(«).

for a11

=

G U be such that KiuK-i

«

€

u

-

= v"u We have

= (-i)B«rii», or equivalently, These formulas are proved by checking on generators. Proposition 37.2.5. Let i / j in I. For any m G Z, we have (a) -^i.eC^iJjl.mje) = xi,j;l,— (i,j')-m;e> (b) 37'_e(aJ*,j;l,m;c) = ^i,j; 1, — < i ) — m.;e * The two formulas are equivalent, by 37.1.2(d), so it suffices to prove (a). For m < 0 or m > —(i,j'}, both sides of (a) are zero. So we may assume that 0 < m < —(i,f). We argue by descending induction on ra. For ra = — (i,j'), (a) follows from 37.2.2(b). Assume now that (a) is known for some ra such that 1 < m < ~{i,j'). We show that it is then also true for ra — 1. Recall from 7.1.2(b) the identity x

ITI 1]iK—eixi,j',l,m—l;e'

i,j;l,m\eFi — [ (i, j )

Applying to it the algebra isomorphism a : U —> U o p p , which interchanges x i,j;i,m;e and x'jj.i )Tn;e , we deduce that ~Xi,j-,l,m-,eFi + FiXi,j;l,m;e = H » , / } ~

m

+ 1]

We apply T[ e to this; using the induction hypothesis, we have H i , j') - m + 1] =

Tle{xfi j ; l

=

x

r n . e )EiK_ e i

- E i K ^ l M ^ e )

i,j\\, — (ij')— m,eEiK— ei — EiK— eixi,j;l,

/

-rp

e((ij')+2m) jp

— {ij')—m;e

\f>

37.3. Relation

of the Symmetries

with

Comultiplication

293

The last equality follows from 7.1.2(a). Since [— ( i , f ) — m +1]* 0, we deduce that T / e ^ m _ 1 ; e ) = x i ) j ; i ) _( i ) j /)_ m + i ; e . This proves the induction step. The proposition is proved. 3 7 . 3 . RELATION OF THE SYMMETRIES WITH COMULTIPLICATION 37.3.1.

We define two elements 14' = £ ( - i ) X B ( B " 1 > / 9 { » } i * i B ) ® L/ =

^vn(n-l)/2{n].F(n) ^

E(n)

of (U V J (see 4.1.1 and 4.1.2). P r o p o s i t i o n 37.3.2. The following equality holds in (U ® UJT, for any uG U : (a)

( T ^ t t T ^ A i T ^ u ) = L5A(ti)L?.

It suffices to show that for any integrable U-modules M, N, the two sides of (a) act in the same way on M7V. Let x G M, y G N. Using 5.3.4 twice, we have

L$A(u)L"(x ® y) = L'!-\uL'l(x

The proposition is proved.

® y))

Symmetries and Inner product on f

38.1.1. In this section we fix i £ I. For any j € I distinct from i and for

(fihj'i7?1)

is the same as the element /i,j ; i,m;-i in 7.1.1.)

Let f [i] (resp. af[i]) be the subalgebra of f generated by the elements f(i,j;m) (resp. f'(i,j-,m)) for various j £ / , distinct from z, and various ra £ Z. Since a : f —> f interchanges f(i,j;m) and f'(i,j\m), we have

Note that (b) follows from (a) by applying cr. We prove (a). We consider a product y\V2" 'Vn of elements in f in which each factor is either Oi or is of the form / ( i , j\ ra) for some j different from i and some ra £ Z. Assume that there are two consecutive factors ya = / ( i , j\m) and ya+1 = Oi. Using

which follows from 7.1.2(a), we see that we may replace yaya+1 by a scalar multiple of ya+i2/a plus a scalar multiple of f(i,j;m + 1). Using this procedure repeatedly, we see that y\y2 • • • yn is equal to a linear combination of products y[y2 • • • y'n, which do not have consecutive factors of the form described above, hence have the property that for some s > 0, we have various j,ra. Thus, y\y2 •••y n belongs to Y 2 t > a n y wor< ^ the generators 0j ( j £ I) is of the form yxyi- — yn with ya as above, since

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © Springer Science+Business Media, LLC 2010 2011

294

38.1. The Algebras f[i),

a

f\i\

295

L e m m a 38.1.3. There is a unique algebra isomorphism x > g(x) of f[t] onto ai[i] such that T/ / 1 (x + ) = g(x)+ for any x e f[i]. We have Vi_1(x,+) = g~l{x')+ for any x' E ai[i).

Since T"\,T'i _l are inverse algebra isomorphisms, it suffices to note that

and Tl,-i(f(iJ;m)+)

-(i,j')-m)+,

= f(i,j;

for any j ^ i and any ra E Z. (See 37.2.5.) L e m m a 38.1.4. Assume that x e f satisfies T"1(x+) G U + . Then ir(x) = 0. We may assume that x is homogeneous. By 3.1.6, we have (a)

=

Vi-Vi

By 38.1.2, we can write ri(x)/(vi~v71)

=

J2e\% t> 0

and t>o

where yt,zt belong to f[i] and are homogeneous. By 38.1.3, we have T"\{yt) e u + and Tl'^zi) e U+ for all t > 0. Applying T^ to both sides of (a), we obtain - T^ix^K-iEi (b)

= £(-1) t> o

+

k.iEiTl'il(x+) ~

KiTl'M))-

By our assumption, the left hand side of (b) is in K - i U + , hence so is the right hand side. Using the triangular decomposition in U, we deduce from (b) that Tl'x{y+) = 0 for all t > 0 and T ^ z f ) = 0 for all t > 0. Since T'A is bijective, this implies zf = 0 for all t, hence zt = 0 for all t and »r(:r) = 0.

296

38. Symmetries

and Inner Product on f

L e m m a 38.1.5. Let xt (t > 0) be elements in the kernel of ir, which are zero for all but finitely many t. Assume that 0^xt = 0. Then xt = 0 for all t. This follows from 16.1.2(c) in the setup of 17.3.1. P r o p o s i t i o n 38.1.6. (a) The following three subspaces o f f coincide: f[t]; {x G f\Tl'yl(x+) G U + } ; and {x G f|
G U + } ; and {x G f | n ( x ) = 0}.

Note that (b) follows from (a) by applying a. We prove (a). By 38.1.3 and 38.1.4, the first space in (a) is contained in the second and the second in the third. Let x G f be such that ir(x) = 0. It remains to prove that x G f[i]. By 38.1.2, we can write x = ] T ) t > w h e r e xt G f[i] for all t. By 38.1.3 and 38.1.4, we have ir(xt) = 0~for all t. We then have 0 = (x0 -x) + J2t>i ^ x t where x0 - x and xt for t > 0 are in the kernel of ^r. Using 38.1.5, we deduce that XQ — x = 0 and xt = 0 for t > 0. In particular, we have x = x0 G f[i]. The proposition is proved. L e m m a 38.1.7. r(f(ij;m))

= 1® m m—t—1

f(i,j;m)

+ E II (i-vrafc+am-2a-2K(m"t)/(i,j;i)®^ra-<), t=0 h=0

r(f'{hj;m)) = f'(i,j-,m)® rrt m — t — 1 + E

t=0

II

h=0

1

(! - v72h+2m-2a-2)vlim~t)elm't)

We set a = —(i,j'}.

0 f'(i,j;

t).

Using 1.4.2 and the fact that r is an algebra

38.1.

The Algebras f[i], a f [ i ]

297

homomorphism (see 1.2.2), we compute r(/(i,i;m))=

£

( _ 1 ) r' + r" f ; r (r' + r")(a-m + l) + r'r" + .V

x {e{p ® e f ) ) ( e j ® 1 + 1 ® omoW

® e\s,,))

r'-f-T^' + s' + s" —m r" + s"

+r'+r"+s'+s"=m E s"=m ("D >+ "--{T'+r"){cL-m+\)+r'r"+8's"+2r"s'-9'cx T V

— ^^^jjr'^f'Cm-r'-s'J-r'Ca-m+l) ^ ( —l)f r',s' r" ,s";r"+s"=m—r'—s' r" + s" r"(r'+s'-a-(a-m+l)) X Ve ^ B f ^ e f ^

V jS

E

(-i)'^-1^

e t ' ^ v e p e ^

m—r' — s' — 1 JJ (1 _ h—0

_ r',s' x e f W ' ) ®

r ' + s'

+

£

v-2h+2m-2a-2^

(-ir'Vr"(a~m+1)i ®

flp

m m—t— 1 = i®/(i,j;m) + £ n (i-®r2fc+2ra-2a-2K(ra~t)/(<,i;i)®e<(ra_t)t=0 /i=0 This proves the first formula of the lemma. The second formula follows from the first, using the formula r(a(x)) = (a ® a)lr(x) (see 1.2.8(a)). L e m m a 38.1.8. Let x G f[i] and let x' — g(x) be the corresponding element in ai[i] (see 38.1.3). (a) We have r(x) G f[i] ® f and r(x') G f ® ai[i]. (b) Let 'r(x) G f [i] ® f[i] be defined by r(x) - V(x) G f[i] ® Q{f. We use the direct sum decomposition f = f [«] © Oii (see 38.1.5).

298

38. Symmetries

and Inner Product on f

Let"r(x') G ^ [ i ] ® ^ ] be defined by r(x) ~"r{x') G ffl* ( g ^ f [i]. We use the direct sum decomposition f = We then have (g®g)('r(x)) = "r(x'), where g is as in 38.1.3. We prove (a). If x\,x2 G f satisfy r(xi) G f[z] f and r(x2) G f[i] ® f, then r(x\x2) G f [i] f since r respects products and f [«] is closed under multiplication. Hence to check that r(x) G f[i] <8> f for x G f[i], it is enough to check this in the special case where x = f(i,j; ra). But this follows from the previous lemma. This proves the first assertion of (a). The second assertion of (a) is proved in the same way. We now prove (b). Let (Zh)heH be a basis of the vector space f[i], consisting of homogeneous elements; then (g(zh)heH) is a basis of the vector space af[i] (see 38.1.3). Let f(h) = \zh\,f'(h) = \g(zh)\ be the corresponding elements of N [/]. Using (a), we can write uniquely r X

()

=

C

n>0;h,h'EH

n;h,h'Zh <8> Oin)Zh>, dnih,h'9{zh)9\n)®g{zh,)t

r( x')= n>0;h,h'€H

where c(n; h, h'), d(n; h, h') G Q(v) are zero for all but finitely many indices. By 3.1.5, we have (c) A(x+) = (d) A(x'+) = Zn>0;h,h>eH dn,h,h. g{zh)+E^ Knht) ®g{zh>)+. By definition, we have A(V+) = A ( T ^ ( x + ) ) . Applying (d) we obtain

to

n>0;h,h'eH

By 37.3.2, we can replace ( T j . j 07; , ) _ 1 )A(r/ , 1 (rE + )) by L / i A(x + )Lf, which by (c) equals n>0,h,h'eH

Thus, we have n>0;h,h' = L5(

J2 n>0,h,hfeH

cn.hih,z+km)+ni&E^z+)b':

38.1.

The Algebras f[«], ai[i]

299

or equivalently, n>0;h,h.'(zH n>0;h,h'EH

(equality in (U ® UJ). We consider the U ® U-module u MQ ® w Mo, where Mo is a Verma module. Both sides of the previous equality act naturally on this U ® Umodule. Applying them to the vector , where f G w Mo is the canonical generator, we obtain the equality

E

n,t>0;h,h'eH

vf-^m-irvr^dn-,^

(e) n,t>0;h,h'eH

(equality in u M Q ) . Since F = 0, only the summands with n = t — 0 contribute to the left hand side of (e). Let 7r : " M Q / E ^ M Q be the canonical map. After applying 1 7r to (e), the summands with (n,t) ^ (0,0) (in the right hand side) go to zero; hence (e) implies the equality

h,h'EH

h,h'eH

U

in M 0 ( " M 0 / E i " M 0 ) . The vectors are linearly independent in u MQ; hence we deduce that (d0;h,h' ~ co;hthf)Azh'0

=

0

h'eH

in "MQ/E^MQ), for all h. Hence y:

(do;h,hf ~ co-h,h')zh' G Oii,

h'EH

for all h. Using the fact that f [i] n ( ^ f ) do,h,h' ~ co;h,hf = 0 for all h, h!. By definition, we have 'r(x) =

=0 c

(see

38.1.5), we

deduce that

o-,h,h'Zh ® zh>, and "r{x')

=

Ylh,h'eH do-,h,h'9(zh) ® 9{zh<) and the equalities d0.h,h> = CQ.h,h> imply that (9 ® 9)('r(x)) — "r(x'). The lemma is proved.

300

38. Symmetries

38.2.

and Inner Product on f

A COMPUTATION OF INNER PRODUCTS

P r o p o s i t i o n 38.2.1. For all x,y G i[i], we have (a) (g(x),g(y))

= (x,y).

Assume that we are given two elements z', z" G f[i) such that (g(z'),g(y')) and

= (z',y')

W),g{y")) = (z",y")

for all y', y" G f [?]. We show that we then have (g(z'z"),g(y))

=

(z'z",y),

for any y G f[i]. By definition, we have (z'z",y)

=

{z'®z",r(y))

and (g(z'z"),g(y))

=

(g(z')®g(z"),r(g(y))).

We have (z",0,f) = 0 since ir(z") = 0 (see 1.2.13(a)); hence (z'®z",r{y)) = (z'®z"!r{y)) (with the notations of lemma 38.1.8). Similarly, ( g ( z ' ) , f 0 i ) = 0, hence W ) ®9{z"\r{g{y)))

= (g(z') ® g(z"),"r(g(y))) = ({9®9)(z'®z"),(g®g)'r(y)).

The last equality comes from lemma 38.1.8. By our hypothesis, we have (( z"), (g ® g)'r(y)) = (z! <8> z", 'r(y)). Combining the equalities above, we obtain (g(z'z"),g(y)) = (z'z",y), as claimed. Since the algebra f[i] is generated by the elements f(i,j; m), we see that it is enough to prove (a) under the assumption that x = f(i,j; m). We can assume also that y is homogeneous of the same degree as / ( i , j ; ra). Since y G f[i], this forces y to be a scalar multiple of /(«, j ; ra). Thus, we are reduced to verifying the identity (f(hj-,rn)J(iJ;m))

= U'ihj;™!),

f'(i,y,m'))

38.2. A Computation of Inner Products

301

for any ra, m' such that ra + ra' = a = —{i,f). (See the proof of 38.1.3.) We have f(i,j; ra) = 0j0\m^ mod 0jf; since ( f ( i , j ; ra), = 0, as above, we have (f(hj;m),f(i,j;m)) = (/(«, j;ra),0j0- m ) ). This is equal, by definition, to (r(f(i,j;m)),03; ® 6><m)) and, by 38.1.7, this equals

(b) nr=o (1 -t;rafc+am-9°-2)(fli,flj)(«{M),«Jm)).

Similarly, we have f{i,j',m!)

=

^Oj mod f ^ ; since

we have

= (/'(z\ j; ra'), 0 ^ ) . (f(ij;m')tf(i,j;m')) This is equal, by definition, to (r(/'(i,,7; ra')), 0| m ) <8> 0j) and, by 38.1.7, this equals

(c) r c ^ a -

We substitute and in (b) and (c) by the expressions given by 1.4.4; we see that the expressions (b),(c) are equal. The proposition is proved. 38.2.2. We say that a sequence h = (ii, 12, • • •, in) in I is admissible if for any a, b such that 1 < a < b < n, we have (a) € U+ and Assume that an admissible sequence h as above is fixed, and that we are given 0 < p < n. An element x € f is said to be adapted to (h,p), if for any a such that 1 < a < p we have and for any b such that p + l < 6 < n , we have Given such x and given a sequence c = (ci, C2,..., cn) G N n , we define L(h,c,p,x) G f by L(h,c,p,x)+ ( 7Ti(cp+2) \ npl rpl Tp(cp+l) i rpf i P + i , - U ^ i p + 2 )• ' •- £ t p + 1 ,-l i t p + 2 > -l -

I

rpl ( jp(cn) \ in-l,-l\^in )

The definition is correct, since the factors in the right hand side are in U + , by (a),(b).

302

38. Symmetries

and Inner Product on f

Proposition 38.2.3. Let c = (ci, c 2 , . . . , c n ) G N n , c' = (c'j, c^,..., G n N and /ei x , i ' 6 f be adapted to (h,p). We have the equality of inner products (L(h,c,p,x), L(h,c',p, x')) = (x,x')Y[ns=1{0^\0{£)). For any i G I, t,t' G N and y,y' G f[i] = ker(jr), we have (a) ( f i f W V ) = ( ^ . f f ^(y,y')Indeed, we may assume that y, y' are homogeneous. We can write r(yf) 2/i ® 2/2 with 2/1,2/2 homogeneous, and \y[ \ £ {«, 2 i, 3i,... } (see 38.1.8(a)) and our assertion follows easily from the definitions. Similarly, if y, y' G a f[i], we have (b) ( t f i V f l p ) = {e?\eV){y,y'). Assume first that p < n and that the proposition is true when p is replaced by p + 1. Let c, c' be the sequences defined by c p + i = Cp+1 = 0 and cs = c s ,c' s = c's for s ^ p + 1. Let x = g{x) G f , i ' = g(x') £ f, where g is as in 38.1.3 (with i = ip+1). Let p = p + 1. It is clear that x,x' are well-defined and that they are adapted to ( h + 1). We have from the definitions L(h,c,p,x)+

=

Efc*ll>rir+u_1(Hh,t,p,x)+).

By our assumptions, we have i ;

hence

H l r l

( L ( h , c , p , i )

+

) 6 U

+

;

L(h,c,j5,x) G af[ip+i]

and 7*, + l i _ 1 (Z,(h,c,p,x)+) = (g_1(L(h, c,p, x)))+ where g is as above. Similarly

and L(h,c',p,x') G

CT

%+1],7^p+i)_1(L(h,c,,p,x/)+)

= (^" 1 (L(h,c , ,p,x , ))) + .

Using (a), we have (£(h, c, p, x), L(h, c', p, x'))

= ( ^ ' ' . ^ " ' j W h . 2. P . ^ ( h . e'.P. *'))•

38.2. A Computation of Inner Products

303

The last equality follows from 38.2.1. Using now this hypothesis, we see that the last expression equals We now replace (g(x),g(x')) by and we see that the formula in the proposition holds in this case. Using this argument repeatedly, we are reduced to the case where p = n. In the rest of the proof we assume that p = n. We now have L(h, c, n , » ) + = x+1ZaTZ,_u1

•• •

•••

We now assume that n > 1 and that the result is known for n — 1 instead of n. Let h be the sequence (ii,i2, • • • ^ n - i ) - Let c = (ci,02,... , c n _ i ) , c ' — (cj, , . . . , c j ^ ) . Let x = g~1x,x' = g~lx' where g is as in 38.1.3, with i — in. It is clear that x,x' are well-defined and they are adapted to (h,n — 1). We have from the definitions L(h, c, n, x)+ =

i(L(h, c, n -

1,X)+)E£>\

and similarly

By arguments almost identical to the ones above (using (b) instead of (a)) we see that (L(h, c, n, x), L(h, c', n, x'))

= (g(L(h,c,n

- 1,x)),g(L(h,c',n

- 1,

= (L(h, c, n - 1, x), L(h, c', n - 1, x ' W ^ ^ ) (g-\x),g-Hx,))fl(e£)A<;°)) S— 1

=

s=1 Using this argument repeatedly, we are reduced to the case where n = 0. We now have L(h, c, n, x) = x and the result is obvious. The proposition is proved.

39.1.1. S t u d y of t h e s y m m e t r i e s T[e on A\. In this and the next subsection, we assume that the root datum is X-regular and y-regular. L e m m a 39.1.2. Let i = (ii,i2>--- AN) be a sequence in I such that silsi2 - • • siN is a reduced expression of an element of W. We have

We argue by induction on N > 0. For N = 0, there is nothing to prove. Assume that N > 1. Let 7](i) be the left hand side of (a). Let i7 = (^2^3,... ,iAr). Then the induction hypothesis is applicable to i' and

Note that r/(i') belongs to the si2si3 • • • siN(A)-weight space of A^, and we have, by definition, a\ = (ii,Si 2 Si 3 - — SiN(A)}. Using 5.2.2(a), we see /

Since Eil(rf(i )) belongs to the Si2Si3 - -Si N (A) -h Zj-weight space, it is enough to prove that this weight space is zero. If this weight space were non-zero, then the SiN • • * si3si2(si2si3 • • • siN(A) + i'i)-weight space, which is the (A + siN * • • Si2(i[))-weight space, would also be non-zero. But then A, hence S{N • • • Si3Si2 (ii) < 0 which we would have A + SiN • • • contradicts the assumption that SixSi2 • • - S{N is a reduced expression. The 39.1.3. A l e m m a on monomials. Assume that s^s^ • • • sin is a reduced expression in W. Let A e X be such that (ip, A) > 0 for p = 1,2,... , n.

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © Springer Science+Business Science+Business Media, LLC 2010 2011

304

39.2. Braid Group Relations for U in Rank 2

305

L e m m a 39.1.4. For any i e I, we have EiX = xEi + yiiK-iV^-1

- KiV-{i>X)+1)

for some

Vi

G IT.

We argue by induction on n. For n = 0, there is nothing to prove. n) J Assume that n > ^ F 13^ • - • F^ By the induction — 1. Let x' = F %i ln . hypothesis, we have = x'Ei +y[(k-iv\i'x)~l

-

for some y[ G U~~. Hence =

EiF^x'

= I F ^ * ' 1

* '

= F^x'Ei

'

V l - V ;

+F ^ y ' i K - ^ -

1

-

Ki«r+1)

vv - v r 1 It remains to note that ai =

—02^2 ~

+

which follows easily from the definitions. 39.2.

B R A I D G R O U P RELATIONS FOR U

IN R A N K 2

39.2.1. In this section we assume that I consists of two elements i ^ j is and that the Cartan datum is of finite type or, equivalently, (i,jf)(j1if) equal to 0,1,2, or 3. We define accordingly h = h(i,j) to be 2,3,4 or 6 (see 2.1.1). In this case, the root datum is automatically X-regular and F-regular. By interchanging if necessary i and j, we may assume that either (i,j') = (j, i') = 0, or ( j , i') = —1 and (i, j'} G {—1, —2, —3}. We shall consider each case separately. 39.2.2. Assume that ( i , f ) = - 3 , (j,i') = - 1 , so that h = 6. Note that Vj — v f . As in 37.2.1, we set si,m i e =

£ ( - 1Y v r ( 4 ~ m ) E ^ E M ' ) 6 U+, r+s—m

=

£ ( - l ) X e r ( 4 " m ) E \ S ) E J E \ T ) G U+. r+s—m

306

39.

Braid

Group

Relations

Recall from 7.1.2 that -vf^~2m)

EiXiyrn.e + X\,myeEi = [m + l]i£l,m+l;e

and -EiXi^m-e

+

X i

f

m .

e

F i

=

[4 -

m]iK-eiXi,m_i;C.

Applying cr, which interchanges xi )Tn;e and

m;e,

we deduce that

- t ,, e ( 3 - 2 " , ) x' l i m ; e E i + £?ix' lim;e = [m + l]ix' 1>m+1;e and -x'l,m,eFi

+ Fixl,m;e = [4 ~

™]i*'l,m-l;e^et.

By 37.2.5, we have T ^ E J ) = xly3.e and T['_E(EJ) = x'1)3.e. By 37.2.5 with the roles of i j interchanged, we have T^ e (Ei) = - v ^ E j E i + E i E j = x'1Ve and similarly, T^_ e (Ei) = x M ; e . By 37.2.5, we have Ti[_e{x M ; e ) = x' i a . e and T/ e (xi 1;e ) = xi,2 ;e . We have 2;e) = [ ^ ^ - . ( - t ^ l . l ; ^ +

=

P l r ^ - v f ^ l . l j e

=

E ^ )

a?i > 2 ;e-

We set ® = [3]r 1 (- v t rCa? l,2;c®l,l;e and We have 2le(*l,3;e) = =

P l r ^ . e t " ^ ' ^ ! ^ [3] r

1

e

n e (^i

+

X,,2;e^) 1,2;e ) +

^ ( x ^ e ) ! ^ ) )

3 l - t « ! i t ) = [3]," = [3]; = [3]'"'(-fr'^lAe®!,!;® + ^l.lse^lAe) =

39.2. Braid Group Relations for U in Rank 2

307

From the previous formulas we see that rpt

(a)

Ej

rpl

• xi ) 3 ; e

rpt

•x

rpt

•x

rpt

• x13.e

• Ej

rpt

T'-

and (b)

Ei

T'3,e -> X l,l;e

T'.

> Xl,2;e

i

>

Ei.

Thus, T[ J''J ETL ERJ.ETI.E(E]) = EJ and T ^ X X , X X J E > 1 = plying UI and using 37.2.4, we deduce that T ^ T ^ T ^ T ^ T F ' ^ F J ) = T leT"eT",eT"eTjAFi) = Replacing e by - e we deduce '

K X X . X X , ^ )

= FJ

F,

Apand

and

From (a),(b), it follows that

T

rpf rpf m/ rpt / Tp \ rpt rpf rpt rpt rpf ( r> TP \ T*/i,erpt j,e i,e j,e i,e j^K^j) = 1 i,eJ j,eJ i,e2 j^1 i,e\~Kej^ j) ~ rpt rpf rpf rpt ( TP \ rpt rpt rpf rpt rpf ( if Tj ) erpt i i,e i j e i t,e i j,e i t,ei i l / i) = 1 ^e1 i,eJj^1 i,eJj,e\-Kei^i) )

TP \ ~

IS TP Tf TP

-Kei^i,

n e ^ n e ^ e W e t ^ ) = K e ^ i = ~ k e i F i . ThuS, the autOHlOrphisms T l ^ e T U T ^ T l e T ^ e and T ^ X ^ ^ i ^ ^ U coincide on the generators Ei, Ej; similarly, they coincide on the generators Fi, Fj and one checks easily that they coincide on each K^, hence are equal. Taking inverses, we see that the automorphisms

rpll rpll rpl I rpll rpll rpll are equal.

308

39. Braid Group

Relations

39.2.3. Assume that (z,/) = - 2 , (j,z') = - 1 , so that h = 4. Note that vj = v f . As in 37.2.1, we set r-\-s—rri r+s—m Recall from 7.1.2 that EiXi t m ; e

+

Xiyrn.eEi

= [m +

l]i®l,m+l;e-

Applying cr, which interchanges xi )Tn;e , and a:i)Tn;e, we deduce that - v f - ^ x ' ^ E i + £
= x'1)1;e. We have

^-e(®i,2;e) = P ] " ' ^ ^ ' ! , ! ; ^ +

Prom the previous formulas we see that / \ (a) and

n Ej

(b)

Ei

irri/

t,e

m/

• Xly2;e

rpt

J

j,e

rrif

f

• Xlt2;e

rpf

1

i,e

j-, • Ej

rpf

xi !•„ — ^ Xl,l;e

Ei.

Thus, T^T^JEJ) Tj^Tj^Et)

= Ej = E„

and

39.2. Braid Group Relations for U in Rank 2

309

Applying u and using 37.2.4, we deduce that K E ^ T L ' J F J ) = FJ

T'iJlZ^UFi)

and

= Fi-

Replacing e by — e, we deduce that TleT^TleiFj)

= Fj

T'j<en,eTUFi)

and

= Fi-

From (a),(b), it follows that W ^ ^ U E j ) = TjeEj =

-KejFj,

-KejF„ n ^ l e T ^ E j ) = T'ieTjeT'ie(—Kej Fj) = T T T T E T T T KeiFi = K 3,e ie j,e x,e( i) = j,e i,e j,e(~ ^ ~ eiFi, riterjteriteT}te(Ei)

=v ^

=

-keiFx.

Thus, the automorphisms T[ eT'- eT'i eT'- e and T'j eT'i eT'- eT[ e coincide on the generators Ei, Ej; similarly, they coincide on the generators Fi, Fj and one checks easily that they coincide on each K h e n c e are equal. Taking inverses, we see that the automorphisms T"_eT"_eT"_eT"_e and T}[_Jil_ e Tj[_ e Tt' t -e are equal. 39.2.4. Assume that («,/) = (j,if) = - 1 , so that h = 3. Note that Vj = Vi. As in 37.2.1, we set *l,l;e = E ( - 1 YvfE^EjE^ r+s=1

6 u+,

( - 1 YvTEfEjEP


€ u+.

r+s=l

By 37.2.5, we have T/ jC (£j) = x1>1;e and T('_e(Ej) = x'11]e. By 37.2.5, with the roles of interchanged, we have Tj e(Ei) = x[1.e and T"_e(Ei) — Xl,l;e> From the previous formulas we see that T-

T'

e

(a)

Ej

Xi,i;e —^ Ei,

and (b)

Ei ^

x'1>1;e

Ej.

310

39. Braid Group Relations

Thus, T;jeTle(Ej)

= Ei

and

T i ^ E i ) = Ej. Applying LJ and using 37.2.4, we deduce that T'leTl'AFj)

= Fi

KeTUFi)

and

= Fi•

Replacing e by — e, we deduce that W J F , ) = Fj

and

T j M P i ) = F>From (a), (b), it follows that r ^ J i J E j ) = riteEi

=

-KeiFu

Tj&eTjJEj)

= rjterite(-KejFi)

riterjterite(Ei)

=t ^ t ' ^ - k ^ ) =

rjteriterjte(Ei)

= t^EJ

=

=

-K«FU -kejFh

-kejFj.

Thus, the automorphisms Tj e 7V e Tj e and T ^ T ^ T ^ coincide on the generators Ei, E j ; similarly, they coincide on the generators Fi, F j and one checks easily that they coincide on each K h e n c e are equal. Taking inand T(f_eT^_eT(f_e verses, we see that the automorphisms T^_eTl'_eT^_e are equal. 39.2.5. Assume that (i,j') = (j,i') = 0, so that h = 2. By definition, we have TlJEj)

= Ej,T;;_e(Ej)

= Ej.TjJJE,)

= Ei, T"_e(Ei)

= Eu

and similar formulas with Fi, F j instead of E i , E j . We have T^TIJE,)

= T^(Ej)

=

TUTUEj)

= Tie(-KejFj)

—KejFj, =

-KejFj.

Thus, T ^ T l J E j ) = T l e ^ J E j ) . Similarly, Tj e T(e(Fj) = ^ . . ( J J ) . Thus, TJ e T' i e ,T' i e T'^ e coincide on Ej,Fj and, by symmetry, also on Ei,Fi;

39.3. The Quantum Verma

Identities

311

they clearly coincide on K h e n c e they are equal. Taking inverses, we see that the automorphisms T'^_eT!i[_e,T!i'_eT^_e are equal. 39.3.

T H E QUANTUM VERMA

IDENTITIES

39.3.1. In this section, we preserve the assumptions of 39.2.1. Prom 39.2.2-39.2.5, we see that for any p such that 0 < p < h — 1, the following elements belong to U + (each product contains p factors T' or T")\ (••• UX.JtJiE^ (... T'^T'^m)-, (• • • (• • •V/_e7l'_eT<'_c)(Ei). It follows that the sequences ...) and (.h h h • • • )> w ' t h h terms each, are admissible in the sense of 38.2.2. Consider the following four sets of elements of U + ; each element written below is a product of h elements of U+ and (ci,C2,..., c/,) runs through Nh: (a) { E ^ T l ^ E f ^ T ' ^ T ' ^ E ^ ) • • • };

(c) { E ^ T ^ E f ^ T ' ^ i E ^ ) • • • }; (d) { E ^ T ^ E ^ T ^ A E ^ ) •••}. Using 38.2.3 (with x = 1 and p = n or p = 0) we see that each of the sets (a),(b) is an orthogonal set for an inner product on U + , hence each of the sets (a), (b) consists of linearly independent vectors. This implies, by the results in 37.2.4, that each of the sets (c), (d) consists of linearly independent vectors. L e m m a 39.3.2. Each of the four sets (a)-(d) in 39.3.1 is a basis of the Q(v)-vector space U + . These sets consist of homogeneous elements. The number n(y) of elements in the intersection of one of these sets with U+ is the same for any of the four sets. By 39.3.1, it suffices to show that dimq^) U+ < n(v) for all v. Using 33.1.3(b), we see that it suffices to show that (a) dimQ(qf J/ ) < n(v) for all v (notations of 33.1.1; Q is regarded as an ^-algebra with v —• 1). We shall examine the various cases separately. Assume first that (i, j') = ( j , i') = 0 . Then 0j commute; hence any word in 0j can be expressed in Qf as a linear combination of monomials 6"6\. Assume next that (i,jf) = — l , ( j , i') = — 1 so that h = 3. We set Qij = —9j0i -j- $i6j G Qf. Then commutes with both 6i and 6j by the

312

39. Braid Group

Relations

Serre relations. Hence any word in 9{, 9j can be expressed in q ? as a linear combination of monomials 9?9^9%. Assume next that (i,jf) = —2,(j,i') = - 1 so that h = 4. We set 9ij = —9j9i + 9i9j, 29aj — —9{j9i + 9i9ij (in Qf). Using the Serre relations, we see that any word in 9i, 9j can be expressed in Qf as a linear combination of monomials 9a39h^9ciij9di. This is a special case of the Poincare-BirkhoffWitt theorem which can be checked directly. Finally, assume that { i y f ) = —3, (j, i') = — 1 so that h = 6. We define the following elements in Qf inductively: 9ij = —9j9i + OiQj, 29aj = —9ij9i + 9i9ij, 39mj = 9uj9i 9i9uj, S9uijj = Qij0nj -I- 9aj9ij. Using the Serre relations, we see that any word in 9i,9j can be expressed in Q? as a linear combination of monomials 9"9^• •Of^9f U j 9{. This is again a special case of the Poincare-Birkhoff-Witt theorem which can be checked directly. The estimate (a) follows. The lemma is proved. 39.3.3. R e m a r k . It is easy to see that U + is a noetherian domain. Indeed, it is enough to show that Qf is a noetherian domain and for this we note that the associated graded ring for a suitable filtration is the algebra of (commutative) polynomials in finitely many variables. 39.3.4. Let J (resp. J') be the Q(v)-subspace of U + spanned by the elements in 39.3.1(b) such that (c2, C3,..., Ch-i) = (0,0,..., 0) (resp. such that L e m m a 39.3.5. J' is a two-sided ideal of U + . We may identify canonically the quotient algebra U + / J ' with the Q(v)-algebra defined by the generators Ei,Ej and the relation EiEj — v~lEjEi. We identify f = U + via x 1—• x+. Then we may regard (,) as a bilinear form on U + . From 38.2.3, we see that for any 1/, each of the subspaces J fl U+ and J' fl U j is the orthogonal of the other with respect to (,). To check that J' is a left ideal, we must check that Ei J' C J' (the inclusion EjJ' C J' is obvious). From the definition of we have that ir(EjE^) is a scalar multiple of EjEHence *r(J) C J. This implies, by taking orthogonals, that EiJ' C J'. Thus, J' is a left ideal. Similarly, J' is a right ideal. Hence the quotient algebra U + / J' is well-defined. Clearly, J maps isomorphically onto it. Hence the products E^E\ with a, b G N form a basis of U + / J ' . It remains to observe that EiEj = vJlEjEi holds in U + / J ' since in U + , we have —VjEjEi + EiEj = T'j eEi G J'. The lemma is proved.

39.3. The Quantum Verma

Identities

313

Let A G X+. We define two sequences (oi, 0 2 , . . . , an) and h ( 6 1 , 6 2 , . . . , bh) in Z by ax = (• • • SjSiSj(i), X),bi = (• • • SiSjS^j), A); both products have (h — 1) factors s; a
Let x = F ^ F ^ F ^ ... and y = F$bl) F?2) Fjba)...; have h factors F. Note that x, y belong to U~ where v =

aii

+

a

2

j

+

a3z

+ • • • = bij + b2i

both products

+ 63j H

Both sums have /i terms. Then v is given by (4i + 6 j , X)i + (2i + 4 j , X)j if h = 6, (2i + 2j, A)i + (i + 2j, A>j if h = 4, (i + A>i + (t + j, X)j if h = 3, (i,X)i + (j,X)j iih = 2. The following result is a quantum analogue of an identity of Verma [10]. P r o p o s i t i o n 39.3.7. x = y. The result is trivial when h = 2. We assume that h > 3. For any i\ G / , both E i l x and E Xl y belong to the left ideal of U generated by Eix and ( K ^ v ^ " 1 - K i ^ * 1 ' ^ * 1 ) (see 39.1.4). We may assume that Y is generated by i,j. Then there exists X' G X such that (ii,A') = (ii,A) — 1 for i\ G I. Applying x and y to the generator 1 of the Verma module My, we obtain vectors xl,yl G M\> such that Eit(x 1) — O^Ei^yl) = 0 for i\ G I. Note that xl,yl belong to the A"-weight space of M\>, where X" = X' — Vii' — Vjj'. Hence there are well-defined morphisms of U-modules (3,7 : My —* M\> which take the generator 1 G M\» to #1,2/1, respectively. Using the explicit expression for Vi,Vj in 39.3.6, we see that (u,A") = — («i,A) — 1, where i\ = ii if h = 4 or 6 and i = j,j — i if h = 3. In particular, we have (ii,X") < — 1 for i\ G I. From 6.3.2, it then follows that the Verma module My is simple. Since x and y are non-zero (see 39.3.3), we have that x l ^ 0,2/1 ^ 0, hence /?, 7 must be isomorphisms onto their images. These images may be regarded as non-zero left principal ideals of U ~ , if we identify My with U~ in the obvious way. By 39.3.3, U + , hence also U ~ , has the Ore property, hence these two ideals must have a non-zero intersection. This intersection is a non-zero submodule of the simple U-module /3(My), hence it coincides with it. Similarly, it coincides with 7 ( M y ) . Thus, we have (3(My) = 'j(My). Since these two modules have the same (onedimensional) A"-weight space, it follows that xl = f y l for some / G Q(^) —

314

39. Braid Group Relations

{0}. Thus, x = f y in U . We shall identify the algebras U and U + via Fi —> Ei, Fj —> Ej. We now apply to both sides of the equality x = yf (in U+) the natural homomorphism U+ -» U+/J' (see 39.3.5). Since in U+ U +/J' we have E i E j = v~ 1 EjEi, we see that x, y are mapped to VjEi

,Vjhi

Ej

respectively, where s = ^apaq,t p
=

bp>bq> p'
and p is even, q is odd, p' is odd, q' is even. It follows that Vj = fVj. Prom the definition of the sequences (ai, . . . , a^), (b\, b2,.. •, bh), we see that s = t, hence / = 1. The proposition is proved. 39.3.8. R e m a r k s , (a) The use of non-commutative ring theory in the proof of the fact that the two left ideals considered above have non-zero intersection can be avoided as follows. Let v' G N [/]. The intersection of either of our left ideals with U~ +l/ , has dimension n(v') (notation of 39.3.2); on the other hand, dimU~ + I / , = n(v-\-v'). To show that the intersection is non-zero, it suffices therefore to show that 2 n ( y ' ) > n{y-\-v') if v' has large enough coordinates (here v is fixed as in 39.3.6). But n(v') is explicitly computable and the previous inequality is easily checked. (b) In the simply laced case there is a much shorter proof of the quantum Verma identities (see 42.1.2(h)). 39.4.

P R O O F OF THE BRAID GROUP

RELATIONS

In the following lemma, we preserve the assumptions of 39.2.1. L e m m a 39.4.1. Let M be any integrable U-module. T' T' T'

= Tf Tl T'. ... - M

We have M-

both products have h factors. By the complete reducibility theorem 6.3.6, it suffices to prove the lemma in the case where M = A\ with A G X+. Let rj G Aa be as in 3.5.7. Using 39.1.2, we see that (a) ( T i ^ j , J t , e •••)V = ( F ^ F ^ F ^ •••)>? and

39.2. Braid Group Relations for U in Rank 2

315

(b) ( T j X e n * - ) « / = ( F ^ F ^ F ^ •••)»? • • • , &/») are as (all products have h factors) where (ai, a2,... , a/*), (&i, in 39.3.6. By 39.3.7, the right hand sides of (a),(b) coincide. It follows that so do the left hand sides: (c) ( T l ^ J t . e ••')>?= ( T I X . X , , • • • K Let u € U. Using (c) and the equality in U P T ^ X . ' ' ' )(«) = (see 39.2) we see that (KXeTL

• • • )(«•>) = (W.X.e =

• • •

)(«)

• • • )M((Tl,eTj,eTl,e •••

...)»») -to)

Since any vector in AA is of the form urj for some u € U we see that T'i eT'j eT'i e • • • = T'j eT'i,eT'j,e • • • : Af -> Af. The lemma is proved. 39.4.2. Now (/, •) is again an arbitrary Cartan datum. T h e o r e m 39.4.3. For any i ^ j in I such that h = h(i,j) < 00 (see 2.1.1), we have the following equalities (all products have h factors): J J 1 T' T' T' ...> \a/ T' -Li,eT' -j,eT' -i,e ... — ~ *j.ei.e^j,e (b) Tl[_eT>[_X[-e ••• = Tj-Xi-Xi-e ••• as automorphisms of U and (r\ T'i,eT'j,e T'i,e ... ~ — *T' T' T'j,e ...' VW /J\ rpll rpU rpt! ^ rpll rpU rpU V / i,—e1j,—e1i, — e ' ' ' 1 j,—e1 i,—e1 j,—e as linear maps M —> M where M is any integrable U-module M. We prove (c). Let U ' be the algebra defined like U by replacing I by /' = {i,j} and keeping the other data X, Y, ( , ) , . . . unchanged. We may restrict M to an (integrable) U'-module in an obvious way. Since (c) is true for this restriction, by 39.4.1, it is also true for the original M. Thus (c) is proved, (d) follows from (c) by taking inverses. We prove (a). Let u E U. Let ux = ( ^ e T j e T / e • • • ){u) € U and u2 = (T'j eT'i eT'j e • • • )(u) € U. For any integrable U-module M , and any m € M we have (using (c) twice): Ul((Tj,eT[,eTj,e - ) " • ) = "1

''' M

= (T'iteT'j,en,e • • • )(«»») = ( ^ t ^ e ' ' ' )(«"»)

316

39. Braid Group Relations

Since Tj e T / e T j e • • • : M —• M is an isomorphism, it follows that u\ — u2 acts as zero on M. Since M is arbitrary, it follows (see 3.5.4) that ui = u2. This proves (a). In the same way we deduce (b) from (d). The theorem is proved. 39.4.4. Let e = ±1. Let w G W. We define algebra isomorphisms U U and :U U by

•*-w,e

ii,e

'''

e

:

i;\r,e>

where SixSi2 • • • SiN is a reduced expression of w. (T^ e,T'w e are independent of the choice of reduced expression, by 2.1.2 and 39.4.3.) Prom the definition, we have ww' ,e

w,e w'ww',e

w,e w',ei

if w,w' £ W satisfy l{ww') = l(w)l(wf). Thus we have four actions of the braid group on the algebra U. 39.4.5. By 5.2.3 and 37.2.4, we have Ke = C U T ' ^ U

=

: U - U, oT Wte a =

: U ^ U.

Let us define, for any linear map P : U —• U, a new linear map P : U —> U by P{u) = P(u) for all u E U . With this notation, we have m/

w,e

rpt

rpff

^ w, — e*) w,e

rpff

^ —

(see 37.2.4). 39.4.6. Let u e U be such that, for any i e l , we have KiuK-i — v^u for a b some integer n*. For any w G W, we have e(u) — (—1 ) v T'w e(u) where a, b are integers depending only on w and (rij) but not on u. This follows from 37.2.4. 39.4.7. For any integrable U-module M and any w G W, we define Q(v)~ linear isomorphisms T^ e : M —> M and T^ e : M —> M by rpff

1

w,e

rpff

rpff

rpff

ii.e1 i2ye ' ' * 1

rpt

1

w,e

rpf

1

rpf

rpf

ii .e1 i2,e ' # ' 1 ijsr ,e >

39.2. Braid Group Relations for U in Rank 2

317

where s^s^ - • SiN is a reduced expression of w. e, e are independent of the choice of reduced expression, by 2.1.2 and 39.4.3.) From the definition, we have rpl

ww' ,e

rpt

w,e

rpl

io',e>

rplt

ww',e

rpll

w,e

rpll

if w, w' G W satisfy l(ww') = / ( i y ) / ( w ' ) (identities as maps M —> M). Thus, we have four actions of the braid group on M. We have e = -1 {T'w_x _e) : M —• M for all w. For any u G U , m E M, we have rwJum)

= T'w<e(u)(T'mem),T^e(um)

=

T^e(u)(T^em).

L e m m a 40.1.1. Assume that we are given i ^ j in I. Let h = h(i,j) < oo be as in 2.1.1. Let p be an integer such that 0 < p < h. We denote by T"j.p U ; both products have p factors. Then the subalgebra of U generated by E i , E j ;

T"j T ( j

( E j ) ( E j )

and T j \ . and T j ^

p

( E i ) ( E i )

belong to belong to

In the case where h < oo, the lemma is contained in 39.2. In the rest of the proof we assume that h — oo, or equivalently, that aa' > 4 where

Let Z be the subalgebra of U+ generated by the two elements x(i,j\a) and x(i,j;a — 1). Let Z' be the subalgebra of U + generated by the two elements x(j, i\af) and x(j,i;a' — 1). We prove by induction on m the

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © Springer Science+Business Science+Business Media, 2011 Media, LLC 2010

318

40.1. Preparatory

Results

319

This equality, together with the induction hypothesis, shows that G Z'; (a) is proved. Tj _1(x(i,j;m)) We have (b) C Z\ provided that a > 2 and T!_X(Z') C Z, provided that a ' > 2. The first assertion of (b) follows from (a); the second one follows from the first by symmetry. We show by induction on p > 1 that (c) if a > 2 and a' > 2, then T(j.p(Ej) is contained in Z' (if p is even) and in Z (if p is odd). For p = 1, we have T'^.^Ej) = = x(ij;a) G Z. The induction step is given by (b). This proves (c). In particular, under the assumptions of (c), T[ j. p (Ej) belongs to the subalgebra of U generated by Ei,Ej; by symmetry, we have also that Tj i;p (Ei) belongs to the subalgebra of U generated by E^ Ej (for all p). We next consider the case where one of ct, a ' is 1 and the other is > 4. We may assume that a > 4, a ' — 1. Let Z\ be the subalgebra of U generated by the elements x(i,j;a),x(i,j;a — 1) and x(i,j;a — 2). Let Z'x be the subalgebra of U generated by the two elements x'(i,j;l) and x'(i,j;2). From T- x(x'(i,j-,rn)) = x(i,j\a — ra) we see that (d) T[^(Z[) C Z,. We show by induction on ra that (e) Tj _1(x(i,j;m)) G Z[ for any ra > 2. We have T^ixiij;

i)) = T^OC'CM; i)) = *CM;0) = E{.

We also have rjt_1(Ei)

= x(j,v,l)

=

x'{i,j;l).

It follows that r ^ x i u ^ ) ) = iq.iT^f-v-^-VEMifiV

+xfrr.im

= [ 2 ] ~ \ - v - ( Q - 2 ) x ' ( i , r , 1 )Ei - Eix'(i,j; 1)) = x'(i, j; 2). Thus Tj ^l(x(i,j; 2)) G Z'x so that (e) holds for ra — 2. We now assume that ra > 2. Then + x(i,j;m-

1 )Ei)

= MrV^^^^'tu'iiin-iW^';"1-1)) +

rji_1(x(i,j;m-l))x'(i,j;l))

320

40. Symmetries

and U +

and this is in Z{, by the induction hypothesis. Thus (e) is proved. We show by induction on p > 1 that (f) T[ j.p(Ej) is contained in Z[ (if p is even) and in Z\ (if p is odd); Tj i . ^ E i ) is contained in Z[ (if p is odd) and in Z\ (if p is even). We have T(J;1(Ej) = T ^ E j ) = x(i,j;a) e Zx and T ^ E i ) = e Z[. Thus (f) holds for p = 1. The T'j _xEi = x(j,i; 1) = x'{i,j;l) induction step follows from (d),(e). To apply (e), we use that a > 4. In particular, T(j.p(Ej) and T j f i . p ( E i ) belong to the subalgebra of U generated by Ei, Ej. This completes the proof of the second assertion of the lei. ma. The proof of the first assertion is entirely similar. L e m m a 40.1.2. Let w E W and let i € I be such that l(wsi) = l(w) We have

- I...

(a) T ' l A E i ) € U + ; (b) r w „ ( E i ) € U + . We prove (a) for e = 1 by induction on l(w). When l(w) = 1, the result is trivial. Assume now that l(w) > 1. We can find j £ I such that l(wsj) = l(w) — 1. Note that i ^ j. By standard properties of Coxeter groups, there is a unique element w' E W such that l{w'sj) = l(w') +1, l(w'si) = Z(u/) +1 and w = w'y where either (c) y = SiSjSi ••• Sj (p> 2 factors) and l(w) = l{w') + p,l(y) = p or (d) y — SjSiSj -

Sj (p > 1 factors) and l(w) = l(w') +

l(y) = p.

We have necessarily p < h(i,j), since l(wsi) = l(w) + 1. According to 40.1.1, Ty X(Ei) belongs to the subalgebra of U generated b\ ,, Ej. Then T is i n w,i(Ei) = Tw',iTy[i(Ei) the subalgebra of U generated by T^t l(Ei) and T^ JEj). Since, by the induction hypothesis, we have T^, x(Ei) e U + and T^A(Ej) e U+, it follows that T^x{Ei) € U + . This proves (a) assuming that e = 1. We apply " : U ^ U to T^ A (Ei) e U+; using 39.4.5, we obtain T w - \ ( E i ) £ u + - Thus, (a) is proved. The proof of (b) is entirely similar. P r o p o s i t i o n 40.1.3. Let h = (21,1*2, • • • ,in) be a sequence in I such that s^s^ • • • Sin is a reduced expression for some w £\V. T h e n T t i ^ . - . r ^ M € U+ and T>u^e-• *Tln_ue(Ein) e U+ This follows immediately from the previous lemma.

40.2. The Subspace U + ( w , e ) of U +

321

4 0 . 2 . T H E SUBSPACE U + ( W , E ) OF U +

P r o p o s i t i o n 4 0 . 2 . 1 . Let w G W and let e = ± 1 . Let h = (i\,i2, "-An) be a sequence in I such that s^s^ — - Sin is a reduced expression for w. (a) h is admissible (see 38.2.2). (for var(b) The elements E ^ J E ^ ) • • • T ' , ^ •.. Tin_ue(E^) n ious sequences c = (c\, c2,..., cn) G ~N ) form a basis for a subspace U +(w,e) of U + which does not depend on h. (c) The elements E^T['ue(E^) ious sequences c = (ci, c2,...,cn) U + ( w , e ) of U+ defined in (b).

• • • 7 ^ 7 ™ e ' •• K - . J ^ ) Vor varG N n j form a basis for the subspace

(d) Let i G I be such that l(siw) = l(w) - 1.

Then EiU+(w,e)

C

+

U (ii7,e).

(a) follows from definitions using 40.1.3. We prove (b) assuming that e = —1. We shall regard (,) as a pairing on U + , via the isomorphism f —• U+ given by x t—• x+. The fact that the set of vectors in (b) is linearly independent follows from the fact that it is an orthogonal set with respect to (,) (we use (a) and 38.2.3, with p = 0 and x = 1). Let U + ( h , e) be the subspace spanned by this set of vectors. To show that U + (h, e) depends only on w and not on h, it suffices, by 2.1.2, to check the following statement: if h' is obtained from h by replacing h consecutive indices i, j, i,... in h by the h indices j, i,j,... (for some i ^ j with h = h(i,j) < oo), + / then U+(h, e) = U ( h , e ) . Using the fact that the Tns are algebra automorphisms of U satisfying the braid relations, we see that the last equality would be a consequence of the analogous equality in the case where I is replaced by {i, j}. But in that case, the desired equality holds by 39.3.2; both sides are equal to U + . This proves (b) for e = — 1. Now (b) for e = 1 follows from (b) for e = —1 by applying ~ : U+ —> U+ (see 39.4.5). Using 39.4.6, we see that (c) follows from (b). We now prove (d). If i is as in (d), then we can find a sequence h = (ii,i2,... ,in) in I such that i\ = i and SixSi2 • • • Sin is a reduced expression for w. Prom the definitions, it is clear that for this h, we have £?iU+(h,e) C U+(h,e); (d) follows. Corollary 40.2.2. Assume that the Cartan datum is of finite type. Then + U + ( t / ; o , e ) = U . Hence, given e = ± 1 and a sequence h = (i\,i2,... ,in) in I such that SixSi2 • • • Sin is a reduced expression for WQ, the vectors E I ^ t u e I ? ) • • • TL*TLe

•••t U A E ^ )

322

40. Symmetries

and U +

(for various sequences c = (ci, C2,..., c n ) € Nn) form a basis for the Q{v)vector space U + ; moreover, the vectors

4ci>r/i,e(4c2)) • • • n^nie (for various sequences c = (ci, c2,..., vector space U + .

••• n u M : ^

cn) e

form a basis for the Q(v)~

For any i e l , we have l(siWo) = l(wo) — 1, hence Ei\J+(wo,e) C U+(w 0 ,e) (see 40.2.1(d)). Thus, U+(w 0 ,e) is a left ideal of U+. It clearly contains 1, hence it is equal to U + . 40.2.3. It would be interesting to find an extension of the statement of 40.2.2 to the case of an affine Cartan datum (/,•)• (For a result in this direction, in the case of affine SL2, see [1], [4].) The following approach may be useful in finding such an extension. Assume that we are given a sequence h = (..., i-2, i-\, «o, ii, i2, • . . ) of elements in I (infinite in both directions) such that for any integers a p + 1 and x e f such that T"Z , 1 1 , 1, • • -T" Zp-1-1,1 x

r

T

x+

6 u +

for a

s

n.,-iil+„-i • • - L-i " ^ pFor any sequence c = ( . . . , c_2, c_i, Co, c\, c2,...) of numbers in N (infinite in both directions and such that c n = 0 for all but finitely many n) we define L(h, c,p, x) e f by the following formula

This is well-defined, since the factors on the left and on the right of x+ belong to U+, by 40.1.3. Now let c' = ( . . . , c'_2, c;_l5 Cq, c2,...) be another sequence like c and let x' e P. The following result is a consequence of 38.2.3. Proposition 40.2.4. We have the equality of inner products ( L ( h , c , p , x ) , L ( h , c ' , p , x ' ) ) = (x,x')

sez

40.2.5. Let U + ( > ) (resp. U + ( < ) ) be the subspace of U + spanned by the elements ^ ' ^ . . - i ^ ' X ^ , - ^ U . - ^ t f ) ' ' '

( res P- b y

the

40.2. The Subspace U + ( w , e ) of U +

323

elements • • • ^ . i ^ ^ . i ^ V ^ a ^ f c 0 ) ^ ) f o r v a r i o u s sequences ( c p + i , c p + 2 , . . . ) in N, with cn = 0 for large n (resp. for various sequences (cp, Cp_i,...) in N, with c_ n = 0 for large n.) The previous proposition implies that the natural map (a) U + ( > ) <8> P ® U + ( < )

U+

given by multiplication is injective. It would be interesting to show that, in the affine case, the map (a) is an isomorphism and to describe explicitly the space P .

CHAPTER

41

Integrality Properties of the Symmetries

41.1.1. Let e = ± 1 and let i G I. The symmetry T[e : U U (resp. T/'e : U —• U) induces for each A', A" a linear isomorphism A /UV —• Si(A')USi(A") (notation of 23.1.1; Si : X —> X is as in 2.2.6). Taking direct sums, we obtain an algebra automorphism T/ e : U —» U ill A zndTlJuxx'l')

=

(u)T^(x)rite(J) (rest^fatfu') = is an for all u,u' G U and x, x' G U. Then automorphism of the algebra U with inverse T"_ e . These automorphisms satisfy braid group relations just like those of U. 41.1.2. From the formulas in 37.1.3, we deduce that

It follows that T / e , T " e restrict to ^-algebra automorphisms ^ U —> ^ U .

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © © Springer Springer Science+Business Science+Business Media, LLC 2010 2011

324

41.1. Braid Group Action on U

325

The following result is an integral version of 40.1.3. P r o p o s i t i o n 41.1.3. Let h = («i,Z2,... be a sequence in I such that $i1Si2 • - • Sin is a reduced expression for some w G W; let t G Z. Then

Let u be the left hand side of (a). Let G X; define £ G X by £ = 2 £ .. Hence, by ( O - We have = e e 41.1.2, we have wl^ G ^ U . On the other hand, by 40.1.3, we have u G U + . Thus, to prove (a), it suffices to prove the following statement: if x G f and ( G I satisfy G ^ U , then x G ^ f . This follows immediately from 23.2.2. The proof of (b) is entirely similar. The following result is an integral version of 40.2.1. P r o p o s i t i o n 41.1.4. Let w G W and let e = ±1. Let h = (ii,22> • • • ,in) be a sequence in I such that Si1Si2 • • • Sin is a reduced expression for w. Then (a) the elements E ^ T ^ E ^ ) •. • ... 2 J n _ l i e ( ^ > ) (for various sequences c = (ci, c 2 , . . . , cn) G N n ,) form an A-basis for an Asubmodule ^U +(w,e) ofJJ+(w,e) which does not depend on h; (b) the elements E ^ T ^ E ^ ) • • • e • • • T ' ^ E ^ ) (for varn + ious sequences c = (ci,c2,... ,c n ) G N ,) form an A-basis for AU (w,e) in (a). (c) Let i G I be such that l(siw) E^AV+(w,e)cAV+(w,e).

= l(w) — 1 and let t G Z.

Then

Using the method of 40.2.1, we see that it suffices to prove (a) in the case where I consists of two elements i,j and h(i,j) < oo. In that case, the result follows from the analysis in [7]. (If i - i = j • j, this can also be deduced from Lemma 42.1.2.) 41.1.5. With the notations of 41.1.4, let G ^ f be the element corresponding to B ^ . - i C O ' ' ' K . - i K . - i • • • K ^ E ^ ) under the isomorphism f —• U + given by x •—> x+. P r o p o s i t i o n 41.1.6. Let h , c be as in 41-1-4- Let it : £ ( f ) —• _1 £ ( f ) / v £ ( f ) be the canonical projection. We have 0£ G C(f) and there is a unique element b of the canonical basis B such that 7r(0{;) = ±ir(b). Prom 38.2.3, we have (0£,0j;) G 1 + v-lZ{[v~1}} flQ(v). This implies the proposition by 14.2.2(a).

326

41. Integrality Properties of the

Symmetries

Proposition 41.1.7. Assume that the Cartan datum is of finite type. Then (a) AU+(w0,e)

=

AU+.

This follows from 41.1.4 in the same way as 40.2.2 follows from 40.2.1. 41.1.8. Braid group action on Let R be as in 31.1.1. Let e ='±1. For any i G / , the .4-algebra automorphism T( : ^ U —• ^ U (resp. T"E : ^ U —• ^ U ) induces, upon tensoring with R, an i?-algebra automorphism T/ e : ftU —• R\J (resp. T" e : —> ^U). These automorphisms satisfy the braid group relations on just like they did over Q(v) (this holds over for A by reduction to Q(v), since ^ U is imbedded in U, and then it holds in general by change of rings from A to R). Similarly, we have T'I E~L = T"_E as automorphisms of R\J. 41.1.9. Braid group action and the quantum Frobenius homomorphism. Let R be as in 35.1.3. In the setup of 35.1.9, the homomorphism Fr : R\J —> RU* is compatible with the braid group actions on and The proof is by checking on generators. 41.2.

B R A I D G R O U P A C T I O N ON I N T E G R A B L E

HU-MODULES

41.2.1. In the following proposition we assume that the root datum is F-regular and we consider A, A' G X+. Proposition 41.2.2. The symmetries T'i e, T"e of the \J-module "A* ® Aa> map the AXJ-submodule <S>_4 (^Av) into itself. <S>A GIAA')- By definition (see 5.2.1), the vector T'i e(m) is Let m € given by a sum of infinitely many terms such that all but a finite number of terms (depending on ra) are zero. The finitely many terms that can be non-zero are of the form um where u E ^ U . They belong to AA <8u (^AV) since this is an ^U-submodule. Thus this submodule is stable under T/ e . The same argument shows that it is stable under T"e. The proposition is proved.

41.2.3. Let R be as in 31.1.1. Let M be an integrable object in RC. We define R-linear maps T/ e : M —> M and T"E : M —> M by the formulas in 5.2.1, in which v is regarded as an element of R, by the ^4-algebra structure on R. It is clear that these operators are well-defined.

41.2. Braid Group Action on Integrable

R\J-modules

327

Proposition 41.2.4. (a) The operators T'i e : M —• M satisfy the braid M. group relations. The same holds for the operators T"e : M (b) We have T(e~l

= T['_e as operators M

M.

(c) For any u G RV and any m G M, we have T'ie(um) andTl[e(um) = T<[e{u)T>[e(rn).

= T/ e (w)T/ e (m)

Using the functor in 31.1.12 in the case where {Yr, X',...) is the simply connected root datum of type (I, •), we can reduce the general case to the case where the root datum is simply connected, hence y-regular. In that case, using the characterization of integrable objects given in 31.2.7, we are reduced to the special case where M = ^AA ®R (#Av) with A, A' G X + . Indeed, suppose that M is a sum of RU-submodules Ma and that the proposition holds for each Ma. Then it clearly holds for M. Suppose that M is a quotient of an integrable object M' such that the proposition holds for M'; then it clearly holds for M. In this special case, the proposition follows by change of scalars from the case R — A which in turn follows from the already known case where R — Q(v). The proposition is proved.

42.1.

COMBINATORIAL DESCRIPTION OF THE L E F T COLORED

42.1.1.

GRAPH

In this chapter we assume that the Cartan datum is simply laced

L e m m a 42.1.2. Consider the Q(v)-algebra with generators a, (3 and relaSet 7 = a/3 — v~lf3a. For x = a, (3 or 7, and n > 0, we set

= x n /[n] ! ;

Now (b) is obvious when p < 0 or q < 0. For q = 1, (b) states that + 7a^ this is proved by induction on p > 1, using (a). Assume now that q > 2 and that (b) is known when q is replaced by q — 1. We write (b) for (p,q — 1) and multiply it on the right by f3. Using

This can be rearranged using (a) and yields (b) for (p, q). Thus (b) is To prove (c), we replace c*(m)/?(•?") in the right hand side of (c) by the expression provided by (b); we perform cancellations, and we obtain (c).

G. Lusztig, Introduction to Quantum Groups, Modern Birkhauser Classics, DOI 10.1007/978-0-8176-4717-9_30, © Springer Springer Science+Business Science+Business Media, Media, LLC LLC 2011 2010

328

42.1.

Combinatorial

Description

329

To prove (d), we replace a(p)/?(<7) on the left hand side and a ( p + r ) /? ( p ~ n ) on the right hand side by the expressions provided by (b); we perform cancellations, and we obtain (d). Now (e) follows from (d) by symmetry; (f) and (g) follow immediately from (b) and (h) is a special case of either (d) or (e). Note that (h) is a special case of the quantum Verma identity 39.3.7. 42.1.3. Let H be the set of all sequences h = (ii, 22,• • • >in) in I such that Si1Si2 ' •' Sin is a reduced expression for WQ. (Thus, n = 1(WQ).) We shall regard H as the set of vertices of a graph in which h = • • • ,jn) are joined if h' is obtained from ( i i , i 2 , . . . , i n ) and h' = (ji, h by (a) replacing three consecutive entries i,j, i in h (with i • j — —1) by j,i,j or by (b) replacing two consecutive entries i,j in h (with i • j = 0) by j, i. For such joined (h, h'), i.e., in case (a) (resp. (b)) we define a map R^ : N n = N n as follows. This map takes c = ( c i , . . . , c n ) € N n to c' = ( c i , . . . ,c'n) G N n which has the same coordinates as c except in the three (resp. two) consecutive positions at which (h, h') differ; if (a, 6, c) (resp. (a, b)) are the coordinates of c at those three (resp. two) positions, the coordinates of c' at those positions are (6 -I- c — min(a, c), min(a, c), a + b — min(a, c))

(resp. (b, a)).

It is easy to check that R\J is a bijection; its inverse is R^. From 2.1.2, it follows that (c) the graph H is connected. 42.1.4. Given h = (i\,... (a)

,i n ) E H and c = ( c i , . . . , cn) € N n , we define

££ =

According to 41.1.3, 40.2.2, the elements E^ (c € N n ) are contained in and form a Q(v)-basis of U + . We shall denote this basis by Bh. Hence, given h, h' £ H and c G N n , we can write uniquely EcU=

where

G Q(v).

E

c'eN"

330

42. The ADE Case

Proposition 42.1.5. (a) Assume that h, h' are joined in the graph H. For in Z[v - 1 ]. Its constant term is 1 if c' = (c) and is c, c' G N n , zero otherwise. (b) For h G H, let £h be the Z[v~1]-submodule o / U + generated by the basis 2?h- Then Cu is independent of h G H. We denote it by C. (c) For h G H, let n : C —> C/v~lC be the canonical projection. Then 7r(i?h) is a Z-basis of C/v~lC, independent of h G H; we denote it by B. Assume that the proposition is known in the special case in which I consists of two elements i,j. Using the definitions and the fact that the T[ i : U —• U are algebra homomorphisms satisfying the braid relations, we see that (a) in the general case is a consequence of (a) in the special case. To prove (in the general case) that the objects defined in (b),(c) in terms of h, h' G H coincide, we may assume, in view of the connectedness of the graph H, that h, h' are joined in H, in which case the desired statements follow immediately from (a). Thus, we may assume that we are in the special case above. In the case where i-j = 0, the result is trivial. Hence we may assume that i • j = j -i = — 1. Now H consists of two elements: h = (i,j,i), h' = (j,i,j). Besides £ h s we introduce the Z[u -1 ]-submodule £ of U + generated by the set B' = {E^E^E^q

> p + r] U {E^E^E^q

>p + r}

in which we identify e[p)E^E\r) = E^E^E^ ioxq = p + r. By definition, we have T ' ^ ^ E j ) = EjEi - v ~ l E i E j = T ^ ( E i ) and T'j _x(Ei) = EiEj - v~1EjEi = T^(Ej). It follows that T^T^Ei = 3 E j and T ^ T ^ E j = Ei. Hence, if c = (ci,c 2 ,c 3 ) G N and c' = (c\,c'2,c'3) G N 3 , we have El = ElCl)(EjEi - v ~ l E i E j ) ^ E f 3 ) and Ei = E^\EiEj

-

v-lEjEi)^E\c>\

where the notation x^ is as in Lemma 42.1.2. Let (p, q,r) G N 3 be such that q > p + r. From 42.1.2(f), we have (q)E(r)1 E(p) 1 E 3

=

^ / j

n—0

v-(p-n)(q-n)

p — n + r £jq—n,n,p—n+r p— n

42.1.

Combinatorial

Description

331

where p— n + r G tT(«-n-r>( 1 + v ^ Z l v ' 1 ] ) p— n

,-{p-n)(q-n)

is in v

1

Z[v 1], if n < p and it equals 1 if n = p. Similarly, j

i

j

n + p j^i—n+p,n,q—n r —n

r —

/ -< n=0

where i; - ( q - n ) ( r - n )

r —n + p

r—n

G v -< r - n ) <*- p - n >(i + tr 1 Z[v" 1 ])

is in t; _ 1 Z[t; - 1 ], if n < r and it equals 1 if n = r. These formulas show that £h' = £ and, if TT : £ —» C/v~lC is the canonical map, we have 7R(B') = TT(B\1/); moreover, 7r maps B' onto TT(B') bijectively. By the symmetry between i and j , there is an analogous statement for h (note that B' are symmetric in i,j). Thus, we have C^ = C and TT(B') — 7R(BH). It follows that (b),(c) hold. The formulas above show also that (a) holds. The proposition is proved. Corollary 42.1.6. The A-subalgebra submodule of\J+ generated by C.

of U + coincides with the A-

The fact that C has been noted in 42.1.4. To prove the reverse inclusion, it suffices to show that for any i G / and any s G N, is stable s under x multiplication by E\ \ Now has an ^4-basis formed by the elements E£ where h is a fixed element of H which starts with i and c runs through N n . Multiplication by E[8^ takes each element of this basis to an .4-multiple of another element of this basis. The corollary follows. 42.1.7. From the definitions it is clear that C = ®VCV where C u — £ n U + for any v G N[I]. This induces a direct sum decomposition C/v~1£ = ®vCvjv~xCu. It is clear that B is compatible with this decomposition; in other words, we have B = U^B^) where B(v) is the intersection of B with the summand Cu/v~lCu of C/v~lC. 42.1.8. We consider the equivalence relation on H x N n generated by (h, c) ~ (h',0 7 ) whenever h, h' are joined in H and (c) = c'. Let H be the set of equivalence classes.

332

42. The ADE Case

Lemma 42.1.9. For any h G H, the map f : N n —• H, which takes any c to the equivalence class of (h,c), is a bijection. From 42.1.5(a), we see that the (surjective) map H x N n —• B given by (h, c) •—• 7r(E^) is constant on equivalence classes; hence it factors through a (surjective) map H —> B. On the other hand, for any h G H, the composition N n ^ H B is a bijection (again by 42.1.5). The lemma follows. We have the following result. Theorem 42.1.10. (a) For any b G B there is a unique element 6 G C such that 7r(6) = b and 6 = 6. (b) The set {6|6 G B} is a 7*[v~l]-basis of C and a Q(v)-basis o / U + . We shall regard the pairing (,) on f as a pairing on U + via the isomorphism f —• U + given by x x+. Let h G H. By 38.2.3, the basis + E£ of U , where c is running through N n , is almost orthonormal for (,). Applying 14.2.2(b) to this basis, we see that any element (3 G B satisfies (3+ e £ and 7r(/?+) G ±B. In particular, we have £ ( f ) C C. Applying 14.2.2(b) to the canonical basis B of f = U + and to x = E{;, which satisfies (xyx) G 1 + v ^ Z ^ " 1 ] ] H Q(v) by 38.2.3, we see that x G £(f); hence, by the previous sentence, C = £ ( f ) and n(x) = ±ir(/3+) for some f3 G B. Since (3+ C £ ( f ) = we see that the existence statement in (a) holds. The uniqueness in (a), as well as statement (b) now follow from the known properties of B. The theorem is proved. 42.1.11. We keep the notation from the proof of Theorem 42.1.10. We fix i e l . Assume that h G H starts with i. Let 6 G B be such that 6 = 7t(1?£) where the first coordinate of c is 0. Let c' G N n be such that c' has the same coordinates as c except for the first coordinate, which is s G N. Let b' = tt(££ ) G B. We shall use the following notation. For 6 G B, we define 0b e B by /?+ = 6 (see the proof of 42.1.10). Lemma 42.1.12. (a) Write E£ = z+ where z G f. Then z G f[z] and Ei = (4>iz)+. (b) We have (3b> = 4>f(3b mod (c) We have (3b G Bi;0. (d) We have (3b> = 7r<>s (/?&).

v~lC(f).

42.1. Combinatorial

Description

333

Since c starts with 0, the element E£ G U + is in T[ U + , hence by 38.1.6, z is in f[t], and ir(z) = 0; hence f(z) = O^z. It follows that = E ^ E * = E g . This proves (a). We prove (b). Using (a) and the definitions, we have z = /3b mod v~lC(i) and 4>fz = (3b' mod v~lC(f). Since 4>f maps v~lC(f) into itself it follows that 4>fz = 4>t((3b) mod v _ 1 £ ( f ) , hence <j>l{/3h) = (3b> mod v~lC(f). This proves (b). From the fact that the elements £ £ , where c" runs through N n , form a basis of U+, it follows immediately that (e) for any t > 0, the elements E^' where c" runs through the elements of N n with first coordinate > ty form a Q(v)-basis of E\U+. We prove (c). Assume that (3b G 8i;t with t > 0. Then (3+ G E\U+, hence it is a linear combination of elements as in (e); in particular, E£ appears with coefficient 0, contradicting the definition of fib- This proves G 3, we see from 17.3.7 that (d) follows from (b) and (c). (c). Since This completes the proof. Corollary 42.1.13. We have B = {b\b G B}. above.)

(We identify f = U+ as

From the proof of 42.1.10, we have that {6|6 G B} C B. We show by induction on N = tr v that b G B, if b G Bu. If N = 0, this is clear. Assume that N > 1. By 14.3.3, we can find i G / and s > 0 such that b G Bi;s. We then have b = niiS/3 where (3 G Bi;0 (see 14.3.2). We have /3 = ±bi where bi G B. By the argument in the previous lemma we have that the sign is +, hence b = iTijSbi. By the induction hypothesis, we have b\ G B; the previous equality then implies that b G B. 42.1.14. The basis {b\b G B} = {(3+\(3 G B} of U+ is in a natural bijection with the set B, which in turn is in a natural bijection with the set H (see the proof of 42.1.9). We thus have a purely combinatorial parametrization of the canonical basis B. The structure of left colored graph on B (see 14.4.7) corresponds to a structure of colored graph on H, which we will now describe in a purely combinatorial way. For any i e I, we define a function gi : H —• N as follows. Let c G H; we choose h G H such that the sequence h starts with i. By 42.1.9, c is the class of (h,c) for a unique c G N n . We set gi(c) = c\ where c\ is the first coordinate of c. To show that this is well-defined, we consider h' G H such that the sequence h starts with i. Let c ; G N n be such that c is the

334

42. The ADE Case

class of (h', c') and let c[ be the first coordinate of c'. We must show that ci = c[. Now the set H* of all sequences in H which start with i can be naturally identified with the set of reduced expressions for S{Wo; applying 2.1.2, we see that H^, regarded as a full subgraph of H is connected. Hence to prove that c\ = c[, we may assume that h, h' are joined in the graph. Then c and c' are related by an elementary move as in 42.1.3(a) or (b). This elementary move operates on coordinates other than the first, since h, h' start with the same element i. Thus, we have c\ = c^, as desired. 42.1.15. For any i G / , we define a partition H = U t >oHi )t by setting Hi t = g^l{t). We define a bijection 7r^t : H;(o = H^* as follows. Let (h, c) be a representative for an element of H^o- Then c starts with 0; let c' be the element of N n which starts with t and has the same subsequent coordinates as those of c. By definition, 7r^(h, c) = (h,c'). One checks that this map is well-defined. From our earlier discussion, it is clear that the partitions of H just described, together with the bijections 7rijt, correspond to the analogous objects for B which are the ingredients in the definition of the left colored graph. 42.1.16. We can also describe in purely combinatorial terms the left colored graph for not necessarily simply laced Cartan data, by reduction to the simply laced case, using 14.4.9 and 14.1.6. 42.2.

R E M A R K S ON T H E P I E C E W I S E L I N E A R B I J E C T I O N S R :

N

n

=

N

n

42.2.1. Let h , h ' G H. We define a bijection : N n ^ N n as a composition „h(i) ( a \ Dh' _ ph(l) ph(2) fl W / t h — - h(0) h(l) ' ' ' ^ ( t - l ) where h(0), h ( l ) , . . . , h ( t ) is a sequence of vertices of the graph H such that h(0) = h, h(t) = h' and such that h(s),h(s + 1) is an edge of the graph H for s = 0 , 1 , . . . ,t — 1; the factors on the right hand side of (a) are the bijections defined in 42.1.3. (A sequence as above can always be found, by 2.1.2.) From 42.1.5, it follows that the definition of R^ is correct, that is, it does not depend on the choices made. Indeed, 42.1.5 gives us an intrinsic definition of this bijection: with the notation in 42.1.5, we have (c) = c' if and only if 7r(£{;) = 7r(£^',). The bijections R^ are piecewise linear, since they are products of factors which are piecewise linear. 42.2.2. In this section we will show that the bijections R^ also appear in a completely different context.

42.2. Piecewise Linear Bijections

:Nn = Nn

335

Let K be a field with a given subset KQ C K — {0} containing 1, and such that the following holds: (a) if / , / ' € K0, then / + /'<= K0,

f f G K0,

J^JT e K0.

For example, we could take (b) K = R, K0 = R>o or (c) K = R((e)) where e is an indeterminate and K0 is the subset of K consisting of power series of the form / = ases + a s + i e s + 1 + • • • such that s > 0 and as > 0; we then set | / | = s. 42.2.3. We consider a split semisimple algebraic group Q over K, corresponding to the root datum, with a fixed maximal unipotent subgroup U + and a fixed maximal torus T normalizing U + , both defined over K. For each i G I, we denote by Uf the simple root subgroup of U+ corresponding to i; we assume that we are given an isomorphism Xi of the additive group with defined over K. Let B~ be the Borel subgroup opposed to U+ and containing T. We shall identify with their groups of K-rational points. We shall regard xx as an isomorphism of K onto U f . Proposition 42.2.4. Let w G W. Let h = (21,22, • • is a reduced expression for w. I such that S{lSi2

be a sequence in

(a) The map KQ —• U+ given by ,Pn) *-+ Xix(pi)Xi2(p2) ' • • Xin(pn) is injective. (b) The image of the map (a) is a subset U+{w) ofU+ depend on h. (c) If w' eW

which does not

is distinct from w, then U+(w) C\l4+(wf) = 0.

Let h) be the image of the map in (a). To prove (b), it suffices, by 2.1.2, to check the following statement: if h' is obtained from h by replacing h consecutive indices i,j, i , . . . in h by the h indices j, i,j,... (for some i ± j with h = h(i,j)), then U+(h) = U+(h'). To prove this statement, we may clearly assume that I consists of two elements i,j. In the case where i • j = 0, we have xi(p)xj(p') — Xj(p')xi(p) for any p,p' G K. Assume now that i • j ~ —1. We have the following identity, by a computation in SL3: xi(t)xj(s)xi(r)

=

xj(t')xi(s')xj(r')

336

42. The ADE

Case

where (d)tf = j £ ,

s' = t + r,

r' = ^-r

or equivalently, (e)t=ji£7, s = t' + r', By the definition of i^o? we have s, t, r G KQ if and only if s' ,t' ,r' G KQ. This proves (b). We prove (c). Let Si be an element of the normalizer of BSIB. T in Q which represents Si G W . If p e K — {0}, we have Xi(p) G Hence if pi,p2,... ,Pn are in K — {0}, then xix(pi)xi2(p2)

• • -Xin(Pn) e

B S I

X

B S I

2

B - ' - S

I N

B

C Bsixsi2

- S

I N

B

by properties of the Bruhat decomposition. Thus, U+(w) C Bsi1si2 • • • SinB so that (c) follows from the Bruhat decomposition. We prove (a). Assume that Xi,{pi)xi2{p2)

• -xin(pn)

= arfl(pi)xf2(pi)

•

-xin(p'n)

where p i , . . . , p n and p[,... ,p'n are in KQ. We prove that pi = p\ for all I by induction on n. This assumption implies XiAPl - Pl)Xi2(P2) ' • ' Xin(pn)

=Xi2{p'2)-"Xin{p'n).

If pi — p[ ^ 0, the two sides of the last equality are in BSi^ Si by the argument above. Pi = p\. Then we have

This is a contradiction. f

Thus, we must have

Xia (p2) • • • xin (pn) = xi2 (p 2) • • • xin

and the induction hypothesis shows that p2 = p2,...

(p'n)

,pn = p'n.

Corollary 42.2.5. The subset U t u e w U + ( w ) ofU+ is closed under multiplication. It coincides with the submonoid (with l ) ofU+ generated by the elements Xi(p), for various IEL and p G KQ. Let i e l and p G KQ. Let h = (?i, i2j... , i n ) be as in 42.2.4. If i i i s i 2 ' ' ' $in 1S a reduced expression in W, then Xi(p)U+(h) C U+(h') where h ' = (i, ii, 12,... , in)- If SiSilS{2 •• - Sin is not a reduced expression in jn-i> W, then we have s^s^ • • • Sin = S{Sj1Sj2 • • • sjn_1. for some ji>j2,..., Clearly, Xi(p)U+(h') C U+(h') and, by Set h ' = (i,juj2,...,jn-i). 42.2.4(b), we have W + ( h ' ) = W + ( h ) . It follows that Xi(p)U+(h) C U + ( W ) . We have thus proved that the set U w e wM + (w) is stable under left multiplication by elements of the form Xi(p) as above. The corollary follows. s s

42.2. Piecewise Linear Bijections

:Nn = Nn

337

42.2.6. From now on we assume that K,KQ are as in 42.2.2(c). Recall that we then have a well-defined map /»-» | / | from K0 to N. For any h = ( i i , i 2 , . . . , i n ) in H and any c = ( c i , . . . , c n ) G N n , we define a subset c) of U+ as follows. By definition, U+(h, c) consists + of all elements of U which are of the form x^ (pi)xi 2 (P2) • • • Xin (pn) where Pi, p2, • • •,Pn are elements of K0 such that \pi | = c x , \p2\ = c2i..., \pn\ = cn. From 42.2.4, we see that we have a partition (a) U+{w0) =

UcU+(h,c).

Proposition 42.2.7. Let h, h' be elements o / H and let c, c' be elements of N n such that i?{j'(c) = c'. We have U+(h,c) = ^ ( h ' j c ' ) . In particular, the partition 42.2.6(a) ofU+(wo) is independent of h. We may clearly assume that h, h' are joined in the graph H. That case reduces immediately to the case where I consists of two elements i,j. The case where i • j = 0 is trivial. Assume now that i • j = —1. Using the identities (d),(e) in the proof of 42.2.4, we see that it is enough to verify the following statement. Let s' = t + r, r' = Then t, s, r, t', s', r' G K0 be such that t' = \t'\ = | a | + | r | - m i n ( | < | , |r|),

\8'\ = min(|t|, |r|),

|r'| = | 4 | + | « | - m i n ( | t | , |r|).

This is immediate. The proposition is proved. 42.2.8. We now see that the set of subsets in the partition 42.2.6(a) of U + (wo) (which is intrinsic, by 42.2.7) is in natural 1 — 1 correspondence with the set H, hence also with the canonical basis B. At the same time we have obtained a new interpretation of the piecewise linear bijections jfh . jvjn ^ ^jn m t e r m s 0 f the geometry of the group Q.

Notes on Part VI 1. The braid group action on U has been introduced (with a different normalization) in [5], in the simply laced case, and in [6], for arbitrary Cartan data of finite type. Another approach (for Cartan data of finite type) to the braid group action has been found by Soibelman [8]. The general case has not been treated before in the literature. T h e fact that the braid group acts naturally on integrable modules over arbitrary ground rings (see 41.2) is also new. 2. The paper [3] of Levendorskii and Soibelman contains several results relating braid group actions (for finite type) with comultiplication and with the inner product. In particular, an identity like 37.3.2(a) appears (for finite type) in [3]. Our lemma 38.1.8 is closely related to [3, 2.4.2]; however, neither of these two results implies the other. Propositions 38.2.3 and 40.2.4 are generalizations of [3, 3.2]. 3. Corollary 40.2.2 and Proposition 41.1.7 appeared in [6] and [2]. 4. Most results in 42.1 appeared in [7]. The results in 42.2 are new; in obtaining them, I have been stimulated by a question of B. Kostant.

REFERENCES 1. I. Damiani, A basis of type Poincare-Birkhoff- Witt for the quantum algebra of SL(2), J. of Algebra 161 (1993), 291-310. 2. M. Dyer and G. Lusztig, Appendix to [6], Geom. Dedicata 35 (1990), 113-114. 3. S. Levendorskii and I. Soibelman, Some applications of quantum Weyl groups, J. Geom. and Phys. 7 (1990), 241-254. 4. S. Levendorskii, I. Soibelman and V. Stukopin, Quantum Weyl group and universal quantum R-matrix for affine Lie algebra Al, Lett, in Math. Phys. 27 (1993), 263264. 5. G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237-249. 6. , Quantum groups at roots of 1, Geom. Dedicata 35 (1990), 89-114. 7. , Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498. 8. I. Soibelman, Algebra of functions on a compact quantum group and its representations,, (in Russian), Algebra and Analysis 2 (1990), 190-212. 9. J. Tits, Normalisateurs de tores, I. Groupes de Coxeter et endues, J. Algebra 4 (1966), 96-116. 10. D. N. Verma, Structure of certain induced representations of complex semisimple

Tn1nphrn.fi Riill. Amer. Math Snr 7A MQfiJ^

Index of Notation 1.1.1. (/,•); vi/ 1.1.2. Pi (for P E Q(v)); vv (for i/ G Z[/]); tr v 1.2.1. 'f; 'f„; \x\ for x G 'f; 0,; N[J] 1.2.2. r —• 'f 'f 1.2.3. (,) on 'f 1.2.4. J 1.2.5. f;f„; (,)on f; |x| for x G f 1.2.6. r : f —» f <8> f 1.2.7. a on 'f; 1.2.9. cr on f 1.2.10. ~ on Q(v), on 'f; f : 'f — W f , {,} on 'f 1.2.12. ~ on f; f : f — f(g>f; {,} onf 1.2.13. r i ; ir 1.3.1. A; [?] 1.3.3. [n]; [n]! 1.4.1. 0\p) 1.4.7. ^f, ^ f , 2.1.1. h(ij); W 2.1.2. i(u>)

2.1.3. w0 2.2.1. V; X; (,); i ^ z; i 2.2.3. A < A' 2.3.1. 2p\ n 2.3.2. p(i) 3.1.1. 'U; 'U + ; ' U " ; U; U+; U"; Ei] Fi\ K F(p).

(P).

' f1 i ' K^ Ku\

* a: 3.1.3. u; on U; cr on U 3.1.4. A o n U / U 3.1.12. " on U x

x+; x

i'

3.1.13. ^ U 3.2.4. U° 3.2.6. U+; U " 3.3.1. 5, 5' 3.4.1. C 3.4.4. wAf 3.4.5. Mx 3.4.7. 3.5.1. C' 3.5.5. X+ 3.5.6. AA 3.5.7. -A a ; *?a; 4.1.1. A 4.1.2. 9; e„ 4.1.4. {n}i 4.2.i. e 1 2 , e 2 3 , e 1 3 5.1.1. M 5.2.1. :M 5.3.1. LI; L'l 6.1.7. ttE : M M 7.1.1. /ij;n?m;e 3:1 /n,m;e 8.1.1. V(X)\

D(K);

8.1.11. CT\

$(K)

HnK;

K\j]

8.1.2. M(AT); -M(X)[n]; HnK; r< n ; 8.1.7. A4G(X); M g P O H ; A * 8.1.9. T>j(X,G\AyB) 9.1.2. 9.1.3. 9.1.3. 9.2.1. 9.2.5.

V; V„; G v ; E v * , i/, ^Fi/j Lu TV; Qv P T ,w; QT.W IndT, w ; v nes T ^yj Ind^ w 9.2.10. Res^ yv 9.3.1. PV;I';>7; 7V,I';
G. Lusztig, Introduction to Quantum Groups, Modern Birkhäuser Classics, DOI 10.1007/978-0-8176-4717-9, © Springer Science+Business Media, LLC 2010

/C(QV); £(7V)

339

Index of Notation

340

25.4.1. mcab>, mf

13.2.1. ^k 13.2.6. k 14.2.1. A 14.2.3. B 14.2.5. BV 14.3.1. 5 i; > n ; '£<;>„; Bi-nl a Bi, n 14.4.2. B; B„ ff 14.4.6. Bj ;n ; Bj;n

26.1.2. ( , ) on U

27.1.1. M[A]; M[> A]; M[> A] 27.2.1. £[A] 27.2.3. B\A]w; B[A]io 27.2.5. M* 27.3.2. hOb^ 29.1.1. B[Ai] 29.1.2. U[> Ai]; U[> Ai] 31.1.1. (j>: A-+ R\ j?U; Rf 31.1.5. RC 31.2.4. RC' 31.1.13. RMX

14.4.11. B(A); B ( A a )

15.1.2. 11 16.1.1. Vi 16.1.3. i\ li 16.1.4. Fi\ Ei 16.2.3. A(Z); A 17.3.3. C(f)

31.2.1.

18.1.1. (A)„

18.1.2. B{A) 18.1.5. £(f)„ 18.1.8. L(A); L'(A) 19.1.1. p : U —• U 19.1.2. ( , ) on A

19.3.1. 19.3.4. 23.1.1. 23.1.2.

^A a " : Aa - Aa a'UA"! U; 1 a U(^)

23.1.8. ~ : U

23.3.3. P(C,A,A'); 24.3.3. hOx^'K 25.2.1. 60 c 6" 25.2.4. B

RAX,

31.3.2. RLX 32.1.5. /T^m.M' 33.1.1. R f 35.1.3. A' 35.4.1. f 36.1.2. flf 36.2.1. RU 36.2.5. 37.1.2. T'i e,Tl[e 37.2.1. Xn rn^e\

U AP{T,A,A')

38.1.1. 38.1.3. 39.4.4. 41.1.1. 42.1.8.

2/n,ra;e> 2/n,m;e

f[i]; CTf[i] g : f[i] ^[i] TweJTwe T'i e,T['e : U - > U H

Index of Terminology adapted basis, 16.3.1 admissible automorphism, 12.1.1 form, 16.2.2 almost orthonormal, 14.2.1 antipode, 3.3.2 skew, 3.3.2 based module, 27.1.2 basis at oo, 20.1.1 braid group, 2.1.1 Brauer's eentralizer algebra, 27.3.10 canonical basis of f, 14.4.6 of A a , 14.4.12 of " A a (g) A A /, 24.3.4 of U , 25.2.4 signed...of f, 14.2.4 topological...of (U~"
induction, 9.2.5 integrable object, 3.5.1 Kashiwara's operators, 16.1.4 left graph on B , 14.4.7 length of w, 2.1.2 linear category, 11.1.1 morphism of root data, 2.2.2 orientation, 9.1.1 periodic functor, 11.1.2 perverse sheaf, 8.1.2 quasi-classical case, 33.2.1 quasi-7£-matrix, 4.1.4 reduced expression, 2.1.2 restriction, 9.2.10 root datum, 2.2.1

y-regular..., X-regular..., simply connected..., adjoint..., 2.2.2

semisimple complex, 8.1.3 Serre relation (quantum), 1.4.3 signed basis, 12.6.4 sink 9.3.4 structure constants 25.4.1 symmetries, 5.2.1 traceless, 11.1.4 triangular decomposition, 3.2, 23.2.1 unital module, 23.1.4 Verma module, 3.4.5, quantum .... identity, 39.3.7 weight space, 3.4.1 Weyl group, 2.1.1

Comments added in the second printing 1. Let M G C' and let i G / , e = ±1. We define two Q(v)-linear maps S t , e , S ? e : M ^ M by

a,6,c

a,b,c where a, b, c run over N; although the sums are infinite, on any given vector in M , all but finitely many terms in either sum act as zero. Several readers have asked me about the relationship between these operators (which appeared in [3, 3.1]) and the operators T!e,T"e : M —• M in 5.2.1. The relationship is as follows: D

iye ~ 1i,eIxi > T>e

D

i,e ~ 1i,eJ^i

e

•

To prove this, we may replace C by C[ and we may assume that M is a simple object of C-. Then the desired identities are checked by calculations similar to those in 5.2.2. It follows that the braid group relations 39.4.3(c),(d) remain valid if T ' , T " are replaced throughout by S',S". 2. In the last sentence on p. 183, it was stated that it should be possible to remove the signs ± in Corollary 26.3.2. This is indeed true, as was shown by Kashiwara [2]. 3. The question raised in the last sentence of 40.2.5 has been answered by J. Beck [1] in the untwisted case. REFERENCES [1] J. Beck, Convex bases of PBW type for quantum affine algebras, Communications in Math. Phys. (to appear). [2] M. Kashiwara, Crystal base of modified enveloping algebras, Duke Math. J. 73 (1994), 383-414. [3] G.Lusztig, Problems on canonical bases, Amer. Math. Soc., Proc. Symp. Pure Math. (1994).

Errata to Introduction to Quantum Groups (1994 printing) p. 3 line 13: after "homomorphism," and before "then" insert: preserving Z[/]-gradings p. 9, line -5: inside the last bracket [ ], replace o! by a" and t' by t". p. 11, in the numerator of the first fraction on line -4, insert a factor v f v j p. 12, in the numerator of the fraction on line -8, insert a factor vj p. 13, just before 1.4.7, add the following text. Here is another proof of Proposition 1.4.3 which is not based on 1.4.5 but is based, instead, on the following statement which is easily verified. (a) For any k € I the left-hand side of the equality in 1.4.3 is in the kernel of r* : f —> f. Note that 1.4.3 follows from (a) in view of 1.2.15(a). p. 31, in 3.4.6 line 6: delete (a) (twice) and in 3.4.6 line -2, replace p. 32 line 3.5.3(b): replace EjF? + QtyF?-1

by E j F i 1 + Q{v)K i F^~ l +

p. 32 line -1: add: See also 23.3.11. p. 33, line 13, replace M\ by A^. p. 40, line -3: replace n + 2m by n + 2m — 2. p. 40, line -1: replace n — 2m by n — 2m + 2. p. 41 lines 5,6: replace If x 6 Mn(0), then by definition, (c — sn)x = 0 by We have Mn(0) = 0. p. 41 lines 9, 10, 11, 12: replace (four times) n + 2m by n + 2m — 2. p. 42: delete lines 1,2. p. 42 line 6: replace roman m,n by italic m,n. p. 45: the equalities in line -2 of 5.2.6 should read p. 45: the line "This follows immediately from the previous lemma" in 5.2.7 should read: This has been stated in the proof of the previous lemma.

p. 49, line -5: insert 2 in front of the second p. 49, line -7: insert 2 in front of the second p. 50, line 4: replace 2 by 4 in front of the second p. 56, Lemma 7.1.4 should read: Assume that m = an + 1. We have _ v371-1 j p. — K p.f+ _ f+ f+ ft" 3Jn,m\e Jn,m;e -J ^», _ 01 -1 •'n-l.mjl Vj

Vj

n+1 3v~

Vj

f+ - 1 in-l,m;-l•

V-

p. 57 line 3 and 4: the factors v ® r ( c m - m + 1 ) should be be deleted. p. 68 at the end of 9.1.2: insert the line: "Here g\ : Vi —> Vi is the restriction of g" p. 71 line -8: replace G xP F

by G XQ F.

p. 72, line 7: replace T{ by TV p. 77 in 9.3.1 line -8: replace Y(i) > 7(i) by ^ > yx and 7 r (i) > 7p) b y 7i > 7ip. 85, line 11: insert ) before = . p. 87, lines 2,3 of 10.3.2: delete "and TV = 0 for all i' 6 I — I' " p. 91, lines 7, 8, 12, 18: replace . . . (a*) by • • • 0 (a*) (in five places). p. 113, lines -13, -12: replace "a-orbit"

by "orbit".

p. 118, line 8 of 14.2.2: replace pb € Z by pb e Q. p. 124, line 2 of 14.4.11: replace

by

ef'X)+1.

p. 128: Ref. 6, add: I.H.E.S. before 67. p. 132, line 1 of 16.1.2(b): replace t r ^ * " 1 ) / 2 p. 138 line -7, -5, -3: replace last C{M))

by

by t ^ " 1 ) / 2 . C{M).

p. 138 line -2: replace all C{M)) by C{M) (three times). p. 164, line 6 of 19.1.1, replace Sp\ by piS. p. 171, line 4: replace Let be A a by Let b e AA, b / 0. p. 177 line -2: replace w(u)x{ji) (following the second =) by u){u) • x(v)-

p. 181 in lines 2, 5, 6: replace df,}9itb',n by p. 181 in lines 2, 6: replace \b'\ by \(5'\. p. 185 in line - 8 , - 9 : replace AI, AI', A'2J A'2' and any t G U(Ai - A"), s G U(Ai - A"), by the following: z^ in Z [I] such that A'1? A", X'2, X'i in X, any 7 A; - A'/ = vu Ai - A / = v2 (in X) and any t G U(i/i), s € U(i/ 2 ), p. 190 line -1: replace M\ T

by

U

M\ T.

At the end of p. 191 (after "...from 23.3.6") add a new subsection: 23.3.11. We give a more detailed proof of Proposition 3.5.4 based on results in this chapter. Let 7TA',A" be as in 23.1.1. Prom the definition we see that if ,A2 (w) = 0 for any AI, A2 in X then u = 0. u E U satisfies Let a = G N[/], b = G N[/], X e X, J(\a,b) = Ei + ^ + E ^ y U ^ - v ^ ) . Let « G D a , M J(A, a, 6). By 3.5.3 it is enough to show that u = 0. Hence it is enough to show that 7TA1,A2(w) = 0 for any AI,A2 in X. If (i, A2) = ai — hi for all i G I we have 7r\lix2(J(X2,a,b)) C P(X2,a,b) (notation of 23.3.3). Hence it is enough to show that for any £ G X we have HJP(£, a, b) = 0 where the intersection is taken over all a, b such that (i, (} — a^ — bi for all i. Using the isomorphism <j) in 23.2.5 we see that we may assume that our root datum is simply connected. We identify X+ with N[J] by A M A where A = ^ enough to show that for any £ G X we have + n A ' , A € A : + ; C = A ' - A - P ( C j A', A) = 0. For A', A G X such that £ = A'-A we have U an isomorphism U l c -> f ® f = M\ My (see 23.3.1(c)) which by 23.3.5 carries P(C> A'5 A) onto T\ f + f (g> Ty where T\ (resp. Ty) is the kernel (resp. f = My —> Ay). of the canonical homomorphism f = UM\ It is then enough to show that nA,A'ex+;C=A'-ApA <8> f + f ® Ty) = 0. Now assume that x belongs to the last intersection. We have x — EI/GF I/'GF' ^ ® & where F, F' are finite subsets of N[/]. We can find A) > tr(i/) + tr(i/), Ei<*> A + C) > A G X+ such that A + C e X+, t r ( v ) + tr(i/) for any v G F, z/ G F'. If x ^ 0 then for such A we have x^T A (8)f + f<S) 7x+ c . This shows that x = 0. This proves 3.5.4. p. 200 line -5: replace last A' by A

p. 220: part (d) of Theorem 27.3.2 and the first two lines of the proof are missing. Thus, at the bottom of p. 220 (after 27.3.2(c)) add: —> C/v~xC is (d) The natural homomorphism C fl an isomorphism. The element b\ ob[ = J2b2,b'2 ^fei.&i,62,63® satisfies the requirements of (a). This shows the existence in (a). It is also clear that the elements p. 228: the diamond in line 2 of 28.2.6 should be bigger (of the same size as the diamond in line 4 of 28.2.6). p. 237 in line 2 of 29.5.1: replace: on U[> A] for some A e X+ by on some two-sided ideal of finite codimension of U. p. 240 first line of 30.1.7: remove one dot after "Example". p. 246 line 8: replace x 6 iu

by

x E r$u

p. 320 in line 5 of 40.1.2: replace l(w) = 1 by l(w) = 0. p. 320 in line 6 of 40.1.2: replace l(w) > 1 by l(w) > 0. p. 325 line -10: replace [7] by [6].