Lectures on Lie Groups

Let V be a finite dimensional real vector

= Aut

V

I

which is an open subspace of

V). We ca n identify G with Hom 0J e

I

A E Hom (V

I

V).

exp A

Proof.

=

I

V).

Let

Then we as sert 1 +A+

A2

2"

An

+

+-+

n!

2 2 A t Consider 1 + At + -2- +

This is

easily seen to be a smooth homomorphism from Rl to Aut V and is the I-parameter subgroup corresponding to A. A2

exp A

2 .13

= 1 + A + 2"

EXAMPLE.

+ ••• +

Consider G

An

AT

I

Thus

+ ....

= Tn = Rn /Zn.

Then G

n and exp can be identified with the covering map R

-t

e

= Rn

Tn.

I

ONE-PARAMETER SUBGROUPS

THEOREM.

2.14

13

exp is a diffeomorphism of a neighbourhood

of 0 E G with a neighbourhood of e in G. e Proof.

This is immediate from 2 • 10 a nd the Jacobian theorem.

(See [12

I

p. 12

I

Theorem 1J.)

2.15

THEOREM.

by ~ (Vl

I

Let G

v 2 ) = exp

Vl

exp v 2.

a neighbourhood of 0 E G

Proo f •

.

~lSt

I

eV2

Then

I

C{)

and define ~ : G

e

- G

is a diffeomorphism of

with a neighbourhood of e in G.

e

lI:I V exp x exp ~ G xG-?>G, IJ. h ecomposltlonV1'07 . . 2

and so is differentiable. and V2

= V1

e

Further

I

~

I

is the identity on both V1

and so is the identity on G • We may proceed as in e

2.14.

2.16

PROPOSITION.

of G, and let S

C

Let G1 denote the identity component

G1 be a neighbourhood of e. Then the sub-

group generated by S is G 1

•

Pro 0 f.

G1

Clearly gp{S1

C

•

Now gp[S} is an open sub-

group of Gl , so all its cosets are open. Thus gp{S} is also closed, so gp[S}

2 .17

THEOREM.

= G1 . If G is connected, a homomorphism of Lie

LECTURES ON LIE GROUPS

14 groups 9

1 :

G

e:G - H • e

e

Proof.

Thus

H is determlned by the induced homomorphism

By 2. 11 we have the commutative diagram:

e is

determined by 9 at least on the subgroup of G 1

generated by the image of exp. e in G, so

2.18

e is

determined on G.

LEMMA.

Let

~

sends 0 E V to e E G. xy = x + y +

0

But this is a neighbourhood of

a

: U

a

- G

a

be a chart on G which

Then, omitting

~

a

,we can write

(r) in a neighbourhood of e in G, where r = r (x, y)

denotes the distance of (x, y) from (e, e) in G x G under a metric.

Proof.

Since the product in G is differentiable, there is a

constant vector a and constant linear functions b, c such that xy

=a +

bx + cy +

0

(r).

Set x

=e

and we find that

y = a + cy +

0

(r) so a = 0, c = 1. Similarly b = I, so

xy = x + y +

0

(r) •

15

ONE-PARAMETER SUBGROUPS

2.19

THEOREM. a

form T

Proof.

A connected Abelian Lie group G has the

b xR •

We show first that exp : G

e

- G is a homomorphis m.

Well, exp s exp t = (ex p

~)N (ex p ~)N

t s ( exp N exp N

)N

since G is Abelian

by 2.18 and 2.14, where we consider s , t fixed and N varying

= ex p (s

+

t

+

0

(1 ))

= exp(s + t).

Thus exp is a homomorphism, and, by 2.16, exp is onto. Consider K = Ker exp. morphis m, K is dis crete.

By 2.14, since exp is a homo-

Now a dis crete subgroup of a rea 1

vector space is a free Abelian group, with generators gl, ••• , gr which are linearly independent over R.

(This is

proved by induction over the dimens ion of the vector space.) Extend this to a basis of G • Then K is expressed as the set e of points with coordinates (nI, ••. ,nr,O, •.• ,0), each n n r r Thus G = G /K = T x R - • e

i

E Z.

16

LECTURES ON LIE GROUPS

2.20

COROLIARY.

A Lie group which is compact, connec-

ted and Abelian is a torus.

2.21

EXERCISE.

2 .22

DEFINITION of submanifold .

Classify the compact Abelian Lie groups.

Let W c V be a real finite dimensional vector space and subspace.

Let M be a smooth manifold, and N a subset of M.

Then a chart (i)

M

(ii)

~

a

a

r

~

a

N

: V

a

=¢

sends V

a

-+

M

a

is good if

(the empty set) , or

nW

onto M

a

n N.

An atlas is good if all its charts are good. N is a submanifold if there is a good atlas in the differential structure of M. Equivalence of good atlases is defined by the identity map being smooth, as before.

2.23

PROPOSITION.

If N is a submanifold of M

I

then N can

be given a differential structure as a manifold so that (i)

the inclusion of N in M is smooth, and

(ii)

P -. N is smooth if and only if the composition

P

-+

N ..... M is smooth.

ONE-PARAMETER SUBGROUPS is clear and left to the reader.

Proof

I

2.24

REMARK.

2.25

EXERCISE.

Nl

X

17

It follows that T(N) is embedded in T(M).

If Nl

N2 are submanifolds of Ml

I

N2 is a submanifold of Ml x M2

I

1M2

then

and its differential

structure as a submanifold is the same as its differential structure a s a product.

PROPOSITION.

2 .26

If G is a Lie group

submanifold and a subgroup

Proof.

and H is both a

then H is a Lie group.

Apply 2.23 and 2.25 to the maps of pairs

Ii : G x G H x H I

2.27

I

I

-+

THEOREM.

G Hand i: G H I

I

-+

G H. I

A closed subgroup H of a Lie group G is a

s ubmanifold •

Proof.

LEMlvtA.

2.28

oI h

n

h ->

The next three lemmas constitute a proof.

n

In 2.27

I

suppose G

e

has a norm.

EGis a sequence of points such that exp h e

1 0, and P h h I nn I n

->

Suppose n

E H,

v E G . Then exp (tv) E H for all t E R. e

LECTURES ON LIE GROUPS

18 t Ih

I h n - tv and Ih n I - 0 so we rna y choose inten gers m such that m Ih I - t. Then exp m h - exp tv. But n n n n n Proof.

I

exp m h = (exp h) n n n

mn

I

E H

I

and H is closed. So exp tv E H,

as required. Let W be the set of such tw in G • Then exp We H. e

2.29

LEMMA.

Proof.

W is a vector subspace of G • e

Clearly w E W implies tw E Wall t E R.

So

I

suppose WI w 2 E W I

I

10.

and suppose WI + W2

We will show that WI + W2 E W. Consider exp(twI )exp(tw2). This is in H.

For t suffi-

ciently small we can write exp(twI )exp(tw2) = exp (f(t))

I

where

f (t) is a smooth curve in G and f (0) = O. e Now exp(twI )exp(tw 2 ) - exp t(w 1 + w 2 ) so

~f(t)

hn

= (~),

- WI + W2 as t - O.

= oCt)

I

by 2.18

Thus we may apply 2.28 with

for n sufficiently large, and v

= Iw,

! w,1 (w, + w, l,

and deduce that w 1 + W2 E W.

2 .30

LEMMA.

Proof.

exp W is a neighbourhood of e in H.

Split Gas WI e

e Wand

cons ider the diffeomorphis m

cP (Wi w) = exp (Wi )exp (w) between a neighbourhood of 0 in G I

I

e

19

ONE-PARAMETER SUBGROUPS and a neighbourhood of e in G (2.15). not hold.

Suppose the lemma does

Then there is a sequence of pa irs

exp(w )exp(w ) E H, exp(w )exp(w ) n n n n l

l

-+

(Wi , W

e, and

n

Wi

n

n

)

such that

10.

Since

exp(w ) E H,exp(w l ) E H. Then we can find a subsequence of n n 1 Wi such that - - Wi' -+ Wi E WI, for some such Wi, and n Iw~1 n Iw II = 1. It follows from 2.28 that

Wi

E W, which is a contra-

diction. Thus exp W is a neighbourhood of e in H. It is now clear that exp provides a good chart for a

neighbourhood of e. any other point of H.

2 • 31

Left translation gives a good chart round This completes thE: proof of 2.27 •

EXAMPLES. O(n)

C

GL(n, R)

U(n)

C

GL(n, C)

SP (n)

C

GL(n, Q)

are closed subgroups, and so submanifolds, of Lie groups. Thus they are Lie groups. In each case, the tangent space at e of the subgroup consists of the matrices X such that

XT

= -X.

LECTURES ON LIE GROUPS

20 Proof. (i)

Suppose X is in the tangent space at e of the subgroup.

Take a smooth curve of the form f(t) = 1 + tX + o(t) in the subgrouP. Then f (t) Tf (t) = I, by definition of the subgroup. That is,

-T

(1 + tX

+ o(t)) (l + tX + o(t))

= 1.

Thus xT+X=o. (ii)

-T

= -X.

Suppose X

Then (exp tX) T=

(~

= ~t =

00

o

n

t n Xn In!

)T

by 2.12

n

(-X) In!

(exp tX)-l.

Therefore exp (tX) lies in the subgroup, and X in the tangent space.

2 .32

EXAMPLE.

If G is a compact LIe group, and H is a

closed connected Abelian subgroup

2 .33

PROPOSITION.

I

then H is a torus.

A closed connected subgroup H of a

Lie group G is determined by its tangent space at e.

Proof.

See 2. 1 7 .

21

ONE-PARAMETER SUBGROUPS

2.34

DEFINITION.

closed subgroup. cosets gH.

Suppose G is a Lie group and H a

Then the quotient space G/H is the set of

We have the projection p : G

-+

G/H, and give G/H

the quotient topology.

2.35

EXERCISE.

2.36

PROPOSITION.

G /H is Hausdorff.

If H is a closed subgroup of a tie

group G, we can give G/H a differential structure as a manifold so that (i)

p is smooth,

(ii)

f: G/H

-+

M is smooth if and only if fp : G

-+

M is

smooth.

Proof.

SplitG

e

asW'E9WwhereW=H

e

as before.

be a small neighbourhood of 0 in W', and define l/J : U by U

-+

W'

-+

G

e

exp!» G ~ G/H.

with a neighbourhood of eH.

Proof.

PROPOSITION.

H

See [17, 1.7.5J.

-+

G/H

Then l/J is a homeomorphis m

The rest of the proof is left as an

exercis e for the reader.

2 .37

-+

LetU

G

->

G/H is a fibration.

Chapter 3

ELEMENTARY REPRESENTATION THEORY

In this chapter we set up elementary representationtheory.

The ba sic definitions a nd constructions occupy 3. 1

to 3. 13. Then we introduce integration. the usual consequences

I

including complete reducibility (3.15

to 3.21). Then comes Schur's Lemma quences

I

From this we draw

I

some of its conse-

and the definition of the representation ring (3.22 to

3.28). Then traces

I

characters and the orthogonality relations

(3.29 to 3.37). Then the Peter-Weyl theorem and the completeness of characters (3.38 to 3.49). Then the usual material on real and symplectic representations (3.50 to 3.64).

Next

comes the behaviour of the representation ring for products (3.65 to 3.67) and coverings (3.68 to 3. 70). Finally we have the representation-theory of the torus (3. 71 to 3.78). 22

ELEMENTARY REPRESENTATION THEORY

3.1

DEFINITIONS.

23

Let A be one of the classical fields R

(the real numbers), C (the complex numbers) or Q (the quaternions).

Let G be a topological group.

Then a AG-space is a

finite-dimensional vector space V over A provided with a continuous homomorphism

e:G

- Aut V.

(Such a V is also called a representation of G over A or a Gspace over A.) Alternatively, for each g E G and v E V we are given gv E V, and the following conditions are satisfied. (i)

ev = v and g(g'v) = (gg')v.

(ii)

gv is a A-linear function of v.

(iii)

gv is a continuous function of g and v. By choosing a base in V we can regard

e as

taking

values in GL(n,A). We then speak of a matrix representation.

In the case A = Q, if we wIsh to write our matrices on the left, it will be prudent to arrange that V is a right module over Q.

Fortunately we can make any left module over Q into a right module over Q, and vice versa, by the formula qv

= vq

(q E Q, v E V).

Here the conjugate of a quaternion is defined as usual:

if

LECTURES ON LIE GROUPS

24 q

=a

+ bi + cj + dk, then

q=a

- bi - ij - dk.

Let V and W be I\.G-s pa ces. A G-map is a function f : V - W which commutes with the action of G, that is, f (gv)

= g (fv).

A I\. G-map is a G-map which is I\.-linear: mostly we deal with such.

The set of such I\.G-maps is written HomI\.G(V, W), or

sometimes simply Hom tor space over R if I\.

G

N, W)

= R or

if I\. is understood.

Q, over C if I\.

It is a vec-

= c.

A I\.G-isomorphism is a I\.G-map which has an inverse. As usual, we say that two I\.G-spaces are equivalent if they are isomorphic.

3.2

DEFINITION.

Let V be a G-space over C. A structure

map on V is a G-map j : V - V such that (i)

j is conjugate-linear, that is ,

j (zv) = z(jv)

(z E C)

I

and

= ±1.

(ii)

j2

3.3

EXPIANATION.

If V is a G-space over Q, we may

regard it as a G-space over C with a structure map such that j2

= -1.

Actually we may do so in two ways.

On the one hand

we can take the C-module structure given by i acting on the

ELEMENTARY REPRESENTATION THEORY

2S

left and the structure map given by j acting on the left.

On

the other hand we can take the C-module structure given by i acting on the right (-i acting on the left) and the structure map given by j acting on the right (-j acting on the left).

It makes

no difference which we take, because we can define an automorphism a : V - V taking one structure into the other, for example a (v) = kv. Conversely, given a G-space over C with a structure map such that j2 = -I, we can clearly reconstruct a G-space over Q. Similarly, it is often convenient to regard a G-space V over R as being equiva lent to a G-space V lover C provided with a structure map such that j2

= +1.

To pass from V to VI

we take V I = C ®R V provided with the obvious operations and structure maps: Z (Zl ~

v) =

g(z ®v)

ZZI

=z

®v

(z , Zl E C)

® gv

j (z ® v) = z ® v.

To pass from V

I

to V we split V into the +1 and -1 eigenI

spaces of j; these are G-spaces over R which are isomorphic under i. These operations are clearly inverse to one another,

LECTURES ON LIE GROUPS

26

up to isomorphism.

3.4

DEFINITION.

Given I\G-spaces V and W, we can form

the direct sum of the two vector spaces

V (f)W and make G

I

I

act on it by g(v w) = (gv gw). I

I

Equivalently we may take two G-spaces V and Wover I

C with structure maps j ,j

v

w

such that j2 = j2

w

v

I

and put on

V ffi W the structure map \, (f) jw • The next five operations start from a G-spa ce V over 1\ and construct a G-space over some 1\'.

are displayed in the following diagram

I

The possibilities

which is not commuta-

tive.

3.5

DEFINITIONS.

(i)

If V is a G-spa ce over R define cV = C 0 R V regarded I

I

as a G-space over C as in 3.3. (ii)

Similarly

I

if V is a G-space over C

I

define

qV = Q ®C V and regard it in the obvious way as a G-space I

ELEMENTARY REPRESENTATION THEORY

27

and a left module over Q. (iii)

If V is a G-space over Q

let c V have the same under1

I

lying set as V and the same operations from G but regard it I

as a vector space over C. (iv)

Similarly

I

if V is a G-space over C

I

let rV have the

same underlying set as V and the same operations from G

I

but

regard it as a vector space over R. (v)

Let V be a G-space over C.

We define tV to have the

same underlying set as V and the same operations from G

I

but

we make C act in a new way: z acts on tV as z used to act on

V. Let us adopt the viewpoint of 3.3; then both c and c' act on a G-space over C provided with a structure map j

I

and

they act by forgetting the structure map. All these constructions are natural; given a J\G-map

f : V- W

I

we can construct maps cf

I

qf

I

c' f

I

rf and tf.

All these constructions commute with direct sums

3.6

PROPOSITION. rc

=2

cr

= 1 +t

qc' = 2

~.

28

LECTURES ON LIE GROUPS c'q

=1 + t

tc

= c

rt

=r

tc'

c'

qt

=q

t2

= 1.

These equations are to be interpreted as saying that rcV';: V ~ V crV ~ V Ei) tV

for each V over R for each V over C,

etc. Proof.

Most of this can safely be left to the reader; we

show that cr

= 1 + t.

Let V be a G-space over C. We want to study C 0 with C acting on the first factor and G on the second.

R

V

Let C

act on the first factor of C ®R C, and let C ®R C act on C

~

V in the obvious way.

Then C ®R V is a G-space over

the C-a Igebra C ®R C. We will now split the unit 1 ® 1 of C ®R C into orthogonal idempotents, and so obtain a splitting of C ®R V. detail, let e1 =

t(l

® 1 + i ® i)

In

ELEMENTARY REPRESENTATION THEORY e2

=

Then ei = e 1

t(l ,

29

® 1 - i ® i) •

e~ = e 2 , e 1 e 2 = 0 and e1 + e2 = I, as required.

So

where the isomorphism is a G-isomorphism over C.

Further,

v ~ e 2 (C ®R V)

by, for instance,

v"'" e 2 (1 ® v)

tV ~ e 1 (C ®R V)

by, for instance,

v ..... e 1 (1 ® v) •

and

Thus crV ~ V ~ tV.

3.7

DEFINITION.

Given G-spaces V and W over C, we

can form the tensor product of the two vector spaces, V ®C W, and make G act on it by g (v ® w) = gv ® gw. Suppose now that V and W admit structure maps such that j~ = map j

= jv

f.

V'

j~= f.

\r,jw

Then V ®C W admits a structure

W

® jw such that j2

= f.Vf. W •

We can separate three

cases. (i)

f.

V

= f. W = +1.

sentations is real.

The tensor product of two real repre-

The construction amounts to taking two

G-spaces V, W over R and forming V ®R W. (ii)

f. V = + I, f. W

= -1.

The tensor product of one real

representation V and one quaternionic representa tion W is

30

LECTURES ON LIE GROUPS

quaternionic.

The construction amounts to taking V ®R Wand

making Q act on it by q(v

~

w)

=v

® qw.

The case (V = -I, (W = + 1 is similar. (iii)

(V

= (W = -1.

The tensor product of two quaternionic

representations V and W is real. construction in terms of V

«b W.

It is natural to interpret the

For this to make sense we

must consider V as a right module over Q and VV as a left module over Q. If we use the resulting structure maps, then the -1 eigenspace of the structure map jv ® jw on V 0

C

W

coincides with V ®Q W.

3.8

PROPOSITION.

(i)

All our tensor products are compatible with the maps

c, c' .

(ii)

The tensor product ® is bilinear over the direct sum EB.

3.9

DEFINITION.

Given G-spaces V and W over the same

field A, we can form Hom from V to W.

A

(V, W), the set of A-linear maps

It is a vector space over R if A = R or Q, over

C if A = C. We can make G act on it by (gh)v

= g(h(g-lv))

(h E Hom

A

(V, W))

31

ELEMENTARY REPRESENTATION THEORY or eq 11ivalently

(Note that Hom AN, W) is covariant in Wand contravariant in v.) The subspace of elements in Hom

A

(V,W) which are in-

variant under G is precisely Hom I\G N, W). We may also proceed as in 3.3, 3.7.

Let V and W be

G-spaces over e which admit structure maps jv,jw such that j~

= f. V ' j ~

= f.

W

• Then Home(V, W) admits a structure map j

given by

(Note that Home(V, W) is covariant in Wand contravariant in V.)

= f.Vf. W •

We have j2

(i)

f.

real.

V

= f. W

= +1.

We can separate three cases.

The Hom of two real representations is

The construction amounts to taking two G-spaces V, W

over R and forming Hom (V , W). R

In fact, if we form

Home (cV, cW), then the +1 eigenspace of j may be identified with Hom

R

tv, W);

thus

Home(cV, cW) ~ cHom (ii)

f.

V

= +1,

f.

W

= -1.

RN

, W).

The Hom of a real representation

into a quaternionic representation is quatemionic. f.

V

= -I, f.

W

= +1

is similar.

The case

We leave it to the reader to

32

LECTURES ON LIE GROUPS

interpret the construction in these cases along the lines of 3. 7 (ii). (iii)

f.

V

= f. W

= -1.

tations is real.

The Hom of two q aternionic represen-

The construction amounts to taking two G-

spaces V, W over Q a nd forming HomO (V, W).

In fact, if we

form HomC(c'V,c'W), then the +1 eigenspace of j is HomO (V, W); thus HomC(c'V ,c'W) ~ cHom

O

N , W).

3.10

COROLLARY.

(i)

If V and Ware two G-spaces over R then

dimCHom (ii)

CG

(cV , cW)

= dimRHom RG (V, W).

If V and Ware two G-spaces over 0 then

dimCHom

CG

(c' V, c' W) == dimRHom

OG

(V, W).

This follows immediately from 3. 9(i) and (iii), by looking at the subspaces of elements invariant under G.

3 • 11

PROPOSITION.

(i)

All our Hom's are compatible with the maps c, c' •

(ii)

Hom is bilinear over the direct sum ffi. A particular case of 3.9 is important.

33

ELEMENTARY REPRESENTATION THEORY

3.12

DEFINITION.

Given a G-space V over C, we define

its dual V* by V* = HomC(V, C). Here the target space C is given the trivial operations from G: gz = z for all g E G and z E C. The G-space C is real; it follows that the dual of a real representation is real, and the dual of a quaternionic representation is quatemionic. The general case 3.9 may be reduced to the special case 3.12.

3.13

LEMMA.

We have an isomorphism

HomC(V,W} ~V* ®C W commuting with the action of G and with the structure maps j (if any).

Proof.

The isomorphism sends V*

~

W into the map h,

where h (v) = (v*v}w. In dealing with compact topological groups, one of our best weapons is integration.

3.14

INTEGRATION.

Let G be a compact topological group.

Then for each continuous function f : G - R we can define a

34

LECTURES ON LIE GROUPS

real number

r

JG

f =

r

f(g)

.)gEG

so as to satisfy the following conditions. (i)

S ha s the usual properties of an integra I, tha t is,

it

G

is a pos itive linear functiona 1. (ii)

~

1 = 1. G

(iii)

The integral is invariant under left and right transla-

tions; that is, for each x E G we have

=

\ f(xy) , yEG

SyEGf(y)

SyEGf(yx) = SyEGf(y) • Similarly

I

we may integrate functions which take

values in any finite-dimensional vector space over R so as to obtain values lying in that vector space; and if we do so, integration commutes with linear maps. If G is a Lie group then the integral is slightly easier

to construct than if G is a more general topological group.

We

will not disc 1SS this here, but refer the reader to [13,14,20]. 1

The first function which asks to be integrated is

e:G

- Hom AN, V) •

35

ELEMENTARY REPRESENTATION THEORY

3. 15

e:G

PROPOSITION. - Hom/\ (V, V). I

= SeE

S uppos e gi ven a repres enta tion

Then

Hom /\ (V V) I

G

= I)

is idempotent (I2

and its image is VG' the subspace of

elements invariant under G.

Proof.

For each fixed v E V, the function Hom

(V V) - V I

given by h - h (v) is linear (over R); so it commutes with integretion.

That is, Iv =

S

gv.

gEG

It is now clear that 1m (I) c VG;

for

g' (Iv) = g'S gv gEG =

S

g'gv

(since 9 acts linearly)

gv

(invariance of integration under

gEG =

S aEG

left translation)

iv; so Iv E V . Also we have I\V G G

= v. The propos ition follows.

= 1;

for if v E V ' then G

LECTURES ON LIE GROUPS

36

Propositions 3. 16 and 3. 18 may be viewed as applications of the principle embodied in 3. 15; they could equa lly easily be proved directly.

PROPOSITION.

3 .16

Let G be a compact topological group

and let V be a G-space over C.

Then we can give V a positive

definite Hermitian form H which is invariant under G. over, if V carries a structure map j

I

More-

we can choose H so that

H(jv,jw) = H(v,w). The reader who wishes to do so may check that if V has a structure map j

I

then a Hermitian form with the property

stated amounts to a Hermitian form over A = R or Q according to the case.

The statement we have given avoids separating

cases, and is convenient for later use.

Proof.

Consider the space L of Hermitian forms H on V.

This is a vector-space over R, and G acts on it by

By 3. 15, if we take any Hermitian form H and integrate gH, we get a Hermitian form invariant under G, given by K (v , w)

=S

H ( g-1 V , g-l w) .

gEG

If we start by choosing H to be positive definite

I

then K is

37

ELEMENTARY REPRESENTATION THEORY positive definite.

Now suppose that V has a structure map j, and that we begin by choosing a positive definite Hermitian form H invariant under G.

Then we can construct a new form by integrating

over the 22 or 24 group generated by j; the formula is K(v, w) = t(H(V, w) + H(jv ,jw)). This form ha s the required properties. If we impose on V an invariant Hermitian form, then

we can choose in V an orthonormal basis.

e:G U (n).

Thus we can regard

Aut Vas taking values not merely in GL(n,C), but in We then speak of a unitary representation.

Similarly

for orthogona land symplectic representations in the cases A = Rand Q.

3.17 Proof. form H.

COROLIARY.

If G is compact and A = C, then V*?!!tV.

Impose on Van invariant positive definite Hermitian To be explicit, suppose tha t H (v, w) is conjugate-

linear in v and linear in w.

Then we can define

a : tV - V* by (av)w=H(v,w), and a is a G-isomorphism over C.

38

LECTURES ON LIE GROUPS

PROPOSITION.

3 .18

G-space V is projective.

If G is a compact group, then every

That is, suppose given the following

diagram of AG-maps, in which

~

is onto.

x

!~

V~y

Then there is a AG-map y : V - X such that the following diagram is commutative.

P fO

0

3.9.

f.

Consider HomA (V, X), made into a G-space as in

By 3. 15, if we take any A-map 0 : V - X and integrate

gO, we get a A-map y which is invariant under G, that is, a It is given by

A G-map.

y =

S

(9 g) 0 (BVg-l ) .

gEG

x

We can choose 0 to be a A-map such that have

~y

=

~s

(9 g)0(9 g-1 )

gEG =

S gEG

=

S gEG

x

v

~ (9xg) 0 (BVg- 1 ) (9yg)~0 (9 Vg-1 )

~O

= a.

Then we

39

ELEMENTARY REPRESENTATION THEORY

= a.

3. 19

DEFINITION.

A non-zero G-space V is reducible if

some proper subspace of V is a G-space; otherwise irreducible.

3.20

THEOREM.

If G is a compact group, every G-space

V is the direct sum of irreducible G-spaces.

Proof.

By induction over dim /\ V; so assume the result

true for G-spaces W with dim /\ W < dim/\ V.

It will now be

sufficient to show that if V is reducible, then it is the direct sum of two subspaces of less dimension.

Suppose that V has

a proper subspace S which is a G-space; then 3.18 shows that the exact sequence

o-

S - V -

viS -

0

splits, so we have a /\G-isomorphism V ~ S ~ VIS.

Alternatively, if /\ = C we may complete the argument by imposing on V a Hermitian form H which is invariant under G, and taking T to be the orthogonal complement of S; then

40

LECTURES ON LIE GROUPS V=SffiT.

If V has a structure map j, and S is closed under j and H is as

in 3. 16, then T is closed under j.

3.21

EXAMPLE.

We wi 11 show tha t 3.20 does not hold for

groups which are not compact.

Embed R' in R2 as the subspace of vectors be the subgroup of GL(2, R) which stabilises Rl.

G is the set of matrices

[~l

Let G

Equivalently,

[~ ~] with ac '10.

Then R2 is a reducible G-space. However, no other proper subspace of R2 is stable under G, so R2 does not split as the direct sum of irreducible G-spaces. Alternati vely, to get a

II

minimal" counter-example,

. [10 b]1 .

take the group G to be the set of matrices

Next we shall need to know to what extent the decomposition of a G-space into irreducible summands is unique (3.24).

For this purpose we need the following classical

result .

3.22

(SCHUR'S LEMMA).

(i)

If f : \' - W is a I\G-map and V, Ware irreducible then

Let G be any topological group.

f is either zero or an isomorphis m.

ELEMENTARY REPRESENTATION THEORY

= C,

(ii)

If A

then fv

= AV for

41

f : V - V is a CG-map and V is irreducible

some constant A E C.

(In the second case we may write f = A.)

Proof. (i)

Since V and Ware irreducible, Ker f is V or 0 and 1m f

is 0 or W. (ii)

The res ult follows.

Consider f - A : V

-+

map is singular for some A.

V, where A runs through C.

This

By (i), f - A is then zero.

Thus

f = A.

3.23

COROLLARY.

(i)

If Vand Ware inequiva lent then Hom

(ii)

If V and Ware equivalent and A = C, then

dimCHom (iii)

CG

Let V and W be irreducible AG-spaces. AG

(V, W)

= o.

(V, W) = 1.

If V and Ware equiva lent and A = R or Q, then

dimRHom AG (V, W) ~ 1.

Proof.

For (iii) we observe that Hom AG (V, W) contains at

least one isomorphism. For the next proposition, let G be any topological group, and let V. run over the inequivalent irreducible 1

AG-spaces (as i runs over some set of indices I).

Let m

i

I

n be i

42

LECTURES ON LIE GROUPS

non-negative integers, of which all but a finite number are zero.

Let m. V. be the direct s urn of m. copies of V., and simi1 1 1 1

larly for n. V.. 1

THEOREM.

3.24 m.

1

1

= n.1

If

m.V. is equivalent to ~n.V., then ill ill

(B

for all i.

Pro 0 f.

Suppose

= EBn.V ..

(Bm.V. ill

ill

Then HomAG(V. ,E8m.V.) ~ Hom \G(V, ,ffin.V.), J i 11 i Jill that is, ~m.HomAG(V"V.) ~ G)n.HomAG(V., V.l.

i

1

J1

i1

J

1

Using 3.23(i), we get m.HomAG(V., V.) ~ n.HomAG(V., V,). J J J J J J Taking the dimension of both sides and using 3.23 (ii) or (i'ii), we get m.

J

= n .. J

If G is compact and A = C we can express the situation

which arises here in the following way (which we need for later use).

For any G-space V over C we can form

This is a finite sum, since HomCG(V , V) is zero for all but a i

43

ELEMENTARY REPRESENTATION THEORY finite number of i, by 3 .20 and 3.23 (i).

IJ. : ~HomCG(V, V) ®CV. 1

1,

1

-t

We ca n define

V

by evaluation:

IJ. (h. ® v.) 1

1

= h.1 (v.)1 .

We make G act on ~HomCG (Vi' V) ®C Vi by g (h. ® v.) = h. ® g v .. 1

1

1

1

Then IJ. is a G-map over C.

3.25

LEMI'v1A.

Assume G compact and A = C.

Then the map

is an isomorphism.

If V is irreducible the result is immediate by 3.23.

Proof.

Pass to direct sums and use 3.20.

3.26

DEFINITION.

Let G be a compact topological group.

Then K/\ (G) is the free abelian group generated by the equivalence classes of irreducible G-spaces over A. Tnus an element of K A (G) is a formal linear combination I; n.V., in which the V. are the equivalence classes of ill

1

irreducible G-spaces over A, and the n. are integers (positive, 1

negative or zero) which are zero for a II but a finite number of i.

44

LECTURES ON LIE GROUPS

By 3.20 and 3.24, the equivalence classes of G-spaces over

A are in 1-1 correspondence with those elements l:n.V. in ill

KA (G) such that n 2: 0 for all i. i An element of KA (G) is called a virtual representation or virtua 1 G-space. The operations c/c· ,r,q and t of 3.5 induce homomorphisms of abelian groups as displayed in the following diagram, which is not commutative.

The equations of 3.6 continue to hold.

3.27

PROPOSITION. c

: KR(G)

c· : KQ(G)

-0

-0

The maps

KC(G) KC(G)

are mono.

Proof.

rc

= 2,

qc' = 2 and KR (G), KO (G) are free abelian.

We shall normally regard KR (G) and KO(G) as embedded in KC(G) by c and c' .

ELEMENTARY REPRESENTATION THEORY

45

3.28

COROLLARY

(i)

If V and Ware two G-spaces over R such that cV

~

cW,

then V ~W. (ii)

If Vand Ware two G-spaces over Q such that

l c V ~ c'W, then V ~

W.

This follows immediately from 3.27, but in ca s e it seems to spring from nowhere we also give a direct proof. Suppose given two G-spaces V, W over C, which admit structure maps jv,jw such that j~ CG-isomorphism f : V

--+

= j~.

We suppose given a

W which does not necessarily commute

with j, and we wish to construct a CG-isomorphism which does commute with j.

We can construct CG-maps which do commute

with j by starting from f

I

or if, and integrating over the 22 or

24 group generated by j. f' = 1.(f + j 2

f"

= ~ i (f

The formulae are

Jrl)

W- V

- j

vl j ~ ).

We have fl - ifll = f.

So det (fl+ z fll) is a polynomial in z which

is not identically zero (for it is non-zero for z

= -i).

fore there is some real x for which det (fl + Xf")

10.

fl + Xf" is a CG-isomorphism which commutes with j.

ThereThen

46

LECTURES ON LIE GROUPS

TRIVIAL EXERCISE.

If V admits a structure map j, then it also

admits -j, and clearly forgetting j gives the same result as forgetting -j.

Display a CG-automorphism sending j into -j.

If A = C, we can make KC (G) into a ring by using the

tensor product of G-spaces over C; we then call it the representation ring of G.

If x U'es in KA (G)

A = R or Q, and y lies in K 1\' (G)

C

C

KC(G), where

KC(G), where 1\

I

= R or Q,

then the product xy behaves as described in 3.7. The standard method of studying KC(G), and indeed the standard method of proving 3.24, is the study of characters. To define these, we need the

3.29

DEFINITION.

~.

Let V be a finite-dimensional vector

space over C, and let f : V - V be a linear map. define Tr f, the (i)

~

of f, in two ways.

Take a base of V, so that f corresponds to a matrix M ..• IJ

Set Tr f = I:M ... i

This is invariant under change of base, since

11

I: T., M 'k (Ti,j,k IJ J (ii)

Then we may

1

)

k'

1

= j,k 1: 1. k M' k = I: M ... J J j JJ

(Bourbaki) We ha ve an isomorphism a : V* ® V - Hom

C

(V, V)

given by (a (v* ® w»v = (v*v)w, as in 3. 13.

47

ELEMENTARY REPRESENTATION THEORY We have an evaluation map ( : V* ®V

-+

C

given by ((v* ®w) = v*w.

Define Tr f =

1

(a-

f.

It is easy to check that the two definitions are equiva-

lent.

The principal properties of the trace are as follows.

3.30

PROPOSITION.

(i)

Tr : HomC(V, V)

(ii)

Consider V

(iii)

Consider ~ ffi Y : V

L

-+

W

C is a linear map.

.L V.

Then

Tr(~y)

W

-+

V E9 W.

®Y : V ®W

-+

V ® W.

Ef)

=

Tr(y~).

Then Tr(~ ffi y) = Tr~ + Try. (iv)

Consider

~

Then Tr(~ ® y) = Tr~ . Try. (v)

Given ~ : V -+ V, define ~ * : V*

as usual. (vi)

Tr~ *

Then

Given

~

: V

-+

=

-+

V* by (~*v*)v =v* ((3v),

Tr~.

V, let

t~

: tV

-+

tV be as in 3.5 (v).

Tr(t~) = Tr~.

(vii)

If

~

: V

-+

V is idempotent, then

The proof may safel y be left to the reader.

Then

48 3.31

LECTURES ON LIE GROUPS

DEFINITION.

Given a G-space over C, we define its

character Xv : G - C by XV(g) = Tre g. It is clear that Xv depends only on the equiva lence

class of V. If V is a G-space over R or Q, we define its character

to be that of the complex G-space cV or clV as the case may be.

(In the case A = R it would be equivalent to consider the

trace over R, but this doesn l t work so well for A = Q.)

3.32

PROPOSITION

(i)

Xv : G - C is continuous.

(ii)

Xv (xyx-1 ) = XV(y).

(iii)

XVffiW(g) = XV(g) + XW(g)·

(iv)

XV®Vj.g) = XV(g) . XW(g).

(v)

X * (g) = Xv (g-l ) . V

(vi)

XtV(g) = XV(g); if v is real or quaternionic then

XV(g)= XV(g)· (vii)

XV(e) = di m C (V) .

Each part follows from the corresponding part of 3.31; the second half of (vi) uses also the equations tc = c tc l = c l I

ELEMENTARY REPRESENTATION THEORY

49

from 3.6.

3.33

PROPOSITION.

(i)

Xv(g-l.) = XV*(g) = XtV(g) = XV(g).

(ii)

SgEGXV(g) = dimCVG , where VG is the subspace of

Assume G compact.

elements of V invariant under G.

Proof. (i)

See 3. 17 .

(ii)

Since Tr is linear, we ha ve

SgEGTrB

9

= Tr =

SgEG B 9

Tr I

(s ee 3. 15)

= dimCIm I

(see 3.30 (vii))

=dimCV

(s ee 3. 15).

G

3.34

THEOREM.

(i)

Let G be compact and let V, W be G-spaces over A.

(Orthogona lity relations for characters. )

Then

SgEGXV(g)XW(g) = dim Hom AG =d

say

(V, W)

I

where the dimension is taken over C if A = C ,over R if A = R or Q.

50 (ii)

LECTURES ON LIE GROUPS If V and W

Now assume that V and Ware irreducible.

are inequivalent we have d = O.

If V and Ware equivalent and A = R or

A = C we have d = I. Q we have d

~

If V and Ware equivalent and

I.

Pro 0 f .

(i)

By 3. 10 the cases A = Rand Q follow immediately from

the case A= C.

So suppose A = C, and consider

H = HomC(V, W). dimCHom =;

We have CG

(V, W) = dimCHG

r

X (g)

by 3. 33(ii)

')9EG H

=S

gEG

=

S

Xv*®w(g)

by 3.13

Xv(g) Xw(g)

by 3.32 (iv),

3.33(i).

gEG

(ii)

See 3.23. Let us choose one irreducible AG-space V. in each I

equivalence class, as in 3.24, and let X. be its character. I

Then the functions X. are orthogonal, and therefore: I

3.35

COROLIARY.

The functions )(. are linearly independen I

ThIS fact can evidently be u'sed to give a second proof

ELEMENTARY REPRESENTATION THEORY of 3.24.

51

If

@m.V. ~ EBn.V., ill ill

then their characters are equa I, so I:m.x. = I:n.x., ill ill

and m. = n. for each}. 1

1

But by using 3.34 (i), we see that this

proof coincides with the first proof. Let C(G) be the set of continuous functions f : G ... C.

3.36

DEFINITION.

Such an f is called a class function if

f (xyx-1 ) = f (y) •

We write CI(G) for the set of class functions.

We

make CI (G) into a ring by pointwise addition and multiplication of functions. Characters are cIa s s functions, by 3. 32 (i) and (ii). We can define a homomorphism of abelian groups

by x(!;n.V.) = l:n.x.· ill ill

For every G-space V we have

xCV) by 3.32 (iii).

=

XV' X is a homomorphism of rings by 3.32 (iv) .

52

LECTURES ON LIE GROUPS

3.37

PROPOSITION.

Pro 0 f.

X: KC(G) ..... CI (G) is a monomorphism.

See 3.35. The image of X is called the character ring of G.

It is

natural to ask how large a part of CI(G) it is; and we will see that it is as large as could be hoped (3.47).

For this purpos e

we need the Peter-Weyl theorem. We recall that classically the Peter-Weyl theorem is stated in terms of component-functions M .. (g) of matrix rep1J

res entations M (g) .

Clearly such a function is obtained by

taking a matrix representation M : G

-+

GL(n, C) and composing

with a linear map GL(n, C) ..... C, namely proj ection onto the (i, j)th component.

We therefore introduce the following

lemma.

3.38

LEMMA.

The vector-space dual to HomC(V, W) is

HomC(W, V), where the pairing between a E HomC(V, W) and

f3 E HomC(W,V) is given by
Tr(a~)

= Tr(j3a).

We have HomC(V I W) ~ V* ® W, and therefore its

dual space is V ®W*~W* ®V~HomC(W,V).

ELEMENTARY REPRESENTATION THEORY

S3

To check that the pairing is as cIa imed is precisely what the student should already ha ve done in proving

Tr(a~) = Tr(~a)

(3. 30 (ii)) .

3.39

THEOREM

(F. Peter and H. Weyl) [15].

compact topological group.

Let G be a

Then every continuous function

f : G - C can be uniformly approximated by functions of the form Tr(a9 (g)), where

e:G -

e runs

over representations

HomC(V, V) and a runs over HomC(V, V).

The proof will occupy 3.40 to 3.44. Actually our line of proof will approximate f by functions of the form Tr(a9 (g-l)); but this makes no difference, since we can begin by replacing f with fl , where fl (g)

= f (g-l) .

The proof is based on the following ideas.

We make G

act on C(G) by (gf) (x) = f (g-l x) . Then C(G) is an infinite-dimensional representation of G. However, we can find certain finite-dimensional subs paces of C(G) stable under G, by using the theory of integral operators

S

k (x ,y)f (y).

yEG

We make G act on C(G x G) by

54

If the

LECTURES ON LIE GROUPS

II

kernel ll k is invariant under G, then the integral opera-

tor gives a G-map from C(G) to C(G) , and hence its eigenspaces are stable under G.

In the case at issue they are

finite-dimensional (3.42) and this provides the necessary representations. We now start work.

3.40

LEMMA.

Let G be a compact group and f E C(G).

Then f can be uniformly approximated by functions of the form v (x) =

S

k (x y)f (y) I

yEG

with k rea 1 symmetric and invariant under G.

Proof.

There is a neighbourhood U of e in G such that

I f (x) and U-1

- f (y)

= U.

Let IJ. : G - R be a continuous function such that

=0

",,(x)

I !S. ( for x ~

U

",,(x) ~ 0, 1

"" (x-

)

= "" (x)

S "" (x) xEG

Let

= I.

and

I

ELEMENTARY REPRESENTATION THEORY Then k is real, symmetric and invariant under G.

everywhere.

55 Also

Integrating over y E G, we get

If(x) - v(x)

I

~

(

where v (x) =

S

k (x, y)f (y) .

yEG It will now be sufficient to approximate such functions

v(x).

3.41

THEOREM.

Assume that k is Hermitian and u E C(G).

Then the function v(x) =

S

k (x, y)u(y)

yEG

can be uniformly approximated by a finite linear combination of eigenfunctions of k corresponding to non-zero eigenvalues.

The eigenfunctions corresponding to the eigenvalue A are, of course, the functions w such that

S

k(x,y)w(y)= AW(X).

yEG

Proof.

See [16, p. 117, 127J.

(Smithies considers integ-

ral equations on [a, b J, but the results are unchanged for integral equations on a compact manifold.) Note also that even if we were to consider some class

56

LECTURES ON LIE GROUPS

of functions larger than C(G), for example L2 (G), the eigenfunctions would be continuous, since k is continuous.

3.42

THEOREM.

Assume that k is Hermitian and A -I O.

Then the vector space V of eigenfunctions corresponding to A Indeed the sum I; I A.

has finite dimension.

i

1

12,

in which

1A 12

is repeated with appropriate multiplicity, is convergent.

Proof.

3.43

See [16, pp. 48,102,112J.

LEMMA.

under G.

In 3.42, assume further that k is invariant

Then every element of V can be written in the

required form Tr(a9 (g-1)) .

Proof. v E V.

The space V is a finite-dimensiona 1 G-space. Define a linear map

as follows: ~

if h E HomC(V, V), then

(h) = (hv) (e) .

Then we have

By 3.38, the element

~

corresponds to an element

a E HomC(V, V) such that

Let

57

ELEMENTARY REPRESENTATION THEORY

LEMMA.

3.44

The set of continuous function G -+ C which

can be written in the form Tr(a9 (g-l)) is closed under linear combinations.

Pro 0 f.

Suppose given

9' :G-+HomC(VI,VI), a I E H omC (V I ' V I ) , and

AI ,A" E C.

a = AI a

l

6" :G-+HomC(VI,V") a II E Hom C ('vTIl ,V")

Form V = VI EB V" and consider

E9 Alia" E: HomC(V, V).

Then we have

This completes the proof of 3. 39; any function f (x) can be uniformly approximated by a function v(x) as in 3.40, which in turn can be uniformly approximated by a linear combination of eigenfunctions by 3.41; a nd this ca n be written in the required form Tr(a9 (g-l)) by 3.42 -3.44.

3.45

REMARK.

If a : V

->

V is a G-map, then Tr(a9 (g)) is a

class function.

Proof.

Tr(a9(xyx- 1

» = Tr(a(9x)(9y)(9x=

1

))

Tr( (9 x-1)a (9 x)(9 y))

(3. 30 (ii))

58

LECTURES ON LIE GROUPS Tr(a9 (y))

=

since a is a G-map. There is a converse to this remark.

3.46

PROPOSITION.

Let G be a compact group.

every class function f : G

--t

Then

C can be uniformly approximated

by functions of the form Tr(j39 (g)), where 9 runs over representations 9 : G

->

Hom

C

(V, V) and j3 runs over Hom

CG

(V, V),

that is, j3 runs over G-maps .

Proof.

By 3.39 we can find 9: G

--t

HomC(V,V) and

a E Hom C (V ,V) s u c h t ha t \ f(x) - Tr( a 9 (x)) \ ~ (. If f is a clas s function we can substitute y-l xy for x and get

\ f (x) - Tr( a 9 (y-l xy)) \ ~ ( . Arguing as in 3.45, this gives \f(x) - Tr(9Y)a(9y-l)(9x)) \ ~ (. Integrate over y; we find \f(x) - Tr(j3(9x)) \ $. ( where j3 ::

S

(9 y) a (9 y-l ).

yEG But as in 3. 18, j3 is a G -rna p .

ELEMENTARY REPRESENTATION THEORY

3.47

THEOREM.

59

Let G be a compact topological group.

Then every class function f : G - C can be uniformly approximated by a linear combination

~

A.X. of irreducible complex

ill

characters.

Proof. 3.25.

~

Let 9 and

be as in 3.46, and let V. and 1

~

be as in

Then the G-map ~ : V - V induces (say) ~i : Hom

CG

(Vi' V) - Hom

CG

(Vi' V).

We have the following commutative diagram.

Therefore Tr(~9 (g)) =

r

(Tr~i) . Tr(9 g),

i

which has the required form I: A.X. (g). ill It is natural to ask for the analogue of 3.47 over R or

Q.

By 3.6 we have tcV ~ cV, tc' V ~ c'V; so by 3. 32 (vi) the

character of a representation over R or Q is rea 1, and by 3.33 (i) it satisfies X(g-l) = X(g) .

3.48

COROLLARY.

Every class function f : G - R such that

f (g) = f (g-l) ca n be uniformly approximated by an R-linear

60

LECTURES ON LIE GROUPS

combination of characters of representations over R, or by an R-linear combination of characters of representations over Qo

Pro 0 f f (g)

Let f : G - R be a class function such that

0

= f (g-l) • 1 £(g)

Since f (g)

By 3047 we can find complex A. such tha t 1

- I;

A.x· (g)

<

1

ill

= f (g-l)

(0

we ha ve

If(g) - I;A.X.(g-l)1 < (, ill

or using 3 033 (i) \f(g) - f\Xi(g) \ <

(0

Since f is real we also have

I f(g)

- I;~.X. (g) 1 < ( ill

If(g) - I;~.X.(g) 1 <

(0

ill

Therefore 1f(g) But here

1 -4(A.

1

+

~-41 (A. + ~.)(X. (g) ill 1

~.)1

+

x.1 (g» 1

<

(0

is a real coefficient and X. + 1

X.

1

is the

character of the real representation rV. or of the quaternionic 1

representation qV. (see 3 6) 1

3 49 0

COROLIARY.

0

0

Every class function f : G - C such that

f (g) = f(g-l) can be uniformly approximated by a C-linear combination of characters of representations over R or by a

ELEMENTARY REPRESENTATION THEORY

61

C-linear combination of representations over Q.

Proof.

Approximate the real and imaginary parts of f by

3.48. We now consider in greater detail which complex representations are real or quaternionic.

3.50

THEOREM.

A representation V over C is real if and

only if there exists a non-singular symmetric bilinear form B: V ® V

->

C which is invariant under G.

A representation V over C is quaternionic if and only if there exists a non-singular skew-symmetric bilinear form B : V ® V - C which is invariant under G.

Pro 0 f. that j 2

First suppose that V carries a structure map j such

=( =±

1.

By 3. 16 we can impose on V a positive

definite Hermitian form H which is invariant under G and satisfies H(jv,jw)

= H(v,w).

Define B(v,w) = H(jv,w). Then B is clearly bilinear, non-singular and invariant under G. Also we have

62

LECTURES ON LIE GROUPS

B (w ,v)

= H (jw ,v) H (v ,jw)

=

= H (jv ,j 2 W) =

(H(jv,w)

=

(B(v,w).

So B is symmetric or antis ymmetric according to the sign of (. We now seek to reverse this argument.

Suppose given

on V a non-singular bilinear form B : V ® V ..... C which is invariant under G and satisfies B(w ,v) where

(=

±1.

=

(B(v, w),

By 3.16, we can also suppose that V carries a

positive definite Hermitian form H which is invariant under G. Then we can define f : V ..... V by B (v , w)

=

H (f v , w) .

The map f is conjugate-linear, a G-map and a 1-1 correspondence.

The property of B gives H(fv,w) = B(v,w)

= (B(w, v) = (H(fw, v) = (H(v ,fw).

Thus

63

ELEMENTARY REPRESENTATION THEORY 3.51

H(fv,w)

= (H(v,fw).

We now define another positive-definite Hermitian form on V by

3.52

K(v,w)

= H(fv,fw).

The form K is invariant under G.

Substituting fv and fw into

3.51, we get

and taking complex conjugates, we get

3.53

K(fv,w)

= (K(v,fw).

The space V now splits as the direct sum of eigenspaces V. for 1

the pair of forms H, K.

The eigenva lues are pos iti ve rea 1

numbers A.; for each such A., V. is the set of v. such that 1

3.54

1

K(v. ,w) = A.H(v. ,w) 1

1

1

1

1

for all w E V.

The eigenspaces V. are stable under G. 1

pres erved by f; for we ha ve K (f v. , w) 1

= (K (v.1 , f w) = (A1 H(v.,fw) 1 = A.H(fv.,w) 1 1

Thus fv

1

E V.. 1

We also have

(3 . 5 3)

(3.54) (3.51).

I claim they are also

64

LECTURES ON LIE GROUPS H{f 2 v. ,w) 1

= (H(fv.1 ,fw)

(3.51)

= (K(v1 ,w)

(3. 52)

=

So f2\ v.

1

=

(A. H (v. ,W) 1

1

(3.54).

(A., where A. is real and positive. 1

1

Let us now define a map j : V - V by

Then j is conjugate-linear, a G-map and satisfies j2 = (. Thus V is real or quaternionic according to the sign of (. This completes the proof. To sum up, the advantage of structure maps is that they come normalised by the condition j2 = ±1; the disadvantage of bilinear maps is that they can be denormalised by a scalar factor for each summand of V.

3.55

DEFINITION.

We say that a representation V of G is

self-conjugate if tV = V.

Evidently representations over Rand

Q are self-conjugate (either using 3.6 or using the fact that the structure map j gives an isomorphism from tV to V) .

3.56

PROPOSITION.

If a complex irreducible representa tion

V of G is self-conjugate, then it is either real or quaternionic, but not both.

ELEMENTARY REPRESENTATION THEORY Pro 0 f.

65

Consider V* ® V*, the space of bilinear maps from

V ® V to C.

It has an automorphism

r defined by

r(v* ®w*) =w* ® v*. We have

r2 = 1. So v*

® V* splits as the direct sum of the +1

and -1 eigenspaces of r.

The +1 eigenspace is the space S*

of symmetric bilinear maps: the -1 eigenspace is the space A* of antisymmetric bilinear maps. Now we also ha ve V* ® V* ~ HomC(V, V*). By 3. 17 we have V* ~ tV, and if V is self-conjugate we have V*

~

V.

If V is irreducible then so is V* I and we have

dim CHom CG (V I V*) = I by 3.23.

That is, for the elements invariant under G we ha ve

dim S* +dim_A* =1 C G GG· Moreover, a non-zero bIlinear map B which is invariant under G corresponds to a non-zero G-map V - V*, iso: so such a B is non-singular.

which must be

We conclude that only two

cases are possible. (i)

dimCS

G

= I, dim~G =

o.

In this case V admits a

non-singular symmetric bilinear form invariant under G, but not an antisymmetric one.

66

LECTURES ON LIE GROUPS

(ii)

dim S* C G

=0

, dim A* C G

= I . In this case V admits a

non-singular anti symmetric bilinear form invariant under G, but not a symnetric one. The result follows by 3.50.

3.57

THEOREM.

Suppos e given a compact group G.

it is possible to choose representations U and W

p

(i)

m

, rV and rc' W .

n

p

The inequivalent irreducible representations over Care

m

n

n

and c' W . p

The inequivalent irreducible representations over Q are

precisely the qcU

Proof.

n

The inequivalent irreducible representations over R Lre

precisely the cU , V , tV (iii)

over R, V over C

over Q to satisfy the following conditions.

precisely the U (ii)

m

Then

m

, qV

n

and W • p

We begin by taking the irreducible complex repre-

sentations V.

First, we can classify them into those such that

tV ~ V and those such that tV

'1 V.

The latter occur in pairs

(V,tV),and we choose one V out of each pair. n

The former are

either real or quaternionic by 3.56; we choose U over Q so that the cU

m

and c'W

this choice makes 3.57 (ii) true.

p

give such V.

m

over R, W

p

It is clear that

67

ELEMENTARY REPRESENTATION THEORY It is also claimed that the representations U

rc'W over R are irreducible, and similarly over Q. p

U

m

is irreducible because cU

m

is so.

m

,rV

n

and

In fact,

We have

crV ~ (1 + t)V n n crc'W ~ 2c'W ,

p

p

and neither can be split into real representations because V

n

and tV

n

are not self-conjugate and c'W

p

is not real. Similarly

over Q. It remains only to prove that there can be no further

irreducible representations over R or Q. For this purpose we introduce:

3.58

LEMMA.

If V and Ware inequivalent irreducible rep-

res entations over R, then no complex irreducible representation can occur as a summand both in cV and in cW. Similarly for c'V and c'W if V and Ware over Q.

Pr 00 f.

dimCHom

CG

(cV ,CW)

= dimRHom RG (V, W) =0

(3. 10)

(3.23).

To complete the proof of 3.57, it remains only to remark that all the complex irreducible representations occur as summands in

68

LECT URES 0 N LIE GRO UPS cU

m

crV = (1 + t)V n

crc'W

p

n

= 2c'W • p

Therefore there can be no more irreducible representations over R.

Similarly over Q. There is a classical criterion for deciding whether a

complex irreducible representation is real or quaternionic. For this purpos e we introduce the following cons iderations •

3.59

DEFINITIO N .

n

Let V

=V

n

(l)V ® ... ®V (n factors). Let X V

n be the summand of V on which the permutation group I; acts by n

pw

=

((p)w,

where (P is the sign of p; that is, XnV is the space of antisymmetric or alternating tensors.

The G-space XnV is called

the nth exterior power of V. Consider the power-sum

in

m 2:. k variables. This can be written as a polynomial

Pk (01

,02 , •••

'Ok) In the elementary symmetric functions

of

,x 2

,x ; the polynomial is actually independent of m,

Xl

, •••

m

a nd the formula is valid even for m < k.

01

ELEMENTARY REPRESENTATION THEORY

3.60

DEFINITION.

69

If V is a complex representation of G,

we define a virtual representation by

The polynomial is evaluated in the ring KC (G) •

3.61

k If W = lJ; V, then

LEMMA.

k XW(g) = XV(g ). k Of course it is clear that Xv(g ) is a class function;

after 3.47 it is natural to as k what it is the character of.

Proof.

Impose on V a positive-definite Hermitian form H.

Then 9 (g) is a unitary map, and we can find in V a V n base of eigenvectors v. with eigenvalues A.. Then V admits

Fix g.

1

1

a base of eigenvectors v.

® v.

11

A.

11

A

12

® ••• ® v.

12

rv .

• •• A. , and similarly for A In

with eigenvalues

In

Hence 9 a cts on AnV

with trace a , the nth elementary symnetric function of the A.• n

1

Hence

k

k

-= Al + A2 + •••

k

= Tr«9 g) )

v

k

= XV(g ).

(by definition of Pk)

LECTURES ON LIE GROUPS

70

3.6

THEOREM.

Let V be a complex irreducible represen-

tation of a compact group G.

Then

1 if V is rea l

o if

V is not self-conjugate

1 -1 if

V is quaternionic

In 3.56, instead of considering V* ® V* ~ S* E9 A*,

Proof.

it is equivalent to cons ider V ® V ~ SeA. We have

and so

= S - A. Therefore

S

XV(g2) =

gEG

S

gEG

Xs (g) - XA (g)

= dimCS

G - dimCAG·

This gives the res ult.

3.63

REMARK.

If V is real then AnV is real.

If V is quater-

nionic then AnV is real for n even, quaternionic for n odd.

Proof. (. Then V

Suppose V admits a structure map j whose square is

n

admits a structure map j ® j ® ••• ® j whose square

is (n, and similarly for Ary.

71

ELEMENTARY REPRESENTATION THEORY

3.64

REMARK.

If V is real then lJ.;kV is real. If V is quater-

nionic then lJ,lkV is real for keven, quaternionic for k odd.

Proof.

If we assign weight ito ui' then Pk(Ul

is a polynomial of weight k.

,U2,· ••

'Uk)

Now use 3. 7 •

We now move on to calculate KC (G x H) in terms of KC(G) and KC(H).

Let V be a G-space and W an H-space (over

C). Then we can form V ® W, and make it a G x H-space by (g , h)(v ® w) = gv ® hw. This defines a homomorphism of rings

3.65

THEOREM.

The map

l.I

is an isomorphism.

More pre-

cis ely , the inequivalent irreducible G x H-spaces (over C) are precisely the products V.

1

(&I

W., where V. runs over the inJ

1

equivalent irreducible G-spaces and W. over the inequivalent J

irreducible H-spaces.

Given the theorem for KC (G x H), it is easy to locate the representations of G x Hover Rand Q; for a n irreducible representation V.

1

(&I

W. is self-conjugate if and only if both V. J

1

and W. are self-conjugate; and then V ® W. is real or quater1

J

J

nionic according to the nature of V. and W. 1

J

I

as in 3.7.

72

LECTURES ON LIE GROUPS Theorem 3.65 will follow immediately from the next

two results.

3.66

LEMMA.

If V is an irreducible G-space and W is an

irreducible H-space (over C), then V ® W is an irreducible G x H-space.

3.67

LEMMA.

in the form

~

i ,j

Any G x H-space U (over C) can be expressed

n .. V. ® W.. 1J 1

J

In particular, the irreducible

G x H-spaces have the form V. ® W .• 1

Fir s t proof

0

f 3. 66 •

J

We ha ve

Thus

S(g , h) EGxH XV®W(g ,h) XV<&IW(g ,h) =

SgEGXV(g)XV(g) ShEH Xv5h )Xw (h)

= 1. By 3.20 and 3.34, V ® W is irreducible.

Pro 0 f

0

f 3.67.

~:

By 3.25 we have an isomorphism over H:

{BHomH(w.,U) ® W. ~U. J

j

Let G act on Hom

H

(VVj , U)

J

by

73

ELEMENTARY REPRESENTATION THEORY

= g(kw)

(gk)w

for

k E HomH(W , U). j

(It is easy to check that gk is an H-map.) Then ~ is a

G x H-map.

But by 3.20 we have an isomorphism of G-

mocules HomH(w.,u) ~ffi n.jV .• i

J

1

1

Thus

u

~ Gl n .. V. ®W .•

i, j

1J

J

1

Finally, if U is irreducible, it is clear that the sum can contain at most one factor.

Second proof of 3.66.

Suppose that Vand W arc irredu-

cible and V ® W has a G x H-subspace S. Then by 3.67 we have S = I; n .. V. ® W .• i, j 1J 1 J

As H-spaces we have V ® W ~ (dim V)W; so the only W. which can have n .. -lOis W.

D

J

only V. which can have n .. -lOis V. ~

1

of (dim V)(dim W), and S

=0

or S

Similarly, the

Hence dim S is a multiple

=V®

W.

We now move on to cons ider the case of a double covering 1T :

G

groups and Ker

->

G.

That is,

11 = 22 =

11

is an epimorph1s m of topological

(1, z}, sa y.

Ker

11

is, of course,

74

LECTURES ON ,LIE GROUPS

normal in

3.68 G

~

~

G,

and even central since Aut 22

THEOREM. X

~

G

1.

A character X : G - C factors as ~

C

for a character X if and only if X factors as a Moreover, X is real if and only if

map of sets.

similarly, X is quaternionic if a nd only if

Pro 0 f.

=

"Only if" is trivial.

sentation of G.

Then z acts on

Xis

X is

real;

quaternionic.

So suppose that V is a repre-

V and

satisfies

Z2

= 1.

So

V

splits as the sum of the +1 and -1 eigenspaces of z, say V = V E9 V • Since z is central, both V and V-are G-spaces. We have

9 (zg)

=

9 (g) ffi (-9 (g))

and taking traces,

--X(zg) = X(g) - X-(g)· If X : G - C fa ctors as a map of sets, then

so X-(g) of G.

=0

and V-

= o.

Clearly V = V is then a representation

If it carries a structure map commuting with the opera-

tions of G, then it carries the same structure map commuting with the operations of G.

3.69

REMARK.

This [heorem is also valid for virtual

75

ELEMENTARY REPRESENTATION THEORY chara cters •

3.70

EXERCISE.

Extend 3.68 to any finite covering, as-

suming G compact connected and A = C. We now turn to consider the representations of the torus.

3. 71

PROPOSITION.

If G is abelia nand

A = C

then every

irreducible G-space V is one-dimensional.

Proo f.

For each 9 E G consider

map because G is abelia n. by some scalar X(g).

e (g)

: V-V. This is a G-

e (g)

By 3.22 (ii),

is multiplication

So every subspace of V is stable under

G and dim V = 1.

3.72

REMARK.

In 3. 71, X(g) E C - {O}.

3.73

REMARK.

Suppose that G is a compact abelia n group

and V an irreducible G-spacc, so that

e may be written as

X : G -- C - {O}. Then X(G) c Sl c. C - {O}, where Sl is the unit circle in C.

First proof. and if

If

I X(g) I = r <

I >..(g) I = r >

I, then

I, then

I X(g n ) I = r n

I X(gn) I -+

O.

76

LECTURES ON LIE GROUPS

Sec 0 n d pro 0 f.

Give V a positive definite Hermitian form

H invariant under G.

Then

H{v,v) = H{gv,gv) = IX{g)1 2 H(v,v), so

I X{g) I =

1.

We recall that Tl was defined to be R/Z.

3.74

PROPOSITION.

A homomorphism a : Tl - Tl has the

form a (x) = nx mod 1 for some integer n.

By 2 . II and 2 . 13, or by the ordinary theory of

Proof.

covering spaces, a lifts to a homomorphism f3 : R - R. j3(l)

==

0 mod 1, so j3{l) = n E Z; and j3{a) = na for a E Z, and

bf3{a/b) = f3{a) 13 (x)

Then

= nx

3.75

= na

for b E Z, so j3{a/b) = na/b.

for a 11 x E R, and a (x)

= nx

By continuity,

mod 1.

A homomorphism a : Tk ..... Tl has the

COROLLARY.

form a (Xl, x 2 for some n 1 ,n 2

3.76

,x ) = n 1 Xl + •.• + nkx mod I k k

, •••

, •••

,n

COROLlARY.

k

E Z.

The irreducible complex Tk -spaces have

the form X{x 1 ,X 2

, •••

I

x ) k

=

Exp 21T i(n 1 Xl + ••• + nkx ) , k

ELEMENTARY REPRESENTATION THEORY

77

where Exp z = e Z •

This follows from 3.71, 3.73 and 3.75, since Tl is isomorphic to Sl under x -- Exp 21Tix. For 1 < j < k, let p. be the Tk -space given by J

Then P j is invert ible, a nd for a ny integers n 1 ,n 2 (positive, negative or zero)

n Pl,l

n2

P2

•••

, •••

,n

k nk . k P IS the T -space k

given by

X(XI ,x 2

3.77

, •••

COROLIARY.

,x ) = Exp 21Ti(n 1 x 1 + ••• + nkx ). n k KC (Tk) is the ring of finite Laurent

series in Pl ,P2, ••• 'P ' and so has no divisors of zero. k We have

t (p

n 1

1

n P 2 2

Thus the only irreducible representation of Tk which is selfconjugate is the trivial representation 1.

3. 78

COROLIARY.

The inequivalent irreducible rea I repre-

s entation of Tk are (i)

the trivial representation 1 of dimension 1, and

(ii)

the representations

LECTURES ON LIE GROUPS

78

r

\1 P2 ...

(cn1 n 2

for (n1 ,n2

, •••

,n ) k

-I

nk)_

~

r. 2 ... -n1 -n 2

-r ~1

P

-nk )

Pk

(0,0, •.. ,0), which are of dimension 2 .

This follows from the above by the discus s ion of 3.57.

Chapter 4

MAXIMAL TORI IN LIE GROUPS

Not ice.

From 4.5 onwards, G will be a compact connected

Lie group.

4.1 g

DEFINITION.

E G.

Let G be a topological group and let

Let H be the subgroup generated by g.

Then g is a

generator of G if cl H = G, where cl denotes the closure. G is monogenic (or monothetic) if it has a generator.

4.2

EXERCISE.

4.3

PROPOSITION.

Monogenic implies Abelian.

The torus Tk is monogenic.

Indeed,

generators are dense in Tk .

Proof.

Let Ul k

sets of T .

,

U2 k

, •••

k

Let T = R /Z

be a countable base for the open k

.

have co-ordInates (Xl' ••• ,X ). k 79

LECTURES ON LIE GROUPS

80 Then a cube is a set (x E Tk; ~,

point

and real ( >

o.

Ix. 1

~.I1 ~ (}

for some fixed

Let Co be any cube. Then we will

define a descending sequence of subcubes whose intersection will be a generator. Suppose, inductively, that we have defined Co

~

C1

::l ••• ::l

C m- 1 and that C m- 1 has side 2(.

Then there

is an integer N (m) such that N . 2 ( > 1, so that the image of C m- 1 under multiplication by N is Tk. such that N . C Let g E

m

We ca n find C m C C m- 1

cU.

m

n C . Then g N{m) c U ,so g is a generator m m

m

k

of T .

4.4

PROPOSITION.

Let G be an Abelian topological group,

k k with T c G such that G/T = Z . Then G is monogenic.

m

Proof.

k Let t be a generator of T .

j ect to a generator of Z

m

.

Choose u E G to prok

IS divisible, so there is s E. Tk with ms = t - mu. g

=u

+ s. Then mg

= m (u

k

Then mu ETa nd t - mu E T . T

+ s)

= t

I

Take

so the powers of g are

dense in T. Translating by rg, the powers of g arc dense in the coset of T containing ru.

This gIves all cosets.

k

MAXIMAL TORI IN LI£ GROUPS

4.5

NOTICE.

81

From now on G is a compact connected Lie

group.

4.6

DEFINITION.

(i)

a subgroup which is a torus, such that

(ii)

if T

4.7

REMARK.

C

U

C

A maximal torus T

C

G is:

G and U is a torus then T = U .

If G is not compact

I

it need not have any

non-trivial tori.

4. B

PROPOSITION.

Any subtorus of G is contained in a

maxima I torus.

Pro 0 f.

Consider a strictly increasing sequence of subtori

Tl cT 2 c .•• eG.

Then L(Td c L(T 2 )c ... cL(G) is a

strictly increasing sequence, and so is finite.

4 •9

PROPOSITION.

Let T be a max ima I torus of G

a connected Abelia n subgroup of G wIth TeA.

Pro

0

f.

TeA c cl A.

a nd A

Then T = A.

But cl A is a closed connected AbelIan

subgroup, and is therefore a torus (2.20). T = A.

I

Thus T = cl A and

LECTURES ON LIE GROUPS

82

4.10 G

e

CONSTRUCTIONS.

by T C G

ric form on G G

e

Ad

~

e

Aut G . e

If T is a torus of G, it operates on Choose a positive definite symmet-

invariant under G

and so under T.

I

Then (3.78)

splits into orthogonal irreducible T-spaces of dimensions 1

and 2.

Those of dimension 1 are trivial.

orthonormal base in those of dimension 2

We can choose an I

and represent T by

T - SO(2).

4.11

DEFINITION.

The integer lattice of L(T) is exp-l (e)

where exp : L(T) - T.

4.12

PROPOSITION.

form Va

e

L(G) = G

e

splits as a T-space in the

~mVi' where T acts on Va trivially dim V.1 = 2 for I

i > 0 and T acts on V. as 1

COS [

Here

211e (t) i

sin 211

e.1 : T --

e.1 (t)

R/Z is given by a linear form

e.I :

integer values on the integer lattice, and no

4. 13

DEFINITION.

e.1

L(T) --- R taking is zero.

If T is a maximal torus, the functions

±BI are called the -roots of G. By 3.24 they are well defIned ---in terms of T.

We will sec that they are Independent of T.

MAXIMAL TORI IN LIE GROUPS

4.14

PROPOSITION.

Proo f. (i) L(T) (ii)

83

T is maximal if and only if Vo = L(T).

It is clear that L(T)

C

Vo.

Suppose Vo = L(T) and T cT'. C

L(T')

C VOl

c

Suppose Vo

Then

, so L(T) = L(T') and T = T·.

Vo

-I L(T).

Then there is X E Vo , X /. L(T).

Now exp (tX) , for t E R is a I-parameter subgroup H of G on I

which T acts trivially, and which is not conta ined in T • Therefore the subgroup generated by T and H is a connected Abelian subgroup strictly containing T, so T is not maximal.

4 • 15

COROLLARY.

4.16

EXAMPLE.

dim G - dim T is even.

Let G = U(n), and let T be the set of

diagonal matrices: D =

exp 211 iXl

. exp 211ix

n

LU(n) can be decomposed into the following summands. (i)

Matrices

with d. real. J

LECTURES ON LIE GROUPS

84

This is L(T). Ma tric es

(ii)

r

M

rs

5

= r

-

-~

5

for r < s. DM

rs

Then D-1 =

w

-w

where w

= exp[21T i(x r

- x ) Jz and 5

ers

= x

r

- x . 5

The matrices (i) and (ii) generate L(G) , so Va = L(T) and T is maximal.

The roots are (x - x ).

4.17

Let G

r

EXAMPLE.

= S U (n).

the previous example are in LSU (n), respect to t of II + tM

rs

I at t

5

Then the matrices M 5

rs

of

ince the derivative with

= 0 is zero.

Similarly, matrices

of type (i) with D:l. = 0 are in LSU (n) • 1

Let T be the set of diagonal matrices

MAXIMAL TORI IN LIE GRO UPS

D

=

85

exp 21Tixl

. exp 211 ix with l:x.

=

1

Va

=

4.18

o.

n

The functlons (x - x ) are still nontrivial, so r

s

L(T), T is maximal, and the roots are (x - x ). r s

EXAMPLE.

Let G

= Sp(n) ,

and let T be the set of

diagonal matrices D

=

exp 211ixl

..

exp 211 ix

n

L Sp(n) splits into the following summands. (i)

Matrices

id 1

with d. real. 1

id (ii)

n r

Matrices M

rs

z

r

s

with z E C.

s

=

-z

86

(iii)

LECTURES ON LIE GROUPS

Matrices r

Nr = zj

r

with z E C.

Here

DN D= r exp(21Tix )zj exp(-21T ix ) r r

= exp(41Tix )zj r

(iv)

Matrices r

s

P rs = r

s

with z E C.

Here

zj

zj

87

MAXIMAL TORI IN LIE GROUPS DP

rs

D-1 =

exp 211 i (x + x ) zj r s ex p 211 i (x + x ) zj r s

Thus Va = L(T) , T is a maximal torus, and the roots are ±2x

r

4.19

I

(x - X ) and r s

EXAMPLE.

:t:

(x + X ) for r r s

-I s. We have U(n) c SO(n).

Let G = SO(2n).

Take T to be the image of the maximal torus we had in U (n) • That is, T is the set of matrices

D

where D. = 1

[COS

211X.

sin

211X.

n

-sin 211X i ] cos 211X.

1

1

1

•

Now LSO(2n) splits into the following summands. (i)

L(T)

I

cons ist ing of matrices

h "'o d

n

-dn

0

88

(ii)

LECTURES ON LIE GROUPS

The rest of LU (n)

I

cons is ting of matrices s

r M

:::::

rs W

r _WT

s

where

W

Then DM

rs

[

D- 1 = M I with rs

XI]

-SIn

[COS 211 (x r - x s)

YI

So T acts with (iii)

=

ers

S

in 2 11 (x - x )

=

x

r

r

21T(X r

-x s)] [X ]

cos 21T(X -x )

s

r

s

- x . s

Let

s E s

..

o s

o

-1

..

and take matrIces E M E- l s rs s

•

In this caS8, T acts with

y

.

89

MAXIMAL TORI IN LIE GROUP8

ers

=

x + x . r s Thus Va

are (x r

4.20

X

s

)

I

= L(T) , T is a maxImal torus and the roots I

± (x +

r

EXAMPLE.

X

s

)

Let G

t-

for r

=

s.

80(2n + 1).

We have

80(2n) c 80(2n + 1) by letting 80(2n) act on the first 2n coordinates.

Let T be the maximal torus we had in 80 (2 n) •

Then L80(2n + 1) splits into the following summands. (i)

ISO (2 n) •

(ii)

Matrices r

F

r

r

Here D acts by rotatIon through x . r

Thus Va = L(T) ±Xr

I

4.21

(x r

X

s

)

I

T IS a maximal torus

and ± (x + x ) for r r s

THEOREM.

t-

I

and the roots are

s.

Let T c:: G be a maxImal torus.

g EGIS contained In a conjugate of T.

Then any

90

LECTURES ON LIE GROUPS

Proof.

(Following A. Wei! [21J; see also [IIJ.) Consider the left coset space G/T, and let f : G/T - G/T

be induced by left multiplication by g, that is, f(xT) = gxT. Then a fixed point of f is a coset xT with gxT = xT, that is, g E xTx-1

•

So we only need to show that f has a fixed point.

We will use the form of Lefschetz' s fixed point theorem given byA. Dold [61.

(This theorem applies to manifolds rather

than simplicial complexes). We summarise what we need: Let f : X -X be a continuous map, and define J\(f) EZ by taking f*:Hq(X;Q)-Hq(X;Q)andsetting l\(f)=I:(-I)q Tr f*. q

Thenl\(f) depends onlyon the homotopy class of f. If f has no fixed points, then I\(f) =

o.

If f ha s only isolated fixed points

(and so a finite number of fixed points), then I\(f) is the number of fixed points counted with multiplicity, which is defined as follows.

Let X be a smooth manifold and x a fixed point of f.

Consider 1 - f

I

:

X - X • If det(l - fl) > 0, then f has multi-

x

x

pliclty + 1 at x: if det (l - f I) < 0, then the multiplicity is -1. We do not need to discuss the case det(l - fl) = O. To compute I\(f) we may replace f with any homotopic map f o • So we may replace g with any other go E G, since G is path-connected. Take go to be a generator of T (4. 1), and

91

MAXIMAL TORI IN LIE GROUPS let fo be the corresponding map.

Then the fixed points of to

are the cosets nT for n in N(T), the normaliser of T in G (as the rea de=- will easily verify).

Let us examine N(T).

N(T) is a closed subgroup of G, and so is a Lie group (2.27, 2.26), and the identity component N(Th is open and so has only a finite number of cosets. see as follows.

Now N(Th = T, which we

N(T) acts on T by conjugation (i. e. ,

n(t) = ntn-1 ) and Aut T is discrete, so N(Th acts trivially. (The reader should verify that N .... Aut T is continuous with this topology on Aut T.

Note that this map arises from the map

NxT .... T which is a restriction of the map GxG -- G given by (g ,h) .... ghg- 1 ). If N(Th properly contains T it contains a 1parameter subgroup not conta ined in T but computing with T, contradicting the maximality of T.

It follows that N(Th = T,

that T has only a finite number of cosets in N(T), and that fo has only a finite number of fixed points. It suffices to consider just one of these fixed points,

say T, as follows. r

n

Let nT be another fixed point.

Define

: G/T .... G/T by r (gT) = gTn. This is a well-defined diffeon

morphis m, commutes with f o , and takes T to hT. multiplicity at nT is the same as at T •

Thus the

92

L[CTURES ON LIE GROUPS Observe that fa can also be defined as fa (xT)

= goxg~l T.

That is, fa is obtained as a quotient of the map G -- G given by X

-0

goxg~l.

This has the merit that e goes to e.

To obtain a basis of (G/T)T' take a basis for Te' extend It to a ba sis of G , and discard the vectors of T • Then (4.12, e e 4. 14) 1 - f~ has the form sin 217 61 (go)

o

1 - cos 217 61 (9 0 )

o

Therefore det (1 - f~)

m

= III

1 - cos 2178 1 (go)

sin 217 91 (go)

-sin 217 8 1 (go)

I-cos 2178 1 (go)

which is greater than 0 unless cos 2178 (go) = 1 for some r. r

But 9 (go) r

(4. 12).

1

0 mod 1, since 8 is a nontrivial function on T r

Hence the multiplIcity is + 1, and I\(f) =

I N (T)/T I

> O.

Thus f has at least one fixed point, and the theorem is proved.

4.22

COROLLARY.

Every element of G lies in a maximal

torus, since the conjugate of a maximal torus is a maxImal torus.

4.23

COROLlARY.

Any two maximal tori, T, U are conjugate.

MAXIMAL TORI IN LIE GROUPS Proof.

93

Let u be a generator of U.

x C G, and thus U c xTx- 1

•

Then u E xTx-1 for some

But U is a maxima I torus, so

U = xTx-1 • Hence any construction apparently dependent on a choice of T is independent of the choice up to an inner a utomorphism of G.

4.24

DEFINITION.

It follows that any two maximal tori

have the same dimension. This IS called the rank of G, and written k or 1.

4.25

PROPOSITION.

Let S be a connected Abelian subgroup

of G, a nd let g E G commute with all elements of S.

Then

there is a torus T conta ining g and S.

Proof.

Let H be the subgroup generated by g and S.

Abella n, so CI H is a compact Abella n Lie group. identIty component (CI H)l is a torus.

H is

Therefore the

CI H/(CI Hh is finite

and generated by g, so CI I~/(CI H)l ~ Z

m

for some integer m.

By 4.4, CI H has a generator h which lies in some maximal torus T.

Then g

~

S

C

H c CI H

= T•

LECTURES ON LIE GROUPS

94

4.26

PROPOSITION.

Let T be a maximal torus of G.

If

TeA c G where A is Abelian, then T = A. That is, a maximal torus is a maximal Abelian subgroup.

Pro 0 f.

Let g EA.

g and T.

But T is maximal so U = T, and gET.

4.27

EXAMPLE.

Then (4.25) there is a torus U containing Thus AcT.

If a E U (n) commutes with all diagonal

matrices it is itself diagonal.

4.28

REMARK.

It is not, in general, true that a maximal

Abelian subgroup is a torus. For example, let G = SO(n) and cons ider the set of matrices of the form ±l

±l

These form a maximal Abelian subgroup.

4.29

DEFINITION.

Let T be a maximal torus of G. Then

the Weyl group W (or C:» of G is the group of automorphisms of T which are the restrictions of inner a utomorphis ms of G. This is independent of the choice of T. Any such automorphism has the form t - ntn-1 , n E N(T).

95

MAXIMAL TORI IN LIE GROUPS

N(T) is a closed subgroup of G, and so compact. Let 2(T) be the centraliser of T, that is, the set of z E G such that ztz-1 = t all t E T.

2(T) is also closed, and TCZ(T) C N(T).

Thus N(T) maps onto N(T)/2(T) ~ W.

N(T)/T is finite (see the

proof of 4.21), so W is finite. Since we are considering G connected, Z(T)

=T

(4.25),

and W = N(T)/T.

4.30

COROLLARY of 4.21. Let V be a G-space. Then Xv

is determined by its restriction to T and is invariant under W.

4.31

COROLIARY.

The homomorphism i* : K(G)

-+

K(T) of

(complex) representation rings is mono, and its image is contained in the subring of elements invariant under W.

4.32

PROPOSITION.

Restriction gives a one-one correspond-

ence between class functions on G and continuous functions on T invariant under W.

P fO

0

f.

We nave already shown that the correspondence is

mono. Suppose given f : T W.

Extend f to

f: G

-0

-0

Y by

Y continuous and invariant under

f (xtx-1 )

= f(t).

To show that

f is

96

LECTURES ON LIE GROUPS

well-defined we need:

LEMlvtA.

4.33

If t1 ,t 2 E T are conjugate in G, then there

is w E W with t2 = wt1 •

T c 2(t 2 ) and, since T c 2(t 1 ), gTg-1 c 2(t 2 ) also.

H is a

closed subgroup of G, and so a Lie group, and so T, gTg-1 are maximal tori of H.

Therefore there is h E H1 such that

T = hgTg-1 h-1 , where Hl is the identity component of H.

But

h E 2(t 2 ) so hgt 1 gh-1 = t 2 . Thus conjugation by hg, which is in W, send s t 1 tot 2 •

Completion of 4.32.

It remains to check that

f

is con-

tinuous. Well, suppose that a sequence 9

f 9 co X

n

-t

• Let 9 X

co

,t

n

n

n

-- 9

ro

f

is not continuous.

Then there is

f 9 n tends to

such that no subsequence of

= x t x-1 and take a subsequence with

-t

nnn

t

for some x 00

a nd so x roro t x-ro1 == 9 ro • Then

ro

,t

ro

f (g n )

which contradicts our hypothesis.

.

Then 9

n

-+

= f (t ) . . . f (t ) n 00

x

t X -1 , rooooo

= f (g ), 00

Thus 4.32 is proved.

97

MAXIMAL TORI IN LIE GROUPS

4.34

LEMMA.

Let N(gh be the identity component of the

normaliser of some 9 E G.

Then N(gh is the union of the

maximal tori of G containing g.

Proof. n E N(gh.

Clearly N(gh contains all such tori.

So let

Then n lies in a maximal torus S of N(gh.

S

commutes with g, so (4.25) there is a maximal torus T of G containing Sand g.

4.35

COROLlARY.

The following two definitions are equi-

valent: (i)

9 EGis regular if it is contained in just one maximal torus, singular if it is contained in more than one maximal torus, 9 EGis regular if dim N (g) = ra nk G,

(ii)

singular if dim N(g) > rank G.

Pro 0 f •

If 9 lies in just one T, then

dim N(g) = dim N(g)l = dim T. If 9 lies in T 1 and T 2 T1

I

T2 , then LTl

dim N(g) > dim T.

I

LTG and LN(g)

=:;

LT1 + LT2 so

,

and

LECTURES ON LIE GROUPS

98

4.36

EXAMPLE.

Let G = Sp(l), which is the set of quater-

Iq I =

nions q with

Maximal tori are circles cos

1.

for p any pure imaginary quaternion with

e + p sin e,

I p I = 1.

The singular points are ±l, with dim N(±l) = 3. All other points g are regular, and dim N(g) = 1.

PROPOSITION.

4.37

Pr a of. w EW T

Ad

~

W permutes the roots of G.

(The notation was introduced in 1.10.) For each we must consider two representations for T, namely

Aut G

w

~

and T

e

Ad

~

T

show that these are equivalent. and then G

Ax

e T

Ad

!

~

G

e

Ax

e

It will suffice to

But w = A

x

I T for some x

E G,

is the required equivalence, since

---~)

AutG

Aut G. e

T

!

Ad

~AutG

e

is commutative, where the bottom map is induced from A I . X

4.38

DEFINITION.

Let U = (t E T ; r

er (t)

=:;

0 mod I}.

U is r

a closed subgroup of T of dimension k - 1, where k = rank G. It is clearly monogenic.

instance:

It need not be cO!1nected.

For

MAXIMAL TORI IN LIE GROUPS

4.39 Xl

==

In Sp{l) , 61 = 2Xl and U l is gIven by

EXAMPLE. 0 or

4.40

I

2" mod

I.

LEMlvtA.

dim N(t)

=k

Proof.

If t lies in exa cUy II of the U , then r

+ 211.

Let V c L(G) be the subspace on which t acts a s the

identity. Then, by definition, dim V= k + 211. N(t)

e

99

We show that

= V.

(i)

The elements of N(t) commute with t, so t a cts as the

identity on N(t) and so on N(t) • Thus N(t) c V. e e (ii)

Suppose x E V.

Then t acts trivially on x, and so on

the I-parameter subgroup H corresponding to x. Therefore H c N(t) and x E N(t) . Thus V C N(t) . e e

4.41

COROLlARY.

t E T is regular if it lies in no U , and r

singular If it lies in some U • r

4.42

COROLLARY.

of dimension

~n

The singular elements of G form a set

- 3, where n = dim G, in the sense that this

set is the image of a compact manifold of dimension n - 3 under a smooth map.

Proof.

Let u bea generator of U • ThendimN(u)Lk+2, r

100

LECTURES ON LIE GROUPS

and, if z E N{u} , z fixes each power of u and so fixes every element of U • r

Define a map f : G/N{u} x U

r

->

G by f{g, t) = gtg-1

•

Then Imf consists of all points in conjugates of U , f is r

smooth, and dim G/N{u} xU'::; n - (k + 2) + {k - 1} = n - 3. r

All the singular points are obtained with r running over a finite set.

Hence the res ult.

Chapter. 5

GE OM ET RY OF TH E Sn' IEF E L DIAGRA M

(Note:

Not ice.

This is not the DYTllkin-Coxeter diagram.)

Throughout this chapter

G is a compact connected

Lie group, and T is a maximal torws of G.

5.1

DEFINITION.

The infinites imal diagram of G is the

figure in L(T) consisting of the hyperplanes L(U). r

The diagram of G is the fig-ure in L(T) consisting of the hyperplanes given by

er (t)

E Z.

under exp of the singular pOInts of

This is the Inverse image G in T.

102

LECTURES ON LIE GROUPS

5 .2

EXAMPLES of didgrams .

(i)

U(2).

Root

Xl

- X2.

~Xl

The integer lattice is marked with asterisks. (ii)

SO (4).

Root s

Xl

± X2.

~Xl

GEOMETRY OF THE STIEFEL DIAGRAM

(iii)

80(5). Roots

Xl

± X2 , Xl,

X2.

---YXI

(i V)

8p (2).

Roots

Xl

±

X2,

2Xl , 2X2.

- - ; ; . Xl

103

104 (v)

LECTURES ON LIE GROUPS SU(3).

Roots

Xl

(01-1)

! 5 •3

PROPOSITION.

Proof.

firs t 1y, Z (G) Now,

Therefore

If 2

er (2)

==

Z (G) =

C

E Z(G)

I

nu r .

Z (T) = T . 2

acts trivially on G and so on G . e

0 mod 1 for ea ch rand 2 E.

nr u r .

105

GEOMETRY Of THE STIEfEL DIAGRAM Conversely

then g acts trivially on G

5.4

nUr

I

and so trivially on G (2.17).

is given by

. centre consIsts

0

(ii)

nU r

SU(n).

+ ••• +

X

wI where w (iii)

e

:= 0 mod 1 for each r

EXAMPLES.

U(n).

Xl

er (g)

if 9 t: T and

I

n

n

Sp(n).

:=

:=:: X

mod 1, so the

n

f matflces . e 21Tix I .

is given by

== 0 mod 1.

=

Xl

;; X

==

Xl

n

mod 1 and

Thus the centre consists of matrices

1.

nur

is given by x. ± x. =- 0 mod 1 all i,j, i.e., J

I

xi == 0 mod 1 all i or xi ==

1

2" mod

1 all i.

Thus the centre con-

sists of matrices ±I. (iv)

SO(2n).

nU

r

is given by x. ± x. I J

==

0 mod 1 for i

I

j.

for n > 1, this is the same a s for Sp (n), and the centre consists of ±I.

Of course, SO (2) is Abelian.

(v)

SO(2n + 1).

r. U is given by r

X

r

:=

0 mod 1 all r.

Thus

the centre consists of just the identity matrix I.

5.5

THEOREM.

Uris then

er

and

es

arelinearlyin-

dependent.

Proof.

U has dimension k - 1. r

We show that

LECTURES ON LIE GROUPS

106

=k

dim N((U) ) r

1

+ 2. The result will then follow from 4.40

applied to a generator of (U) • We need two lemmas. r

5.6

LEMlvtA.

Suppose H

1

C

group which is normal in G.

T, and that H is a closed sub-

Then

(i)

N(T/H) = N(T)/H.

(ii)

T/H is a maximal torus in G/H.

(iii)

W(G/H) ~ W(G).

Proof. (i)

If n preserves T then nH preserves T/H.

Conversely,

if n(tH)n-1 c T then ntn-1 cT.

(ii)

T/H is a compact connected Abelian subgroup of G/H,

and so a torus. Now suppose T/H Then U/H

C

N(T/H)

C

U/H, where U/H is a torus in G/H.

= N(T)/H, so T cUe N(T). Therefore

dim T = dim U, so dim T/H == dim U/H and T/H (iii)

W(G/H) = N(T/H) )'/H = N(T)/H/T/H ~ N(T)/T ~ W(G).

5.7

LEMlvtA.

(i)

n

=1

If dim T = 1 then

and W

= 0,

or

= U/H.

GEOMETRY OF THE STIEFEL DIAGRAM (ii)

n

[Note:

=3

107

and W = 22 .

In fact in (i) G

= S1 ,

and in (ii) G

= SO(3)

If n = 1 then clearly G = T = S1 and W =

Proof.

or Sp(l).]

o.

So sup-

pose n > 1. Take an invariant norm in L(G) and let v be a unit vector in L(T).

Define f : G/f - Sn-

1

L(G) by f(g)

C

=

(Ad g) v .

Then f is well-defined, continuous (even smooth) and is mono

It follows that gIl g2 E T and g1 T = g2 T .

a nd therefore fixes T.

Now G/T is compact and Sn-1 Hausdorff, so f is a homeomorphism of G/T with its image in Sn-l.

But G/T and

sn-1 are both compact rna nifolds of dimension (n - 1), so f is onto.

Then there exists 9 E G such that (Adg)v = -v, and

therefore 9 acts on T by gtg- 1 = t- 1 • automorphisms, so W = 2 2

Now T has only two

•

Let i be the generator of 'lT1 (T). 9 can be joined to e by an arc in G. that is, 21

=

Since G is connected,

So, in 'lT1 (G), i

= -i,

o.

Now we have in fact (2.37) a fibration S1 - G

-+

GI T ~ = S n-1 .

have that 7T2(Sn-l) -

From the exact homotopy sequence we 111

(S1) -

1Tl

(G) is exact. "But

LECTURES ON LIE GROUPS

108 TTl (Sl)

-t

17 1

(G) is not mono, since 2i

o.

--+

I

SO TT2 (Sn-l)

0

and n = 3.

Proof U . r

Consider (U) , the Identity component of

of 5.5.

r

1

This is a torus of dimension k - 1.

V..'e wish to show that u

1. U s

for r

Is,

Let u be a generator.

es

for then

will not be

er .

a multiple of

Consider N(U)lo

T is a maximal torus of N(U)l.

The

elements of N(u) fix u and so fix every element of (U) . r

can apply 5.6 with N(U)l as G, T as T, and (U) r

T/(U) r

1

1

We

as H. Then

is a maximal torus in N(uh/(u) ,and r

1

W(N(U) /(U) ) ~ V..'(N(U)1 ). r

1

1

Now T / (U) r

1

(5.7) N(u) /(U) r

1

has dImension 1, so

has dimension 1 or 3, and N(u)

1

sion k or k + 2. u lies in exactly

1

1

has dimen-

But (4.40) N(U)l has dImension k + 211 where II

of the U. r

Hence

II =

1 and u does not lie

In U • s

5.8

THEOREM.

For each r there is an element

~

r

EW

which is not the Identity but which lea ves every point of U

r

fixed.

Proof.

\;Ve use the

choice of u.

sam~

proof as 5.5

I

but wIth a different

GEOMETRY OF THE STIEFEL DIAGRAM Cuns ider U. r

109

We observed (4.38) that U is monogenic. r

Let v be a genera tor. Now consider N(vh.

T is a maximal torus of N(vh ,

and N(V)1 fixes every element of U.

We can apply 5.6 with

r

N(V)1 as G, T as T, and U as H.

We deduce that T/U is a

r

r

maximal torus in N(vh/u , and N(vh/u has dimension 1 or 3. r

r

By 4.40, dim N(VI )/U 2:. 3, so dim N(Vl )/U = 3 and r

w(N(vh/u) ~ Z2.

r

That is, there is n E N(V)1 which fixes

r

each point of U and which maps T/U by t r

5.9

r

COROLLARY (of the proof) .

cp

r

-t

C1

•

is the inner a utomorph-

ism induced by an element n which can be joined to e by a path of which each point leaves each point of U fixed. r

5.10

Proof.

COROLlARY.

U has either one or two components. r

<." acts on T/(U)

r

r

1

by t -"

t- 1

1 fixed points, namely 0 and -2 mod 1.

, which has only two

But U /(U) r

r

1

is fixed by

C!).

r

5.11

EXAMPLE.

The root 2x of Sp(n) gives U r

r

with two

components.

5 .12

DEFINITION.

For each r let ( = ±1. Consider the set r

110

LECTURES ON LIE GROUPS

{t E L(T);

£

e (t)

r r

> 0

all r}.

This is either empty or is a non-empty convex set.

In the

latter case it is called a Weyl chamber, and its closure IS given by

{t E L(T);

£

e (t)

r r

~

0

all r}.

So we can sa y that the hyperplanes of the diagram divide L(T) into Weyl chambers. A wall of a Weyl chamber is the intersection of its closure with a hyperplane L(U ) when the intersection has r

dimension k - 1. W permutes the planes of the dIagram and the Weyl chambers, by 4.37. For the following theorem, we suppose choser! an invariant norm in L(G).

The word" reflection" is interpreted

by using this norm.

5 . 13

THEOREM

(i)

W permutes the Weyl chambers simply transitively.

(ii)

For each r, W includes the reflection in the plane

L(U ). r

(iii)

Such reflections generate W.

(iv)

More precisely, for any Weyl chamber B, the

GEOMETRY OF THE STIEFEL DIAGRAM

III

reflections in the walls of B generate W. (v)

Let p E L(T) and W

p

be the stabiliser of p.

Then W

P

permutes simply transitively the Weyl chambers whose closures contain p. (vi)

W

p

is generated by reflections in the planes L(U ) r

which contain p. (vii)

More precisely, it suffices to consider those planes

which are walls of a fixed Weyl chamber B

o

such that

p E CI B •

o

Pro 0 f • and (vii) (ii)

By taking p = 0, we see that (v):=;> (i), ~

(vi)

~

(iii)

(iv) , so we need to prove only (ii), (v), (vi), (vii)

For each r, W contains an element cp

r

I

I which fixes

U (5.8), and hence fixes L(U ), and preserves the inner pror

r

duct in L(T). (v)


r

can only be the reflection in the plane L(U ).

Firstly, W

r

p

acts simply. We split the proof into two

lemmas:

5.14

LEMMA.

If v E L(T) is fixed by some lfJ E W, lJ.;

11,

then v E L(U ) for some r. r

Pro 0 f.

Suppose n E N(T), n

f.

T, and n fixes v.

Then n

fixes the I-parameter subgroup H corresponding to v (2.17).

LECTURES ON LIE GROUPS

112

Hence there is a maximal torus U containIng nand H (4.25). Therefore H lies in two distinct maximal tori / so H

UU

C

r

and

v E L{U ) for some r. r

5.15 them

LEMMA. ~

I

=

B for some Weyl chamber B, ~ E: V..',

= 1.

v..'

Proof. v

~B

If

is finite so ~ q = 1 some q >

o.

= lr;; ~/v lies in B and is fixed by lP. If q

shows that v

I

Let v 1JJ

B.

E:.

Then

11, (5.14)

lies in some L(U ) / which contradicts the hypor

thes is .

Continuation of 5.13 (v)

Secondly, v..'

p

acts transitively on the 'vVeyl chambers

whose closures contain p, as follows. Let B , B o

be Weyl chambers containing p in their

I

closure,andletx

o

(B

0

,Xl

EB'.

Since/forrls,

L(U ) i! L(U ) has dimension k - 2 (5.5) r s path from x

o

to x

I

not

me~ting

ther~

I

is a polygonal

any L(U ) ,', L(U ), not meeting r s

any L(U ) unless it contains p / and m~~tlng each L(U ) transr

versely.

r

~ Take

Suppos~

the path x px I, and move it slightly. ~ o

thIS path crosses (LU

k1

), ... / L(U

k1

)

S

ucccss-

ively to get flom B to B1 ••• to B = B Then ~k .. ·
113

GEOMETRY OF THE STIEFEL DIAGRAM

maps B via B o

I

•••

I

1

B

r-1

to B = B r

Thus V..'

I.

p

IS transItive on

the Weyl chambers whose closure contains p. (vi)

Let U E: Wand choose B p

B = lPB • Then B o I

so

lP-1 C!)k

lP- 1 CPk

r

I

Set

CPk B , with the notation above, rIo

= 1 or

1

0

=
••• CD B = B . k rIo 0

••• CPk

such that p E CI B .

0

lP =

But W

W

acts simply, so

p

k r ... C!)k 1 •

Thus these reflections generate V..' . p

Wnte lP = CPk

(vii)

••• <""'k r

as above, and suppose as an 1

inductive hypothesis that we have wntten CPk product of reflections in the walls of B . o possible for s

••• CPk as a s 1 This IS trivially

= 1. Then ct'k

is the reflection in a wall S-t-1 L(U ) of B . But ct'-k 1 ... CP-k 1 maps B to Band L(U ) k S+l k S-t-1 SIS S 0 to, say, L(U m )

I

which contains P. -1

<'ok

s

Then

-1

···ct'k
= CPk

s

•

S+l

Therefore <..'"

Hence of B

o

l,i..J

ks + 1

•••

CPk ,.

= cP

ks

••• ct'k

C!)

I

m

•

may be written as a product of reflections in the walls

which contain P.

5.16

COROLIARY.

Divide L(T) Into orbits under \1\'. Then each

orbit contaIns precisely one pOInt in the closure of each Wayl chamber B. Proof.

Cl B conta ins at lea st one po int of eu ch orbit, uS

114

LECTURES ON LIE GROUPS Let v E L(T).

follows.

Then v E CI B I for some Weyl chamber

B' , and B = wE' for some w E W.

Then wv E CI B.

CI B contains not more than one point of each orbit, a s Let p, q E CI B, and p = wq.

follows. Since W

p

Then p E CI (wB) .

is transitive on those Weyl chambers whose closure

contains p, thereisw' EW

suchthatw'wB=B. Then

p

w 'w = 1 so p = w 'p = w 'wq = q.

5 • 17 (i)

EXAMPLES G=U(n).

G

e consists of the skew Hermitian matrices.

Define an inner product on G by (X ,Y) = tr(XTY) = tr(-XY). This is e

invariant under G. Restricted to L(T) this has the form (up to a factor 4112) x 2 + .•. + x 2

n

1

This is the 'usuaP inner product,

•

so reflection is the 'usual' reflection. The root L(U

rs

ers

= x

r

- x

s

gives the plane x = x for r s

) and reflection in this plane is given by y 1 = x 1 , ...

I

Yr =

X

s

,

•••

I

Ys =

X

r

I

•••

'Yn = x . n

This is indeed induced by an inner automorphism, namely, by conjugation with

115

GEOMETRY OF THE STIEFEL DIAGRAM

s

r

1.

"I 1

0

r

1

1 -1

s

o 1

".

[We write -1 and not +1 for the sake of the next example.] Thus W is the symmetric group on:x , ... ,x . The order 1

n

"

I wi

of the Weyl group W is n!. (ii)

G = SU(n).

The same calculation may be repeated,

and V..' is the symmetric group on x , ... ,x . 1

(iii)

G = Sp(n).

n

IW I

= n! •

This time W consists of transformations

of the form Y 1 = £ 1 X P (l)' ... ,y n = £ n x p(n) ,

where

£

r

= ±l

each r, and p is a permutation.

jw I = n!2n.

= SO (2n + 1). This gives the same Weyl group as

(iv)

G

Sp (n).

IW I =

(v)

G = SO(2n).

n n! 2 . W consists of transformations of the form

116

LECTURES ON LIE GROUPS

where

£

= xl each r,

r

I w I = n! 2 n5.18

11'

n 1

(

r

= + I, and p is a permutation.

1 •

DISCOURSE.

The roots

er

are real linear forms on

L(T), that is, they are elements of L(T) *.

v..' acts on L(T) * by

(wh) (v) = h (w- 1 v) . v..'e have an invariant inner product on L(T), so we may identify L(T) and L(T)* by i: L(T) - L(T)*, where we set (iv I

) (V2)

=

<

VI

,v 2

This commutes with the a ction of W, so

).

all results on the action of W on L(T) can be transferred to L(T)* under the isomorphism i. We put an inner product on L (T) * by copying that of L (T), that is,

or, if you prefer,

=

h (v).

This is, of course, invariant under W. cp acts on L(T) fIxing those vectors v for which r

er (v)

=

o.

for which

Therefore

er (v)

~

r

acts on L(T) * fixing those vectors

= 0, that is,

vectors perpendicular to

er ,

<er ,iv >= o.

Thus (~ fixes those r

and so cp is reflection in the r

IV

117

GEOMETRY OF THE STIEFEL DIAGRAM

plane perpendicular to

er .

Note that reflection in the plane perpendicular to the unit vector v is given by w

5.19

-+

w - 2 (v w)v. I

PROPOSITION

~r

(h) = h - 2(9 rl h)

e.

(erle r ) r

This follows from the discourse.

5.20

A weight is an element of L(T) * which

DEFINITION.

takes integer val ues on the integer lattice. For example

I

each root is a weight.

W sends weights to weights.

5 .21

PROPOSITION.

~ r (A)

Hence:

If A is a weight

I

then

2(e rl X)

=

A -

(e ,e> er r

r

is a weight. W also sends roots to roots

5.22

PROPOSITION. cp

is a root

r

(e s ) = es

±et .

If

es

I

so:

is a root

I

then

LECTURES ON LIE GROUPS

118 cp (9 ) = -9 •

5.23

EXAMPLE.

5.24

PROPOSITION.

r

r

r

In 5.21 and 5.22 the coefficient

-2(9 rl A) (9f' 9r ) is an integer.

Choose v E L(T) so that 9 (v) = 1. Then exp v E U •

Proof.

r

cp fixes U

r

r

I

so

V -

cp (v) is in the integer lattice.

r

A(V) - Alp (v) is an integer. That is r

integer; since cp-l = cP r

r

r

I

I

Therefore

A(v) - (cp A) (v) is an r

this shows that

is an integer which is the required res ult . I

5.25

PROPOSITION.

Let a P be roots with a I

-I ±p.

Then

either

p are perpendicular or

(0)

a

(1)

a,P make an angle of 60 0 or 120 0 and

(2)

a

./2 (3)

I

I

I

I

la I = I p I ,or

p make an angle of 45 0 or 135 0 and their ratio is

or a I p make an angle of 30 0 or 150 0 and their ratio is ../3.

Proof.

We prove this together with:

GEOMETRY OF THE STIEFEL DIAGRAM

5.26

PROPOSITION.

119

Let a, p be roots with a

I ±p,

k be an integer between 0 and -2
p+

and let Then

ka is also a root.

The angle between a and p is given by

Pro of s.

2 ,_ (a,{3)2 co s u.: - (a, a)( p , p> < 1. Therefore

o ~ f=2(a ,(3)2)f=2< 13 ,a >'< 4. \(p,p)

\. (a,a)

J

By changing the sign of a if necessary, we may suppose

(a ,p) ~ O. If (a ,p) 5.26 is trivial.

= 0 then we

have case (0) of 5.25, and

Otherwis8 at least one of

-2(a ,13») (-2«(2 ,a» ( (a,a) ' \ . (p,p)

1s 1. If

~2:~: 5> ) = 1. then f3 + a

is the reflection of f3 in

the plane perpendicular to a, and 5.26 follows ill this ca se. SInce 5.25 is symmetric in a and p, we may assume now that -2(B,a) = 1 (P ,P) .

Let -2
< >

(G,G) ~-=-""'-j;;...-~ (a,a)

= II If

/pl

II,

II =

1, 2 3rT ,o . h en

.,

II

so - - = JII, and cos'" u.: = - so cos u.: lal 4 II

=

= JII - . 2

1 we get case (1) of 5.25, and 5.26 has already

been demonstrated.

We have the followIng diagram:

LECTURES ON LIE GROUPS

120

~------------~

Example. If

1/

SU(3). If we reflect a in the

= 2 we get case (2) of 5 .25.

hyperplane perpendicular to

p

we get

p+

If we reflect

a.

in the hyperplane perpendicular to a we get

p+

a

p+

2a, so

p

p + a,

2a are roots. p+2a

p+a

a

W contains reflections in two hyperplanes at 45 0

tains the dihedral group

l:.: x amp I e s .

If

II

,

and so con-

De.

S p (2) or SO (5).

= 3 we get case (3) of 5.25.

plane perpendicular to

p

we get

p+

a.

Reflecting a in the Reflecting

p

and

in the plane perpendicular to a, we get p + 3a and p + 2a.

p

+ a

121

GEOMETRY OF THE STIEFEL DIAGRAM

2p + 3a

p + 3a

a

W contains reflections in two hyperplanes at 30° , and so contains the dihedral group D12

E x amp Ie.

G2

,

•

which is the group of automorphisms of the

Cayley numbers as an algebra over R.

(It is possible to have

fun examining this example explicitly, but we omit this here')

5.27

DEFINITION.

Choose a Weyl chamber B In L(T) and

call it the fundamental Weyl chamber (FWC). Alter the signs of 6 1

, •••

,8

m

so that 9 (v) > 0 for v E B all r. r

{v (L(T); The roots 9 , ... ,9 1

-8 , ... , -9 1

5.28

m

9 (v) > 0 r

m

Then

all r} = B.

are now called positive roots, and

negative roots.

EXAMPLE.

Let G = U(n).

chamber be given by

Xl

Let the fundamental VVeyl

> x 2 > •.. > x n . Then the positive roots

122

LECTURES ON LIE GROUPS

are the forms x

r

- x

s

with r < s.

In Sp(n) we take the fundamental Weyl chamber to be given by x > ... > x 1

n

> 0, and similarly for SO(2n + 1).

For SO(2n) we take Xl> X2 ... > Xn-l > Xn > -X n- 1

5.29 R.

LEMMA.

•

Let the e be the positive roots and A .2:0 in r

r

ThenI;Ae =0 implies A = o for all r. r r r

Proof.

TakevEB.

Then (I:Xe)v=o, r r

SOL;A(eV)=O, r r

so

each A is O. r

5.30

DEFINITION.

(i)

a is a pos Itive root, and

(ii)

we cannot have a =

5.31

PROPOSITION.

a is a simple root if

p+

Y for

p and y positive roots.

Any positive root a can be written as

a linear combination of sImple roots with non-negative integer coefficients.

Proof.

If a is not simple, then a =

positive roots. process.

p + Y where p, yare

If either is not simple, we may repeat the

If this never terminates, since the number of roots

is finite, there is a n express ion where, contradicting 5.29.

p = p + b 1 + ... + br some-

123

GEOMETRY OF THE STIEFEL DIAGRAM

5 .32

If a, p are distinct simple roots, then

LEMMA.

~O.

Th en 2( a, @> > 0 and'IS an a,a . . a root. T h ere f ore Integer, so 2
f Proo.

S uppose

>0.

< )

< )

p - a or a - p is a positive root, whence either p = (p - a) + a or a = (a - p) +

5 .33

P is

not simple, contradicting the hypothesis.

PROPOSITION.

The simple roots are linearly indepen-

dent.

Suppose v = r;1J. 6 r r

Pro 0 f.

roots, alllJ.

r

,II

s

= r;1I s 6 s

where the 6 are simple r

are non-negative, and the sums run over dis-

jOint sets of subscripts.

Then

!S. O. II

Therefore v = 0 and we may apply 5 .29.

5.34

COROLlARY.

by

6 (v) > 0 , ... ,6 (v) > 0, 1

where 6 , ... ,6 1

Proof.

s

The fundamental Weyl chamber is given S

are the simple roots.

This is clear from 5.31. So simple roots correspond to walls of the fundamental

Weyl chamber.

LECTURES ON LIE GROUPS

124

5 .. 35 is x

EXAMPLE. >x

G = U(n).

... > x .

The simple roots are

n

1:2

The fundamental Weyl chamber

Any other root can be written as a linear sum of these, e. g. , x-x

r

s

=(x -x

r

r-1

)+ ... +(x

S-l

-x)

s

for r < s. And

are linearly independent.

5.36 a =

~IJ.

EXERCISE.

e ,

r r

If a is a simple root and we write

where the IJ. are non-negative numbers and the r

er

are positive roots, then we have wntten a = a.

5.37

DEFINITION.

follows.

The Dynkin diagram is constructed as

Take one node for each simple root a.

Given two

distinct simple roots a, P join the corresponding nodes by II =

0, 1, 2 or 3 bonds, where

5.38

EXAMPLE.

(Xr+1 - Xr-t-2) ,

II

G = U (n).

II

follows 5.26.

Between (xr - xr +1 ) and

= 1. Otherwise

II =

o.

Hence the Dynkin

dIagram is 0---0----0 ... 0---0 with n - 1 nodes.

5.39

LEMMA.

If

er

IS

a simple root then cp permutes the r

GEOMETRY OF THE STIEFEL DIAGRAM

125

positive roots except 9 , which goes to -9 . r

r

v..'e give two proofs.

Proof.

Choose a point v of the diagram such that 9 (v) = 0

(i)

r

and 9 (v) > 0 for any other simple root 9 . Then 9 (v) > 0 for s s t any positive root S other than 9 . t r Let S be a spherical neighbourhood of v not meeting any plane 9 t

=0

for t

-I r.

Let w E: S

n (FWC).

Then tPr(w) E. S.

Therefore (ct' 9 ) (w) = 9 (C!) w) > 0 r t t r

for t (ii)

I

Thus cp 9 is a positlve root.

r.

r t

Let 9 , ... ,9 be the simple roots, and let 9 be a 1 s t

positive root. Write

9 = n S + ... + n 9 . t

1

1

S

S

Then

differs from 9 only in the coefficient of S. Therefore cp (9 ) r

t

has at least one positive coefficient if S

t

-I 9r

r

t

and so (5.31

and 5.33) cp (9 ) is a positive root. r t

5.40

DEFINITION.

The fundamental dual Weyl chamber

(F DvVC) is the s~t of points in L(T} * corresponding under i to

LECTURES ON LIE GROUPS

126

the fundamental Weyl chamber in L(T). the set of h E L(T)* such that (9 ,h» r

That is, the rDWC is

0 for each simple root

9 . r

5.41

DEFINITION.

Define

p E L(T)* by P

Let 9 , ••• ,9 1

1

= -2 (9

1

m

be the positive roots.

+ ••• + 9 ). This is not necesm

sarily a weight.

5.42

PROPOSITION.

P lies in the fundamental dual Weyl

c h am b ere

' I eroot a. In d eed ' /2(a« , S> ) = 1 f oreac hslmp "a,a

Proof.

Let a

than a. (i)

=

9.

Then cP permutes the positive roots other

r

r

There are three cases: CPr (9 ) = 9 • Then <9 , 9 t t r t

>

= 0 so 9

t

contributes 0 to

(a, p). (ii)

9 permutes 9 and 9 , t r t u

-I u.

Then

<9 ,9 + 9 )= 0, r t u so 9 + 9u contributes t (iii)

=

9 • This case contributes t
5.43 groups:

9 t

o.

< 0> > -_

EXERCISES.

1 Work out 2(9

1

+ ••• + 9 ) for the followlng m

127

GEOMETRY OF THE STIEFEL DIAGRAM

(i)

SU (3).

(ii)

SO(5) •

(iii)

G2

5.44

PROPOSITION.

•

In L(T), reflections in the planes

e = k for k E Z cover the a ction of r

~

r

on T.

Pro 0 f. Let v E L(T) be such that er(v) = k. Then the reflection is given by x - ~ (x - v) + v = ~ (x) - ~ (v) + v. r r r But v maps into U , so r

~

r

(v) and v have the same image in T.

Therefore ~ (x) and ~ (x) - cp (v) + v have the same image in T. r r r

5 .45

DEFINITION.

The extended Weyl group

r

is the group

generated by reflections in a 11 the pia nes e = k, k E Z, of the r

diagram. By 5 .44,

ro

= Ker

5.46

(r -

r

covers the action of W on T.

w) •

DISCOURSE.

r

-----7>~

o

r

We have a split extension :> W

1~

W

Define

LECTURES ON LIE GROUPS

128 r

is the subgroup of translations.

o

Each one is the transla-

tion by an element of the integer lattice I, so we can regard r

as a subgroup of 1.

o

(It is not necessarily the whole of 1.)

Our next obj ect is to calculate the fundamental group 111

(G) in terms of the Stiefel diagram.

ant

111

The topological invari-

(G) may be distasteful to some algebraists, and so some

remarks are in order about the use to be made of it. First, one of the main theorems (6.41) is classically stated with the condition

11111

(G)

= 011

,

and some of the subsidiary results used

in its proof use the same condition. going to prove (5.47)

1111

1

(G)

~

However, we are just

I/r II, so it would be possible 0

to rewrite 6.41 with the data in the form II r all is what is used in the proof of 6.41. to use

111

o

= III , which after

Secondly, we propose

(G) to classify the connected covering groups over G,

as is usual in algebraic topology.

For our arguments to pro-

ceed without this (notably at 5.56 below) it would be necessary to construct the double covering Spin(n) of SO(n) without reference to

111 ;

and of course this is possible by pure algebra,

for example, using Clifford algebras. This is an interesting chapter of algebra, but it involves more work without providing so much more insIght.

Sometimes one can buy algebraic purity

129

GEOMETRY OF THE STIEFEL DIAGRAM at too high a price [23]. To continue: we have I ~ I

C

L(T)

to w (1)

-+

T.

The map i : T

THEOREM.

plane

(T), since

111

(T) is Abelian. i

111

(T) ~ 111 (G).

i* is epi and induces I/r ~ o

11

1

(G).

5.48-5.55 will, together, form a proof.

er

Proof.

111

G induces I =

-+

PROPOSITION.

5.48

Consider

Its projection is a closed path in T, and

so represents an element of

Proof.

(T), as follows.

For v E I choose a path w in L(T) from some w (0)

= v + w (0).

5.47

111

= 1.

r

Then r

Let y be the reflection of 0 in the r

is the subgroup of I generated by the y • r

0

contains each y , since reflection in 9 = 0 folo r r

er

lowed by reflection by

= 1 is translation by y .

r

Conversely, we claim that, if y E r, then y(O) = I:n y , r r

whence, if Y E r

, y is translation by I:n y. We prove this o r r

claim by induction on the number of reflections used to build up y. Suppos e suppose () (0)

y = p ()

= I:n s y s •

p (x) = x + (k -

where p is reflection in Now

er (x»

y • r

er

= k, and

130

LECTURES ON LIE GROUPS

Therefore p () (0) =

er O:n 5 y 5 )

But

r. n 5y5

er (I; n 5y5 ) y r •

+ kY r

is an integer, since I;n y 5

5

is in the integer

lattice. Therefore p6(0) has the required form.

5.49

EXAMPLES.

(i)

G = U (n) or SU (n) •

x

- x

r

5

Define

ro

r 5 = 1 (r < 5) is the point (0 ••• 0 1 0 ••• 0 -1 0 1T:

I - 2 by

= Ker 1T.

I/r ~ o

(ii)

The reflection of 0 in

1T (x

1

I

••• I

X )

n

= x

0) •

+ ••• + x . Then n

1

For SU(n) we have I/r = O. 0

For U(n) we have

z. G = Sp (n) • The reflection of 0 in 2x = 1 is r

r

r

(0 ••• 0 1 0 ••• 0). We have I/r = O. o

G = SO(2n) or SO(2n + 1). The reflection of 0 in

(iii)

r X

r

- X

5

(r <

5)

5

is (0 ••• 0 1 0 ••• 0 -1 0 ••• 0).

tion of 0 in x + x r

5

=

r

The reflec-

5

1 is (0 ••• 0 I 0 ••• 0 1 0 .•• 0).

For

r

SO(2n + 1) the reflection of 0 in x = 1 is (0 ••• 0 2 0 ••• 0), r

which gives nothing new. 1T(X

I

1

•••

,x ) n

=x1

+ ••• + x

Define n

1T:

I - 22 by

mod 2. Then r

In the special case of SO(2)

I

r

o

0

= Ker 1T.

Thus

= 0 and I/r ~ 2. 0

GEOMETRY OF THE STIEFEL DIAGRAM 5.50

LEMlvtA.

I

~ 11

1

(T) -

11

1

(G) maps

r0

to

o.

We show that y goes to zero. Well, let w be a

Pro 0 f.

r

rectilinear path from 0 to '}' in L(T). Then r

1 exp w (1 - t) = ~r exp w (t) for 0 ~ t ~ 2".

g E G such that ~ (x) r

= gxg-l,

By 5 .9, we can find

so that

exp ~ (1 - t) = 9 exp w (t)g-l, and such that there is a path from 9 to e each point of which keeps U

r

fixed.

So

exp w (1 - t) is homotopic to e exp w (t) e- 1 = exp w (t) I keeping t = 0, t

=

t

fixed.

Hence exp w (t) for 0

keeping end points fixed.

5.51

NOTATION.

~t ~

1 is contractible

So y goes to zero in r

11

1

(G).

Let GR,TR,L(T)R denote the sets of regu-

lar points in G, T ,L(T) respectively.

Pro 0 f. ~

The complement of G

R

has Hausdorff dimension

n - 3, by 4.42 and standard Hausdorff dimension theory, and

the result follows by standard homotopy theory.

Then fR is a covering with fibre W.

132

LECTURES ON LIE GROUPS

Pro 0 f.

W acts on the left on G/T as follows.

and let n E N(T) represent cp.

Let cp E W

Define

cp (gT) = gTn-1 = gn-1 T. W also acts on the left on T , and so acts on G/T x T • Let R R G/TXWT R be the orbit space.

Since W acts freely on G/T,

the projection G/T x TR

-+

G/TxWTR

is a covering with fibre W.

is a one-one and onto map between manifolds of the same dimens ion, and so is a homeomorphism.

Proof.

Hence the res ult.

Considerthemap fR G/T x TR ----.,. G

R

where fR is a finite cover.

C

G,

Let the components of TR be T~ ;

then since G/T is connected, the components of G/T x TRare G/T x T~; and so each of the following maps is monomorphic. 11 1

(G/T x pt) --;.

111.

i f R* (G/T x TR) ----;.

Now the map G/T x t

o

-+

111

G, given by 9

nullhomotopic by taking a path from t

o

(G ) ~ R -+

to e. So

11 1 (G)

•

gt g-l, is 0

11

1

(G/T) = O.

133

GEOMETRY OF THE STIEFEL DIAGRAM

Hence, from the homotopy exact sequence of a fibration 11 1 (T)

-+

we deduce that

11 1 (G) 11 1

LEMMA.

5.55

then v E r

Proof.

o

(T)

-+

111

(G/T) ,

111

(G) is epi.

If v E I maps to 0 under I ~ 11 1 (T)

-+

111

(G),

.

We may suppose that, for any'}' E.

ro ,

v + Y is not

closer than v to the origin in 1. Then 9 (v) = -1, 0 or 1 for r

each root

er

er , for,

if

er (v)

> 1, then the reflection of v in

= 1 is closer to the origin, and correspondingly if

er (v)

< -1.

Let w be the linear path in L(T) from w (0) = 0 to w(l)

= v.

This does not cross any diagram planes, although it

may lie in some, and may meet others at w (0) and w (1). So there is a linear path

Wi

from

Wi

(0) to

Wi

(1)

= Wi

(0) + v which

is close to wand which meets diagrams planes only close to Wi

(1). ConsIder the diagram fR G/T x L(T)R ~ G

R

r

I,

G/T x L(T)

f

G.

By takIng the identity coset in G/T, the path ~I may be

LECTURES ON LIE GROUPS

134 considered as in G/T lies in G

R

X

L(T).

except near fw' (1).

Then fw l is a loop in G which By 4.42

I

we may move this

loop slightly near fw ' (1) so that it lies in G I and this loop is R contractible in GR. Since G/T x L(T)R - G

R

is a covering I we

may now lift the loop to a path w" in G/T x L(T)R starting near T x

o.

Then w" will be the same a s w' except near

Wi

(1).

Further I since we have altered fw l only near e in G I the projection of w" onto the factor L(T) is close to is contractible in G

R

so w" is a closed loop in L(T)R I and v

I

is approximately zero.

DISCUSSION.

5.56 that

'IT 1

Now f w" R

Wi •

But v is in I I so

V

=

o.

We have now shown (5.47 and 5.49)

(SO(m» ~ 22 for m > 2. Therefore SO (m) ha s a double

cover called Spin(m).

It is clear that the cover of a maximal

torus in SO(m) is a maximal torus in Spin(m). standard maximal torus

T in

maximal torus T in SO(m).

Take as the

Spin(m) the cover of the standard Then L(T) ~ L(T) under the covering

map I though this does not preserve the integer lattices. consists of all (x s ists of all (x

I

I

•••

I

•••

1

,x ) with all x integers n r

,x ) with all x n

I

and I con-

integers and x

r

I

1

+ ••• +

X

n

even. Similarly L(T) * ~ L (T) * I but this does not preserve the lattices of weig hts.

For exa mple

1

I

-2 (x

1

+ ••• + x n ) is not

d

135

GEOMETRY OF THE STIEFEL DIAGRAM weight in SO(m) but is one in Spin(m). Now Ad : G .... SO(n) induces

We distinguish two cases. (i)

Ad* is zero, ar.ld we can lift Ad to get the following

diagram. Spin(n)

G (ii)

~1 Ad

> SO(n)

Ad* is non-zero.

Then Ad defines a double cover G of

G, and we have the following diagram.

For G, (i) applies.

By 3.68, the representation theory of G

determines that of G. So, in what follows, we will assume that (i) applies.

5.57

PROPOSITION. In this case,

P=~(9 1 L.

+ ••• + 9m) (see

5 .41) is a weight.

Pro of.

In 4.12 we split G

e

as a T-space in the form

136

LECTURES ON LIE GROUPS

V EB E o

m 1

Choose bases for V , ••• , V ,V , and put them

V..

1 1 m

0

together in this order to form a base for G • Then the come

~ Aut

position T C G

G

e

maximal torus T' of SO(n).

= SO (n)

sends T into the standard

xr Further, if L(T') ~ R denotes

the rth co-ordinate function, then the composition xr

L(T) ~ L(T')

~

R is the root ±9r, for r

~

m, or zero, for

r > m. With the same sign attached to each 9 , we now have r

±9

1

± ••• ± 9

m

= (Ex )Ad. r

Now Ad lifts to Spin(n)

I

and tExr is a weight for

Spin(n), so (tEXr)Ad is a weight for G. is a weight for G, and so is 1 differs from -2 (±9

5.58

LEMMA.

1

Thus t(±9

1

•••

±9 ) m

1

p = -2(91 + ••• + 9 m), as this

••• ±9 ) by a sum of positive roots.

m

In this case w ....

W

+ P gives a one-one

correspondence between weights w E CI FDWC and weights W

+ P EFDWC.

Proof (i)

If w is a weight and (w, 9 ) .2:. 0 for all simple roots r

9 then (w + p, 9 ) > 0 by 5 .42. r

(ii)

r

If w is a weight and (w, 9) > 0 for all simple roots r

9 then 2 (w, 9 r ) > 0 and is an integer (5.24), so.2 1. r (9r , 9 r )

Now

137

GEOMETRY OF THE STIEF EL DIAGRAM 2 «(3, 9r ) 2(w - (3, 9r ) = 1 so 2! 0 and (9r ,9r ) , (9r , Sr)

--~

W -

(3 is a weight in

CI FDVvC. We showed (5.24) that, if 2 (9[1w) then (9 9) r' r

5.59

is an integer.

PROPOSITION.

If

W

is a weight and 9 a root, r

We now examine the converse.

2 (9 r ,w) (9 9) is an integer for some

r' r w E L(T)* and all simple roots S , then it is an integer for all r

roots 9 • r

2 (9 p Suppose (9 r' 9 and also for the root r

Pro of.

w) 9)

is an integer for all simple roots

r Let

9 •

s

root 9, and let S =~ (9). r t rs 2

~

r

correspond to some simple

Then (9 ,9) = (9 ,9 ) and so t t s s 2 (9r , 9s )

/

(9 ,9 ) \. 9s - (9 ,9 ) 9r s s r r =

2 (9 s'w) (9 ,9)

s

s

which is an integer. But the reflections and any root 9

s

~

r

generate W (5.34 and 5 • 13 (iv))

can be written as

~

9 for some simple root r

9 and some (/) E W, by considering 9 as the wall of a Weyl r s chamber and throwing this chamber onto the FWC (5.34).

138

LECTURES ON LIE GROUPS

Hence the result.

5 .60 some

PROPOSITION.

Suppose

2 (9r ,W)

(9

9) r' r

E L (T) * and each simple root 9.

W

r

is an integer for Then

W

takes integer

values on

ro .

Pro 0 f.

ro is generated by the points y , where 'Y = r r

for any v such that 9 v r

W (y )

r

= 1.

v-

~

r

v

We have

= wv - W (~ v) r = (w - ~ w)(v) r

2 (9 r ,W) 9) 9 r (v) r r

= (9

I

which is an integer.

Thus w is integral on each y

r

and so on

r. o COROLIARY.

5.61

If G is simply connected and

( r

is an integer for each simple root 9

r

Proof.

5.62

I

I

r)

then w is a weight.

r o =I.

THEOREM.

If G is simply connected it has just

k = dim T simple roots 9 1 I ••• 19k I and has weights such that

2 (9r ,W) 9 9

~~:~ e~t) = 6 rt •

binations ~ W

1

+ ••. + nkw

WI

I··· IWk

The weights are then the linear comk

with nr E Z each r.

The rDWC

139

GEOMETRY OF THE STIEFEL DIAGRAM

consists of all points I:n w with each n > 0, and the CI r r r

Fnwc

consists of all points I:n w with each n > r r r1 '2(9 1 + ..• + 9 m) = w 1 + ••• + wk·

Pro of.

Suppose there are just k -

II

o.

slmple roots.

the roots lie in a subspace of L(T) * of dimension k -

ra

lies in a subspace of L(T) of dimension k -

ha s rank at lea st

II

which implies

There are elements

II =

II.

.

be written

W =

L;n

W

,

r r

if and only if each n

r

and then

II,

Then

and so

I/ra

2 (B

r,

Wt)

= 0rt'

(Br' Sr)

t

and they are weights by 5 .61.

Then all

O.

In L(T)* such that

W

Also

Every element w of L (T) * can 2 (Sr w)

' = n so w is a weight (Sr' Br) r'

is an integer.

The statements about

FDWC follow from the definition (5.40).

Set (3 = t(S 1

5.63

+ .•. + B ). Then 2 (9 r ,(3) = 1 (5.42), so m (Br,B ) r

EXAMPLE.

Let G = SU(n).

Take w t = x

1

+ ••• + x for 1 < t < n - I. t

--

so

The elements of L(T)* can be written

Then

140

LECTURES ON LIE GROUPS

since I;x = i

o.

They lie in rDWC if

Thus they may be written

and lie in rDWC if b I

, •••

,b n-

5 . 64

COUNTEREXAMPLES

(i)

Let G

=

U(2), where

but there is only one root.

TTl

I

> O.

(U(2)) ~ Z. We have dim T

= 2,

The rDWC is a half-plane, which

cannot be expressed in the given form. (ii)

Let G

= SO(4),

where

TTl

(SO(4)) ~

Z2.

The dual diag-

ram is as follows:

rDWC

141

GEOMETRY OF THE STIEFEL DIAGRAM Here the asterisks represent weights. quarter-plane shown.

The weights in the CI FDWC do not form

a free Abelian semi-group, and

are not weights.

The FDWC is the

Chapter 6

REPRESENTATION THEORY

Throughout this chapter G is a compact connec-

Notice.

ted Lie group I and T is a maximal torus of G.

6.1

THEOREM.

.

(Weyl Integration Formula.)

There is a

real function u on T such that

r

f(g)

=

.)G

S

f(t)u(t) T

for all class functions f on G. Indeed u (t) =

O=TI

m

j=l

~

e

06/\ W

11 is].

(t)

\

-e

I

where

- 11 i9

o

J

(t))

and 9. runs over the distinct roots of G. ]

Proof.

Define f : G/T x T

-0

G by f(g,t)

Then f factors through G/Txvl:

= gtg-1

•

(See 5.53.)

REPRESENTATION THEORY

143

Now c ha s degree 1, since it is a homeomorphism when restricted to G/TxWTR - G , and b is a R covering.

So f ha s degree

I w 1\ . G

f dg =

I Wi,

I Wi-fold

and

SG/TxTf* dg* ,

where f* ,dg* are the induced function and measure on G/TxT. If f is a class function, then f* is constant along G/T. Now we must evaluate det f' at a general point (g, t) of G/TxT. First, let u run through a neighbourhood of e in T. Then f (g ,tu)

= gtug-1

= gtg-1 gug-1 •

Therefore f (g, t) = Ad g, where we consider the first factor I

fixed.

Second, let v be in a tra nsversal V of T in G. f (gv ,t) = gvtv-1 g-l ,

so f (g, t)(dv) = g(dV)tg-1 - gt(dV)g-l I

so f' (g ,t) = Adg (Adt-1 - I),

Then

144

LECTURES ON LIE GROUPS

where we consider the second factor fixed. det f I (g, t)

= det (Adt-1

-

Thus

I).

Now Adt has the form cos 211 9 1 -sin 211 9 1

so Adt-1

-

I ha s the form

cos 211 9 1

-

1

sin 211 9 1

sin 211 91 cos 211 9 1

-

1 00

and I) = nm(cos 2 2119 -2cos2119 + 1 + sin 2 9 ) 1 \: r r r m (4 . 2 9) =rr(11i9 r 11i9 r ) -11i9~ (-11i9 r 1 s In 11 r 11 e - e ') n \ e - e

det(Adt-1

=rr

-

= 06 , where ( 11 i9 r

o =n,e

- e

-11 iB r )

•

Hence the result.

6.2 ~

DEFINITION.

W acts on L(T).

denote the sign of the determinant.

For

~

E W, let sign

Then we say that

REPRESENTATION THEORY

145

X E K(T) is a symmetric character if ~X = X for each ~ E W,

and is an anti-symmetric or alternating character if ~ X = (sign~) X.

6.3

EXAMPLE.

Then

o = ITlmC e 11 is r

Suppose Ad : G

-11

- e

-+

SO(n) lifts to Spin(n).

i9 ) r

is an antI-symmetric character.

Pro 0 f .

o = I; (

1

... (

Ex p 1T i (( 9 + • • • + ( 9 ) mIl m m

where (. = ± 1 and there are 2

m

1

1 -2 (( 9

terms.

Note tha t, by 5 .57 ,

+ ••• + ( 9 )

11m m

is a weight, so 0 E. K(T) • Let x E N(T) represent (/) E W. given by g .... xgx-1 T

e

•

Then the action of

This induces a map G

e

~

is

.... G which maps e

to T • On T it preserves or reverses orientation according e e

to (sign~). Also ~ permutes V , ... , V (5.5 and 3.22). 1

m

If ~

maps Vj to V preserving orientation, then it sends 9 to 9 ; j k k and if reversing orientation, then it sends 9 to -9 • If it j k reverses orientation But

CD

II

times then (/) 0 = (- 1)11 o.

preserves the orientation of G , since x may e

146

LECTURES ON LIE GROUPS

be connected to e by a path.

Therefore

= +1.

(sign <0)(-1)11 That is,

<00 = (signet' )0.

PROPOSITION. If a character X (of T) vanishes on U

6.4

r

then it can be written

x= [Exp(21Ti9)r

- IJ~,

where lJ; is a character.

Proof (i)

Suppose U

has just one component. Then we may

r

take a basis e follows.

l ' •••

Let e 2

L(U ), and let e r

' •••

1

which 9 (e ) = 1. r

1

,ek of the integer lattice of L(T) as ,e

be a basis of the integer lattice of

k

be a point cf the integer lattice of L(T) for Let ~

1

, ••. , ~k be the characters of the basic

representations of T. Then Exp(21Ti9 ) = ~ r

1

,and we can write

n

X = I; c ~ ,where each c is a finite Laurent series in n n 1 n ~

2 ' ••• ,

~k •

On U , ~l = 1 so I;c = O. The monomials ~2' n r are linearly independent on U , so !: c r

series.

Set

n n

IS

•••

the zero La urent

'~k

REPRESENTATION THEORY

147

This is a finite Laurent series and X = (ii)

Suppose U

has two components.

r

before, but with 9 (e ) = 2. r

Exp (211 is ) r

(~l - 1)l,).

1

=

~

Take a basis as

Then

2 • 1

Consider X = I:c ~n and note that U is given by n 1 r ~

1

=land~

Thus I:

1

=-1.

ThereforeI:c

ddC = 0 and I: c non n even n

before to get X = (~~

-

=OandI:(-l)c

= 0, and

n

=0.

we may argue as

1)~.

PROPOSITION.

6.5

n

n

If X is an anti-symmetric character,

then I[EX P (211i9.) - 1}11, X =n~ J= J "i" where rJ; is a character.

Proof.

It is only necessary to show that, if

x=

[Exp (2 11 i 9.) - 1 ] rJ; 1

and X vanishes on U

r

for r Ii, then ~ va nishes on U , for r

we may then argue by induction using 6.4. Well, lJ; does indeed vanish on U

r

except possibly on U.

1

dim U = k - 1 and dim U. r

on all of U

1

r

n Ur

by continuity.

= k - 2.

n U. r

But

Therefore I/J vanlshes

148

6.6

LECTURES ON LIE GROUPS

THEOREM.

Suppose Ad lifts to Spin(n). Then lj; - 1J,J0

gives an isomorphism from the additive group of symmetric characters to the additive group of antl-symmetric characters.

Proof (i)

0 is anti-symmetric (6.3), so the map goes where the

theorem says. (ii)

1 (' TWT j 00 = I, so 0 -10 and the map is mono (3.77).

(iii)

Suppose X is an anti-symmetric character.

(6.5) X = l/J 0, where lP IS a character.

Then by

Now

(s ign ct' )~ 0 == (s ign ~ )(rtllP) 0 and ct' lP (t)

= lP (t)

except, perhaps, where 0 (t)

UU • Hence by continuity, ct'liJ (t) = r

r

~

= 0,

that is, on

(t) for all t f T and lj,; is

symmetric. Thus the map is onto.

6. 7

DEFINITION.

Let h E L (T) * be a weight, and let Wh

be the orbit of h under W.

Then the elementary symmetric

sum S(h) is gIven by S (h) = !:w EWh Exp 211 iw.

6 •B Then

EXAMPLE.

Let G

= S U (n).

REPRESENTATION THEORY

Sex ) = ~ 1

1

149

+ ••• + ~ , n

where (. = Exp 21T ix.

J

J

Sex 1

+ x2)

=~1~2 +

~

1

~

3

+ ~2 ~3

+ ~n-l ~n S (2 x

1

+x ) = ( 2

2

1

~

2

+

2

2

2

~2 ~ n

+ ~ 2 ~ 1 + ~2 ~ 3

6.9

PROPOSITION.

Let h run over a set of representatives

of the orbits. Then S(h) runs over a Z-basis for the symmetric elements of K(T). This is obvious.

6. 10

EXAMPLE.

In 6.9, h may run over the weights in CI

FDWC.

6.11

LEMr.AA.

Let X be an anti-symmetric character, and

h E L(T)* a singular weight

I

that is

I

h E L(U )* for some r. r

Then Exp 21T ih occurs with coefficient 0 in X.

150

LECTURES ON LIE GROUPS

Pro of.

Suppose

x=

a Exp 21T ih + ..•.

Let cp E W be reflection in L(U )*.

Then

r

-x

= cp X = a Exp 21T ih

+ ....

Thus a = O.

6.12

DEFINITION.

Let h E L(T) * be a weight. Then the

elementary alternating sum A(h) is given by A (h) = !:Ct' EW(s ign Ct' )Exp 211 icp h. If h is singular, then A(h) = O.

Otherwise A(h) contains I

Iwi

distinct terms.

6.13

EXAMPLE.

Let G = SU (n)

I

and let

Then A(h) = det

1

a

~:2 n-l

6. 14

PROPOSITION.

Let h run over a set of representatives

of orbits of regular weIghts.

Then A(h) runs over a Z-basis

for the anti-symmetric characters.

REPRESENTATION THEORY

151

This is obvious.

6.15

EXAMPLE.

In 6.14, h may run over the weights in

FDWC.

6.16

PROPOSITION.

Let X be the character of an ureduc-

ible complex representation of G, and let lJ; 1JJ () =

= XIT.

Then

A (h) for some weight h.

Proof.

If Ai ,A 2 are elementary alternating sums, then

\ii,

I Wi

if

Ai

= A2

=

-iwi

if

Al

= -A2

=

0

if

Al

-I ±A2

A2 =

by 3.34, since any weight is a character for T.

Now lJ;{) may

be expressed as l;n.A. ,n. E: Z (6.14), so I I I

by 6.1 1I S (!:n.A.) - (2:n.A ) = -wi I I J n T

= I:n~ . I

Thus one n. is ±l and the rest zero. I

h this can be made + 1.

6. 17

PROPOSITION.

By a suitable choIce of

Hence the result.

As X runs over the characters of the

distinct irreducible complex representations of G, the

152

LECTURES 0 N LIE GRO UPS

corresponding A(h) are all distinct.

Proof.

If Ai ,A2 correspond to X , X then 1.

2

Hence the res ult . To give a second proof, let K(T)W consist of the symmetric elements In K(T),

that is, the elements invariant

under W; and let K(T) -W consist of the antisymmetric elements.

Consider the following composite.

K(G) - K(T)W - K(T)_W. The first map is mono by 4.31, the second is iso by 6.6.

6.18

PROPOSITION.

Suppose Ad lifts to Spin(n).

Then

every alternating sum A(h) arises as ±¢o for some irreducible representation of G.

Proof. and

0

A (h) =

00

for some symmetric character

0

in K(T),

= fl T for some clas s function f on G (4.32).

Now let X be the character of an irreducible complex representation of G.

Then

REPRESENTATION THEORY

=

=

\'Iw1 lolr A(k) A(h}, 0

153

where A(k) corresponds to X

unless A(k) = ±A(h)

(s ee 6. 16, proof).

By the Peter-Weyl theorem (3.47) this is

not zero for all X.

Hence A (h) = ±A (k) for some irreducible

representation X of G. Summarising, we now have:

6.19

PROPOSITION.

Suppose Ad : G .... SO(n) lifts to

Spin(n). Then we have a 1-1 correspondence between irreducible representations of G and elementary alternating sums in K(T) , given by I"-.J

K(G) ~ K(T)W ~ K(T) -W •

6.20

THEOREM.

If G is compact and connected (but

without the assumption that Ad lifts) then the map K(G) .... K(T)W is an isomorphism.

Proof.

The map is mono, by 4.31. Now, if Ad does not lift to Spin(n) , we have the

following diagram (5.56 (ii)):

154

LECTURES ON LIE GROUPS

Let lJ; be a symmetric element of K(T). symmetric element, so ~)'TT = of G (6.19).

Now

xl

Then

lfJ'TT

€K(T) is a

xfT' for some virtual character X

T factors through G so

X factors

through

G (4.32). Thus (3.68) X = X11 for some virtual character X of G, a nd XI T = l/J •

6.21

REMARK.

Even in the case where Ad does not lift to

Spin, we can define anti-symmetric elements as those X E K(T) such that CPX

6.22 'TTl

(G)

=

(s ign cp)X and X(xz)

DISCUSSION.

=0

I

= -X (x)

for I

I

z E Ker 11.

We are going to show that, when

K(G) is a polynomial algebra.

Classically, the

weights are ordered in a somewhat arbitrary way. When we propose to prove P = Q + .. lower terms" the error must be IIlower" with respect to all choices of ordering. So we will introduce an invariant partial order with which P = Q + .. lower terms" will have approximately this meaning.

6.23

DEFINITION.

Let

Lt:l ,(.(.:2

be weights in L(T)*.

Define

REPRESENTATION THEORY

155

a partIal order on the weights in L(T) * by wnting WI S. lies in the convex hull of the orbIt of W

1

< ~

-

If w

2

cients c

cP

1

,CP

2

cp

1

::: rcp EWccp

with!:c

cp

«j

W

2

)

~2

under W.

~2

If

u';l

That is,

for some non-negative coeffl-

= 1.

It is clear that ~I ::. W2 implies cp I ~I '5....CP

2W2

E W, that is, we are ordering the orbits.

So it will

for any

suffice to consider weights In CI FDWC.

6.24

ALTERNATIVE DEFINITION.

CI FDWC, write

~

::. W2 if

Wi

For

~l'~2

weights in

(v) ::: ~2 (v) for all v (FWC.

We

rna y equally take all v E CI FWC.

6.25

PROPOSITION.

These two definitions are equivalent.

We need:

LEMMA.

6.26

If u E CI FDWC, v E FWC and cp E W, then

(cpu) (v) ::. u (v) with equality only if