Seminar on Micro-Local Analysis (Annals of Mathematics Studies 93)

from ë

non-degenerate

be a point in T N C n . in a neighbor-


„ defines an X

isomorphism

real

N^

of p .

Denote

* onfo i/r~ ë p ^ n C ,p L ,q

M „ (Ö _) J'S compatible izeCn;f>0| C n

with V

SOME APPLICATIONS OF BOUNDARY VALUE PROBLEMS

Prooi.

47

A b a s i c result of S-K-K [8] (Chap. Ill Theorem 2.3.2) a s s e r t s that

there exist a homogeneous canonical transformation from a complex neighborhood of (0, k d z f ) e T ^ Y C T*Y to a complex neighborhood of ( 0 , v T l k d x 2 n _ 1 ) e T * 2 n _ 1 C 2 n - 1 C T * C 2 n _ 1 , a C-Algebra isomorphism $

:

ë

, n i t r. -> d> ~ ë

v

Y,(0,kd z f) *

: e

0„

, an isomorphism

1

C2n-1,(0,VrTkdx2nl)

^

N,(0,kdzf) - ^ ~ l e R 2 n - l ( 0 ^

k d x

}

'

which is

compatible with

$ , and an isomorphism: %Y -^-*
is isomorphic to C L . (S-K-K [8] Chap. Ill Theorem 2.3.6.) Since ë identified with ë m i £

0\ln),

ë*ip ^

^

(5HY'^N^N g

, ,^0) C2n-1

can be identified with the action of 0 ~ ë on

^

L

can be

the action of en<££ (3HV) on ^Y Y

u

(]2n-l

ç

-

£

^

« ^ - 1 ^ | " 1 2 n _ 1 C « o ^ R 2 n - l ) l w + . On C °

the other hand, e m l P (?HV) * — ë „n . . Thus we have an isomorphism C9 Y Y p* C ,p* $ ' ë

« ^ - ^ 0 ~ ë pL which i s compatible with $ . On the other hand, Cn,p* L

$ ' coincides with V up to an inner automorphism (S-K-K [8] p. 429). Hence we may assume without l o s s of generality that <ï>' = *P . Since there e x i s t s an isomorphism 0

from 3)

= eruier) (JR) onto

Cn

ë « ^ (jnY),

J7

x

we can canonically transfer the action of P(z, D z ) 6 9)

on R JCwn()R,® M ) (and hence on R r ~ R K a m f j R , ^ ) ) 0 ( P ) on R Jtamg; O ^ Y ' ^ N ^ N

to the action of

through the isomorphism (12). Hence

choosing *P to be the composite of the isomorphism (12) with $ , we obtain the required result.

Q.E.D.

48

MASAKI KASHIWARA AND TAKAHIRO KAWAI

Since the homogeneous symplectic structure of V PI yf-1 T N , namely, c o Y | v p | / T Y T * N ' * s i ^ e n t i c a l with co_„i n

C 'vnv^îT*N

= co

Cn'T^Cn

kd f(z, z~) (keR*) , the following Theorem 5 is important in applying Theorem 4 to concrete problems. T h i s theorem can be verified by a direct calculation.

THEOREM 5. Let

X and Y be analytic

Z be a non-singular

hypersurface

f(x, y) = 0 j . Assume

that

k grad, x cal tions

of dimension

of X x Y defined by

0


V

d x dyf

(20)

Then

manifolds

n.

i(x,y)fXxY;

+ 0 on Z .

T * ( X x Y ) = | ( x , y , f , r / ) 6 T * ( X x Y ) ; f(x, y)= 0, (f, r,) = .f(x, y)! i s a homogeneous

symplectic

1-form ù) = kd x f. Furthermore the Poisson 0 ( x , y, k) and *A(x, y, k) on T z ( X x Y )

manifold with the canonibracket

\cf>,i//\ of func-

is given by the

following

formula:

0 (21)

Let

{0, <M

k

k

0

5k

: V 0

d iA

5k


dk

d

d

yf

xV

dy<£ d

k ^ d

x*

0


d

y

yf

f

dxdyf

d

xdyf

SOME APPLICATIONS OF BOUNDARY VALUE PROBLEMS

49

REMARK. T h e condition (20) i s necessary and sufficient for

T7(XxY)

to become a homogeneous symplectic manifold with canonical

1-form

kdxf(x,y). This formula i s effectively used to c a l c u l a t e the cohomology groups a s s o c i a t e d with a c l a s s of multiple characteristic operators by the aid of Theorem 4 (Kashiwara-Kawai-Oshima [5]). In order to simplify the presentation, we restrict o u r s e l v e s to the single equation c a s e . The results are as follows:

THEOREM 6. Let in a neighborhood and

Ti = ë x / ë x P

be a micro-differential

of p 6 A = T M X ,

X is its complexification.

where

Assume

that

equation

M is a real analytic P satisfies

the

defined manifold

following

conditions: P has the form P ^

+ Q , with

ord Pj = m-(j = 1, 2) and

ord Q < irij + m 2 - 1 so that the following

conditions

(22)

V-= a ^ ) " 1 « ) )

(23)

{ a ^ W P ^ I I v n ^ O ,

d/-

1 AcüL 7 nx7

yoPJ.oÇPJ]) (24)

(25) (26)

where

iaCPjXaCPj^O,

never

(22)-(27) are

satisfied

(j = 1,2)

vanishes.

V nV

i 2 i<x(P 2 ),a(P 2 ) c S ^ 0 (*>

Vj n vf = v 2 n v 2 c (=w) (*> *(Q) never attains integral ia(P 2 ),a(P 1 )| W

values.

For a holomorphic function f(p) defined on a complexification A of a real analytic manifold A , we denote by f c ( p } , the complex conjugate of f(p) , namely, a holomorphic function defined on A which coincides with f(p) on A . T h e complex conjugate V of an analytic variety V C AC i s , by definition, the variety defined by the complex conjugate of holomorphic functions defining V .

50

MASAKI KASHIWARA AND TAKAHIRO KAWAI

d/< , d/
(27)

Then we have the following result on the structure of ëattc- (H, CM) near p

classified according to the sign of i<j(P), a(Pj)cl (j = 1, 2) .

Case A: { a ^ ) , a ^ ) 0 ! (p*)|ff(P2), a(P 2 ) c i (p*) < 0

(28)

ë « t ^ 0ï,C M ) = 0 /or any j .

Case B(+): Wpj.opJ0]

(29)

(p*), {CT(P2), a(P 2 ) c |(p*) > 0

' 4 ^ g -e^ (0

Case B(-): ia(P 1 ),a(P 1 ) c !(p*), i<7(P2),<7(P2)c!(p*) < 0

(30)

Theorem 6 asserts that both regularity theorem and existence theorem hold micro-locally for the micro-differential equation Pu = f in Case A, while only the existence (resp. regularity) theorem holds in Case B(+) (resp. B(-)). Furthermore it claims that the obstruction against the solvability of Pu = f is described by a flabby sheaf on W.

Here and in the s e q u e l , C w (resp. fem) denotes the sheaf of microfunctions (resp. micro-differential operators) defined on an open s e t of y—1 T R which i s isomorphic to W with the homogeneous canonical 1-form C o | w .

SOME APPLICATIONS OF BOUNDARY VALUE PROBLEMS

51

Next let u s i n v e s t i g a t e the structure of ëacirr (ÎI,(2 M ) when K attains integral v a l u e s . For this purpose we first note that microdifferential operator R on W such that a ( R ) = K i s uniquely determined up to inner automorphisms of ë

w

^*' by the condition (23) (S-K-K [8]

Chap. II. Th. 2.1.2). Hence ^ = ë w / ë w ( R - £ ) (fteC)

i s well defined by

K up to isomorphisms. Actually, ë^icr v^>^jy|) i s described in terms of X t h e s e a s follows:

THEOREM 7. Assume condition

the same conditions

(26). Then we have the following

CaseAj:

WiP^aiP^Kp*)

(3D

g«ti. (n,eM) X

C a s e A 2 : {aÇP^.aÇP^

02)

e

X

e

0 &xi°F ( % e to £=0,-1,-2,--W

- èxtl cfi,eM) -> W

X

W

ëxtl (%e w ). tô

£=1,2,3,-"

w

e £=0,-1,-2,-"

CaseB(-): ia(P1),a(P1)cl(p*)< 0,

The same a s the footnote on p. 50.

W

ia(P2)((r(P2)c|(p*) > 0

CaseB(+): loÇPJ.oiP^Kp*) > 0 ,

0 -

to

|CT(P 2 ), a ( P 2 ) c i ( p * ) < 0

(p*) > 0 ,

'

near p

g^ctjr (£ f ,e w ).

£=0,-1,-2,---

&JF a,eM) W

isomorphisms

for the

ia(P2),c7(P2)c!(p*) > 0

< 0,

tô

(33)

as in Theorem 5 except

) ^

êxtj. (££>ew) - ° <exac<) C

°W

ia(P2),a(P2)cl(p*) < 0

52

MASAKI KASHIWARA AND TAKAHIRO KAWAI

(34) o -> &«â (?i,eM) ^X

©

gxic (f £ ,e w )

£=0,-1,-2,•••

*ei*&«à &

(Jl,CM)X

©

^W ê*ic ( f £ , e w ) - 0

£=0,-1,-2,•••

(exact).

W

REMARK. We want to emphasize the following two interesting points of the above r e s u l t s . Especially the second point s e e m s to us to be very important. For the s a k e of definiteness, we concentrate our attention on Case A r First, both the regularity theorem and the e x i s t e n c e theorem hold for the equation in question even if K attains strictly positive integers. Second, in c a s e K attains 0 or strictly negative integers, say

k,

the structure of ë*cicr (ÎI,(2 M ) i s controlled by the micro-differential equation (R-k) u = 0 on W . We shall give a sketch of the proof of t h e s e results in what follows: The first step is to reduce the operator to a simpler form on A ^ . By a suitable quantized contact transformation, we can e a s i l y s e e that Jl i s isomorphic to ë (z, £ ) e T C n ,

/ë

n(z1D1-z2)

where z« = £1 = 0 .

considered near the point Here we have used conditions (22)

and (23). Again using another quantized contact transformation that preserves Z J D J - Z ^ , we can find a real non-singular hypersurface N s o that T * U = A c A C = T*U

(35)

holds for an open neighborhood U of 0 in C n . of A ^ over C

n

Actually, such a fibering

can be found as follows:

First we consider the problem infinitesimally.

For t h i s purpose we de-

fine a skew symmetric bilinear form E ( v 1 , v 2 ) on S = T *(T X) by .

Define So by T * ( \ / ^ Ï T * M ) . For an R-linear s u b s p a c e

T of S, we denote by T

the orthogonal complement of T with respect

SOME APPLICATIONS OF BOUNDARY VALUE PROBLEMS to R e E , i.e. T 1 = i v e S ; Re E(v, w) = 0 holds for any w e T i . C-linear s u b s p a c e p of S s u c h t h a t p C p

53 For a

, T^ denotes [ ( T f 1 p 1 ) + p ] / p

Note that p / p naturally becomes a symplectic vector s p a c e by E . We now define a linear form f ( j = l , 2) on S by d a ( P | ) ( p * ) . We a l s o define a linear form f3

on S by

A mw * d
^(P2)^(Q)KP*) t — fi

{a(P1),a(Q)l(p*) ,

i^WP^Kp*)

iaCPjXaCP^Kp*)

—U •

Then we have E

(36)

tfl>f3) - E(f2,f3) = 0 .

Since the condition (25) implies that f^

and f^

belong to Cfj + C L ,

we have also (37)

E(fC, f 3 ) = E(fC, f 3 ) = 0 . In what follows we denote f " 1 ^ ) In the sequel, we identify *

feS

by T - .

S with S ( = T *(T X)) by assigning P

to v 6 S so that E(w, v) = f(w) holds for any w e S . By this

identification

T- = Cf- (j= 1, 2, 3) holds. We define p 0 by

R Re &>(=(Re a> = 0) ) and p by Ci1+Ci3

+ Cp0.

Clearly p i s isotropic.

Then it follows from the conditions (24), (25) and (27) that (38)

p n S R = p0 .

In fact, if we assume afj + b ^ + cco = a'f^ + b'f 3 + c'co

(a, b, c, a', b', c V C ) ,

then we have a = a' = 0 by (24), (36), (37) by operating E(fj, *) and E(f^, *) to the above equality. Then (27) a s s u r e s that b = b' = 0 . Hence we get (38). Next we show that there e x i s t s a Lagrangian C-subspace À of S (i.e., \

i

= \)

such that

54

MASAKI KASHIWARA AND TAKAHIRO KAWAI

(39)

A 3 p

and (40)

A H SR = p H SR

hold. Since ( S ? ) 1 = (Sh?

= S?

holds, S£ C p V p

we can find a Lagrangian C-subspace (41)

ù C p /p

is Lagrangian. Hence such that

Ö H S? = 0

holds. Let A be a C-subspace of p

such that & = A/p holds.

Then

it is easy to s e e that A i s Lagrangian in V . On the other hand, (41) implies (42)

An[(sR+P)npJ-] = p

holds. Since A is contained in p

, (42) e n t a i l s that A fl S D C p holds. n

Hence we find a Lagrangian s u b s p a c e A which s a t i s f i e s both (39) and (40). Using this s u b s p a c e A a s an " i n i t i a l c o n d i t i o n , " we will construct the required fibering of A T*C

n

by

= T C . We denote the fiber coordinate of

Ç.

First, s i n c e fj e A , we can define (f>1 which i s homogeneous of degree 0 with respect to £ so that V1 = \1 = 0\ and that i1 = d1(p ) . Secondly we determine ^

on A

by solving

!^lv 2 = oThen i/fj

is n e c e s s a r i l y homogeneous of degree

and d«A 1 (p*)= f 2 / E ( f 1 ( f 2 ) . Thirdly we define 0 2

on

A

by solving

1 with respect to £>

SOME APPLICATIONS OF BOUNDARY VALUE PROBLEMS

55

(\ifrv2\ = \<ßv2\ = 0 1

(43)

(43.a)

(^ 2 l Vl nv 2 = K

( 43 - b >

Since K i s homogeneous of degree 0 with respect to £,

cf>2

1S

a

^so

homogeneous of degree 0 with respect to C,. Furthermore (43. a) implies that E(f 2 , d 0 2 ( p * ) ) = E(f 1 , dc£ 2 (p*)) = 0 . On the other hand, (43.b) entails that d 0 2 ( p ) is contained in Cf 1 + Cf2 + Cf 3 .

Hence, together with (36),

we conclude that d 0 2 ( p ) e Cf3 . Therefore d 0 2 ( p ) e À . Since À is Lagrangian, we can further determine iA(j = 2, • • • , n ) , which i s homogeneous of degree 1 with respect to Ç,

and <£• (j = 3,---,n),

which i s homogeneous of degree 0 with respect to C,

by solving the

following equations:

0
(44)

d<j,k
(i^j, k i = 5 j k (45)

(l<j,k
d0j(p*)6A

d<j
The canonical coordinate system (cß1, •••, 4>n,
to T * C n .

Since A n s R = R R e c o ,

(46)

T *A P

TnCn u

h a s rank ( 2 n - l ) , which implies that A -> C n has rank 2 n - l in a neighborhood of p . T h i s implies that A is a conormal bundle of a real non-singular hypersurface

N of U ,

where U i s an open neighborhood of 0 of C n .

For the s a k e of the simplicity of the s u b s e q u e n t arguments, we assume that U is a polydisc.

56

MASAKI KASHIWARA AND TAKAHIRO KAWAI

Thus we have constructed the required fibering of A ^

over U . Let

f(z, z") be a defining function of N . Since co\\ = kd z f(z, z~) (k> 0) , d_f(z,z) z

(47)

A0 d d_f z z

dzf(z,z)

holds on N . In particular, this implies that the Levi form of N is nondegenerate. Hence Theorem 4 i s applicable to our problem. Therefore it suffices for us to c a l c u l a t e R r z R Karnes ( £ , 0 ) Q [ - 1 ] to c a l c u l a t e + C cn RK<xmg 0 1 , e M ) * , where £ = $ f(z,z^)>0i

/3)

( z ^ - z 2 ) and Z + = JZ6Ü;

in the coordinate system chosen above. For this purpose we

rewrite conditions (24) and (25) in terms of

f.

F i r s t of all, (25) reads

(48)

{z 1 = z 1 = 0 , f ( z , z ) = 0 } = i$L 1 l {dzl

=^r dz l

=0,f(z,z)=oi. )

We may assume without l o s s of generality that f has the form (49) z n + z n + 0 ( z ) z 1 + 0 ( z ) z 1 + i / r ( z ) z 1 + ' A ( z ) z i + ^ ( z " z n , z", z n ) + 0 ( | z | 3 ) , where z " = ( z 2 , •••, zn_\) form. Define 0 o ( z " )

>

an

d 0

are linear and $

i s a quadratic

and a e C (resp. 0 Q ( z " ) and ß € C ) by

0 ( 0 , z", z n ) = 0 o ( z " ) + a z n (resp. 0 ( 0 , z", z n ) = 0 o ( z " ) + j8z ß ). Then comparing the tangent s p a c e s at the origin of the both hand s i d e s of (48) we find that z n + ^ n = 0 entails 0o(z") + a z n + ß z n + 0o(z") = °

(50)

0 o ( z " ) + azn z

« + z« = 0 . n n

+ ßzn

+ 0o(z") = 0

SOME APPLICATIONS OF BOUNDARY VALUE PROBLEMS

57

Setting z n = 0 in (50), we obtain (51)

0 o (z") + if/0(z") = 0 ,

and hence (52) It follows then that a = ß . Therefore f has the form (53) z n + z n + a z 1 z 1 + b z 1 z 1 + b z 1 z 1 + ( z n + z n ) ( a z 1 + a z 1 ) + 0(z",z n ,z",z n )+0(|z| This means that f = f/(l + az1 +az~1) can be chosen to be a defining function of N which satisfies the additional conditions d2i

dz^dz-

j = 2,---,n (z,z>(0,0)

(54)

d2l dz^dz-

= 0,

j = 2,---,n

(z,F)=(0,0)

We denote this f by f in the sequel. Then, by the aid of Theorem 5, we can easily find the following:

^•^--Tor (55)

KiXÏUO)

ka ( 0 ) 1 ^ ( 0 ) L n (0)

where d

L

z

f

n = d

zf

d

zdzf

58

MASAKI KASHIWARA AND TAKAHIRO KAWAI

iL

iL dz.

dz„ 'n-1

dz dz

\ k

2<j,k
dz_

dH

and

<9z? dz}

dz* dz.

Even though several cases are distinguished according to the signs of ! a ( P ) , a ( P . ) c i 0 = 1 , 2 ) in Theorems 6 and 7, the way of the discussion is the same. So we consider only one case here, say case (Aj ) . First define cßiz',^')

by î(z,~z)\z

=-

=Q

, where z = (z 2 , •••, z n ) .

Note that the condition (25) (hence the condition (48)) asserts that zfU; f ( z , z ) = 0, Of

Mr

coincides with \z = (z1 , z')e\3; Zj = 0, <£(z', z')= 0\.

=0 In the sequel we de-

note by F the projection from C^ to C^~ . (See Figure 1.) .- lf=0| {f<0}= Ç Z (^complement of

C={z x =0,0=0}

%2

Fig. 1.

Z)

SOME APPLICATIONS OF BOUNDARY VALUE PROBLEMS

59

dZ„ ÇZQ (= the complement of ZQ)

n-1 z

H + = {<£<0}

Fig. 2.

Note that in this c a s e (i.e., C a s e A x ) , f(z, z ) has the following form (56)

aZlzx

+ bz\ + bz\ +cj>(z',z')+

0 ( | Z l l \z\2

+ |z1|2|z'|) ,

where a > 0 and a > |b| . We next choose a family of closed s e t s j Z c ! Q < ing conditions (57)-(60) are satisfied. (57)

Z0 = Z+

<1

so that the follow-

(See Figure 2 above.)

and Zx = { z e F _ 1 ( U ' ) n U ; <£(z',z') > 0 | ,

where U' is an open ball in C?7 .

H Zc, = Zc

(58)

c'
(59)

(60)

|7 ^c

is proper.

dZ (c< 1) i s non-singular and non-characteristic with respect to £

outside C .

In the sequel we denote by H

the set {z'flT; 4>(z', ~z) > 0 ! . We also de-

fine U ; (resp. \J8 £ ) by i z ^ C

n_1

, \z'\ < ei (resp. i z = ( z x , z')
60

MASAKI KASHIWARA AND TAKAHIRO KAWAI

\zx\ < 8 , | z ' | < el for 0 < e « < 5 .

Wedenote F ^

by Fg

£

for short.

Then we have (61)

RFS(6*(Rrz+(R Ham (£, 0 -

))) 0

R F S *(RT Z (R Hom ( £ , © ) ) ) „ C

VJ

-

RF 5 > E *(lim R r z ( R J t « n ( £ , 0 C cU

))) 0

-

RFg£*(RrZi(RH
On the other hand, it follows from the choice of Z^ that R r z (R H « „ ( £ , 0 ) )

0

= Jim^ Rr(U £ ; R F 5 > E * R r Z i R K o m ( f , e

))

5,640

=

lim

RT(UE;RrgRFSE*RHam(f,e

8jTo

+

'

)) L

Next by the direct calculation concerning the ordinary differential equation with parameters, i.e., ( z , D 1 - z 2 ) u ( z ) = f(z) , we find (62)

lim

Rr(U E ; RF

8,Ei0

R F g E*R JUm ( £ , © _ ) ) L

+

=0,1,2,---

+

<-

^

On the other hand L n l ( 0 ) / L n ( 0 ) > 0 in Case A r

This implies that

the number of positive eigenvalues of the Levi form associated with iz'(flT; cf>(z'f~z')=0\ iz(fU; f ( z , z ; ) = 0 i .

is decreased by one compared with that of Therefore (62) combined with (12) and (19) entails

(63) g**t <)i,eM) ^X

e £=0flf2,---

HJ(e w ^e w )=

e £=0,-1,-2,•••

&*& ( ^ A ^W

This completes the proof of Theorem 5 in Case A and Theorem 6 in Case A j . Other c a s e s can be proved in a similar way.

SOME APPLICATIONS OF BOUNDARY VALUE PROBLEMS

61

REFERENCES [0]

Hartshorne, R.: R e s i d u e s and Duality. Lecture Notes in Math. No. 20, Springer, Berlin-Heidelberg-New York, 1966.

[I]

Kashiwara, M. and T. Kawai: On the boundary value problem for elliptic system of linear differential equations. I. Proc. Japan Acad, 48, 712-715 (1972).

[2]

: Ibid. II. P r o c . J a p a n Acad., 49, 164-168 (1973).

[3]

: Theory of elliptic boundary value problems and its applic a t i o n s . Surikaiseki-Kenkyusho Kokyuroku, No. 238, RIMS, Kyoto Univ., Kyoto, 1 9 7 5 , pp. 1-59. (In J a p a n e s e . )

[4]

: F i n i t e n e s s theorem for holonomic s y s t e m s of microdifferential equations. Proc. J a p a n Acad., 52, 341-343 (1976).

[5]

Kashiwara, M., T. Kawai and T. Oshima: Structure of cohomology groups whose coefficients are microfunction solution s h e a v e s of s y s terns of pseudo-differential equations with multiple c h a r a c t e r i s t i c s . I. P r o c . J a p a n Acad., 50, 420-425 (1974).

[6]

: Ibid. II. P r o c . J a p a n Acad. 50, 549-550 (1974).

[7]

Maire, H. M. and F . T r e v e s : An article on subelliptic systems (in preparation).

[8]

Sato, M., T. Kawai and M. Kashiwara: (Referred to a s S-K-K [8]) Microfunctions and pseudo-differential equations. Lecture Notes in Math. No. 287, Springer, Berlin-Heidelberg-New York, pp. 265-529 (1973).

[9]

: The theory of pseudo-differential equations in hyperfunction theory. Sugaku, 25, 213-238 (1973). (In J a p a n e s e . )

[10]

: On the structure of single linear pseudo-differential equations. P r o c . J a p a n Acad., 48, 643-646 (1972).

[ I I ] T r e v e s , F . : Study of a model in the theory of complexes of pseudodifferential operators. Ann. of Math. 104, 269-324 (1976).

A SZEGO-TYPE THEOREM FOR SYMMETRIC SPACES Victor Guillemin §0.

Acknowledgements T h i s article extends some r e s u l t s of Widom [7] from rank one to higher

rank symmetric s p a c e s . The proof of Theorem 3, which is the main result of this paper, makes u s e of a beautiful trick devised by Widom to prove the analogous rank one result in [7]. I am grateful to him for letting me make free u s e of r e s u l t s from [7] (which was a s yet unpublished at the time this a r t i c l e was being written) and for providing me with some inspiring d i s c u s s i o n s . I would a l s o like to thank Sigurdur Helgason for valuable advice concerning the material in §4.

§1. A Szego

theorem for Lie

groups

Let G be a compact semi-simple L i e group and H a Cartan subgroup of G.

Let

g and Ij be the L i e algebras of G and H . For jSVï)* let

|/31 be the norm of ß

with respect to the Killing form. From the Killing

form we get an orthogonal projection of g on Ï), and by duality an injection i : ï)* ^ g* . G a c t s on g* by its co-adjoint action. Given y e Ï)* we will denote by O

the G-orbit through i(y) in g*.

On O

there i s

a unique G-invariant measure, n , with the property that u (O ) = 1 .

© 1979 Princeton University Press Seminar on Micro-Local Analysis 0-691-08228-6/79/00 0063-16 $00.80/1 (cloth) 0-691-08232-4/79/00 0063-16 $00.80/1 (paperback) For copying information, see copyright page 63

64

VICTOR GUILLEMIN

L e t r: g* -> Ï)* be the t r a n s p o s e of the inclusion map of Ï) into $ and let v

= r^ß

. We can now s t a t e the main result of this paper:

THEOREM 1. Let

W«, W 2 , ••• 6e a sequence

of vector spaces

G a c / s irreducibly

with

to infinity.

N- = dim W^ tending

maximal weight of Wi . Denote

w/iere j 3 £ s ' , s = 1, •••»Ni Wt

(counted

by v^

the following

are the weights

with multiplicity).

Let

measure

on which ß^

of the representation

Then the following

be the

on h :

of G on

two assertions

are

equivalent: i)

ß\r/\ßv\

tends to a limitas

ii) i^k tends weakly

k tends to <x>.

to a limit as

k tends to <x>.

Moreover, if the limit of (i) is y , the limit of (ii) is the measure, defined

v

,

above.

Kostant proves the following theorem in [5]: ( [ 5 ] , Theorem 8.2).

THEOREM 2. The support of the measure,

v , is the convex

of the orbit of y under the action of the Weyl

hull in \)

group.

It is well known that if an irreducible representation of G has maximal weight, ß , then every weight of the representation is contained in the convex hull of the orbit of ß

under the action of the Weyl group.

Theorems 1 and 2 tell us that, asymptotically, the converse of this result is true. We have depicted some p o s s i b i l i t i e s for the support of v

, in

the c a s e of SU(3) , in the figure below: REMARK. Let \ß}^

be the s e q u e n c e ß^ = kß , ß

being a fixed weight.

In [1], Boutet de Monvel and Guillemin prove a somewhat stronger result

A S Z E G O - T Y P E THEOREM FOR SYMMETRIC SPACES

65

y in the interior of the Weyl chamber

y on the boundary of the Weyl chamber

y = the maximal weight of the adjoint representation

than Theorem 1 for the a s s o c i a t e d s e q u e n c e of 1/1 ' s : Namely, for all f e C°°(t)*)

,/

k(f) = 2 4 r)(f)k ~ r ' r=0

with

v (0) y

= v

, y>

66

VICTOR GUILLEMIN

the i ^

s being distributions on h

supported on the convex hull of the

orbit of y under the Weyl group.

§2. A Szego theorem for symmetric

spaces

Theorem 1 is a consequence of a more general result which we will formulate in this section.

T h i s result also generalizes Theorem 1 of

Widom [7]. Let G be a compact semi-simple L i e group and let X = G/K be a symmetric s p a c e . L e t

g and

Ï be the L i e algebras of G and K

and let $ be the orthogonal complement of f in Q with respect to the Killing form. L e t a be a maximal abelian subalgebra of £ and ï) a Cartan subalgebra of Q containing a . T = Ï) Pi f.

We can write ï) = r®ù

where

It is well known that if V is a s u b s p a c e of L (X) on which G

acts irreducibly and ß e £)* i s the maximal weight of the representation of G on V then ß(t) = 0 for ter;

so ß

i s actually an element of a .

From the Killing form we get a projection of £ on a and hence an injection, a* -> £ * .

Since p * is the cotangent s p a c e of X at the identity

coset, o , an element, y , of a* can be thought of as an element of T Q . The action of G on X lifts to an action of G on T X. We will denote by 0

the G-orbit of yea*

in T * X .

As a homogeneous G-space

0

p o s s e s s e s a unique G-invariant measure, \iy , with the property ß ( 0 )= 1

THEOREM 3. Let which

VpV«,--* be a sequence

G acts irreducibly

with

N^ = dim \A tending

be the maximal weight of the representation the orthogonal

projection

y,

i tends to infinity,

in Q* as

differential

operator,

of subspaces

to infinity.

of L (X) on \A . // ß^/\ß^\

r^

k-><x>

l>k

«= / I • «a(B)d M v '

°y

n-

ß^ be

tends to a limit,

then for every zeroth order

B : L (X) -> L (X) ,

lim

Let

of G on V- , and let

trace 771 B771,

(2.1)

of L (X) on

pseudo-

67

A SZEGO-TYPE THEOREM FOR SYMMETRIC SPACES

where

a ( B ) is the symbol

preceding

of B and u

is the measure defined

in the

paragraph.

REMARKS.

1. In the course of the proof we will show that the left hand side of (2.1) h a s a limit for all B if and only if {j8^/|j8j|l is convergent. 2. If X is of rank one, 0* is one-dimensional. Since ß^ e a * , the condition above on the s e q u e n c e i/3^/|/3j|i jS^/ljS^I = y = the positive unit in

is trivially satisfied since

a*.

The proof of Theorem 3 requires some facts about generalized limits which we will describe in the next section and some results about invariant differential operators on X which we will d i s c u s s in §4.

§3. Generalized

limits

Let £°° be the Banach s p a c e of all bounded s e q u e n c e s

c = ic:i, i e Z + ,

with the norm ||c|| = s u p | c ^ | . L e t £Q be the closed s u b s p a c e of l°° cons i s t i n g of all convergent s e q u e n c e s . The functional LQCICJI) = lim

ci

is a continuous linear functional on £Q . A generalized ous linear functional,

L,

limit is a continu-

on £°° whose restriction to £Q is L Q .

The

following lemma, which i s an easy consequence of the Hahn-Banach theorem s a y s that "sufficient m a n y " generalized limits exist.

LEMMA 3 . 1 . Let ized limits,

h1

c be an element and

L2

suchthat

of l°° . / /

c / lQ,

there exist

general-

L1(c)^L2(c).

Therefore, in order to show that a bounded s e q u e n c e ,

C, is convergent

it is enough to show that for all generalized limits, L , L(c) is a fixed number not depending on L .

68

VICTOR GUILLEMIN

§4. G-invariant

differential

operators

on X = G/K

As in section 2 we will denote by g and f t h e L i e algebras of G and K , by p t h e orthogonal complement of Î in g and by a a maximal abelian subalgebra of p. izes to an injection of Q tion of £

The orthogonal projection of £ on a dual-

into p . Also there i s a canonical identifica-

with T Q , t h e cotangent s p a c e of X at the identity coset.

Let u s denote by C ^ T X t h e s p a c e of smooth G invariant functions on T*X and by C ^ p * the s p a c e of smooth functions on £ * which are invariant under the co-adjoint action of K on p .

LEMMA 4 . 1 .

C~T*X

a

C~*>* .

Proof. Since K i s the isotropy subgroup of G at o , it a c t s linearly on T , and it is clear that under the identification of T * with p * described above this action corresponds to the co-adjoint action of K on £ . T h u s if f i s in C ^ T X , i t s restriction to T

i s in C™$ . This gives us a

map of C ^ T X into C ^ p . It i s e a s y to s e e it is bijective.

Q.E.D.

Let M be the centralizer of a in K and M' the normalizer of a in K. Then M i s of finite index in W ( s e e [ 2 ] , page 244) and W = M'/M acts on a a s a finite group of isometries. W i s , by definition, the Weyî group of (G, K) .

LEMMA 4.2.

C£*)* a

C~a* .

Proof. L e t S K £ and SWQ b e , respectively, t h e rings of K-invariant polynomial functions on p

and W-invariant polynomial functions on a .

It is clear that if f i s a K-invariant function on p

i t s restriction to a

is W-invariant; so there are restriction maps, S K £ -> SWQ and ^K^

~* ^ W a ' I n [ 2 ] , it i s shown that the first map i s an isomorphism

( s e e page 430); and it i s also shown that every K-orbit in $* i n t e r s e c t s a ; so the second map i s injective.

By a theorem of G. Schwartz ([6]),

A SZEGO-TYPE THEOREM FOR SYMMETRIC SPACES

every W-invariant C°° function,

f,

on a

can be written in the form,

f = F(g x , •••, g f ) where F i s smooth and g j j ' - ^ g j . the map from S K p to S w a K-invariant function,

g^,

69

are in S w a .

Since

is surjective, each g- extends to a on p ; so f extends to the K-invariant func-

tion V=F(g\r"9gT).

Q.E.D.

Combining Lemmas 4.1 and 4.2 we get an isomorphism, given by restriction: (4.1) Let T

r: C~T*X "||

s

C~Q* .

| | " be the G-invariant metric on T X whose restriction to

= £ is the Killing metric. L e t S*X be the unit co-sphere bundle of X

and Q j the unit cosphere in a .

LEMMA 4.3. The restriction

Proof.

map is an isomorphism,

r : C^S X -> C^Qj .

The only non-obvious point is the surjectivity of r. Let p be a

smooth function on the real line which is zero for x < ~- and one for x > j . Given f e C ^ Q j , extend f by homogeneity to a* - 0 and set fl(f) = p(lfl)f. rg=f

By (4.1)

3

eC~T*X

g l

r g l = lx . Then

where g = g ! | S * X .

Q.E.D.

LEMMA 4.3. The map r preserves

Proof.

suchthat

Clearly sup|rg| < sup|g|

G-orbit in S X i n t e r s e c t s

and by duality

norms.

for each g e C ^ S X.

to an isomorphism r: c £ s * X -> C ° a *

to an

Since every

a^ , this inequality is an equality.

COROLLARY 4.4. r extends (4.2)

sup

isomorphism

of Banach

spaces

Q.E.D.

70

VICTOR GUILLEMIN

ÎILa* -> l s * X

(4.3) where

5lï w û 1

is the space of W-invariant measures

is the space of G-invariant

measures

on d^

and M^S X

on S X .

Let us denote by D G X the ring of G-invariant differential operators on X. According to [ 2 ] , (page 3 % ) , D Q X is a commutative ring. Moreover, as an abstract ring, D G X can be identified with another, much simpler ring. L e t D w a be the ring of constant coefficient differential operators on the vector s p a c e , a .

W-invariant

Then (loc. cit., page 432)

there is a canonical isomorphism ¥:DGX

(4.4)

-

Dwû

We will not attempt to describe here how this isomorphism i s constructed. We do need, however, one important property of it. (Loc. cit., page 430.)

LEMMA 4 .5. The

diagram

¥

DGX

*Dwa

(4.5)

C oo wa

CGT*X commutes,

the vertical

arrows being the symbol

*

maps.

We will also need later on the isomorphism (4.6)

Dwa

given by the Fourier transform.

s

^ a

(Recall that SWQ is the ring of

W-invariant polynomial functions on a .) Let V be a s u b s p a c e of L (X) on which G acts irreducibly, and let P be an element of D r X .

P maps V into itself, and i t s restriction

A SZEGO-TYPE THEOREM FOR SYMMETRIC SPACES

71

to V commutes with the action of G . Therefore, s i n c e the action of G on V is irreducible, P = y v ( P ) Identity on V . y

v

i s a homomorphism of D G X

called the infinitesmal

character

into the complex numbers and is

of the representation of G on V .

By

(4.4) and (4.6) we get an a s s o c i a t e d homomorphism (4.7)

y * : S w a -> C .

The following is an infinitesmal form of the Weyl character formula:

LEMMA 4.6. Let G on V and let a-'s

ß ea pea*

are the positive

polynomial

function

be the maximal weight of the representation be the restriction

of

a

to a* of ^ 5 - » i where the

roots of Q. Then for every Weyl group

invariant

on a ,

(4.8)

.

v *(p)

= P08 + P ) .

For the proof of this s e e [ 2 ] , (X, §6) and [3].

§5. The proof of Theorem

3

The main ingredient in the proof is an argument due to Widom. (See [7], §1.) L e t Vj, V 2 , ••• be a s e q u e n c e of s u b s p a c e s of L 2 (X)

satisfying

the hypotheses of Theorem 3 and let n^ be the orthogonal projection of L (X) on Vj. L e t L be a generalized limit. If N i = dim V^, then, for any bounded pseudodifferential operator,

B,

(trace n^ B n^) N[~

is uni-

formly bounded; so the generalized limit / t r a c e 77- B n\ (5.1)

is defined. L

L

If

B

is a pseudodifferential operator of order < 0 , it maps

compactly into L ; so (trace n- B n-) N[~

tends to zero a s i tends

72

VICTOR GUILLEMIN

to infinity.

This shows that for B of order zero, (5.1) is a function of

the symbol of B alone. Moreover, if || || is the supremum norm on L , sup | a ( B ) | = i n f i | | B + K | | , K T*X-0

compact!

by a theorem of Kohn-Nirenberg ([4]); so (5.1) is continuous

a s a linear

function of the symbol of B . By the R i e s z representation theorem, there e x i s t s a measure, ß , on S X

(

such that

trace TT; BTT;\

C

s*x T h i s measure is a probability measure s i n c e the left hand of (5.2) is when B = I . Given g £ G let y

1

be the diffeomorphism induced on X

by g . If B is a zeroth order pseudodifferential operator, the operator, B^,

defined by

B*f = yjBCy-iff i s also a zeroth order pseudodifferential operator. (5.3)

Moreover,

a(B)8 = a(B8) ,

the left hand s i d e of (5.3) being the symbol of B translated by g. trace n;B&77^ = trace y*^- Bn-(y~

)

Since

= trace n ^ Bn^

we get from (5.2) and (5.3) I

o(ß)Zdß

s*x

=

l

a(B)dß

s*x

for all zeroth order operators, B ; i.e. for all smooth functions cr(B) on S X. This shows that the measure, ß , i s G-invariant.

To determine

we will evaluate the left hand s i d e of (5.2) on certain G-invariant zeroth order pseudodifferential operators.

ß,

A SZEGO-TYPE THEOREM FOR SYMMETRIC SPACES

C : L 2 (X) -> L 2 (X) be the operator which on each irre-

LEMMA 5.1. Let ducihle

G-invariant

\ß + p\ Identity, ß

subspace,

V , of L (X) takes

Moreover,

G-invariant, for all

the

value,

being the maximal weight of the representation

on V and p half the sum of the restricted self-adjoint,

73

first order elliptic

roots.

Then

of G

C is a

pseudodifferential

positive, operator.

(x,
(5.4)

a(C)(x,£) = \e .

Proof.

L e t A be the standard Laplace-Beltrami operator on the vector

space,

a.

Clearly A e D w a ;

so by Proposition 4.6, there is a

G-invariant differential operator, A ' , on X with the property that on each irreducible s u b s p a c e , V , of L (X), A ' = |j8 + p | Identity, ß

being

the maximal weight. Moreover, by (4.5), the restriction of the symbol of A ' to a* is equal to the symbol of A ; i.e. on c* , a ( A ' ) ( x , £ ) = | £ | 2 . Since a (A') is G-invariant, a ( A ' ) ( x , £ ) = | £ | 2 C = (A')

1/2

on all of T*X.

.

Set Q.E.D.

Let q be an arbitrary homogeneous polynomial of degree k on a which i s Weyl group invariant, and let Q be the G-invariant differential operator on X a s s o c i a t e d with q via the identifications (4.4) and (4.6). L e t Q' = QC

, a zeroth order G-invariant pseudodifferential operator.

By Lemma (4.6) trace T^ QVj

/ß[+P

q(ß{ + p)

1

where ß^

i s the maximal weight of the representation of G on V- . This

proves:

LEMMA 5.2. all

The limit of the sequence

\(trace

n- Q'n-)N^

Q' of the form above if and only if the sequence

gent. Moreover,

if l/3^/|/3^ii

! (trace n^ Q'7r-)Nj~ \ converges

converges

to the limit,

to the limit

\ exists

\ß:/\ß:\ y,

cr(Q')(y).

for

! is conver-

then

74

VICTOR GUILLEMIN

Now let us go back to the G-invariant measure, /x , occurring in the formula (5.2). By (4.5) this measure can be canonically identified with a Weyl group invariant measure, v , on the unit sphere û , C û . The s p a c e spanned by the functions,

q|a* , where q i s a Weyl group invari-

ant homogeneous polynomial on a , i s d e n s e in C w ( a 1 ) ; 5.2, the measure, v,

i s just the delta measure at y .

orbit through y in T X - 0 .

s o , by Lemma

Let O

be the

If a G-invariant function v a n i s h e s on O

its restriction to û* v a n i s h e s at y ; s o the measure, \L , i s concentrated on O . Being a probability measure, it h a s to be the same as the measure, \L , occurring in formula (2.1). To recapitulate, we have proved that for all generalized limits, L :

(

trace n- &n\

(*

°y Therefore, by Lemma 3 . 1 , the s e q u e n c e ! (trace n-Bn^N^

\ converges,

and i t s limit is the right hand s i d e of (5.6).

§6. The proof of Theorem 1 Let G be a compact semi-simple L i e group. Then G x G

a c t s on G

by the action, (a, b) g = agb~ . The isotropy group at the identity element i s G itself imbedded a s the diagonal in G x G .

Therefore, we can think

of G as a ( G x G , G) homogeneous s p a c e , and it i s not hard to show that this homogeneous s p a c e i s in fact a symmetric s p a c e . (See [ 2 ] , page 188.) In L (G) we can consider either s u b s p a c e s which are irreducible with respect to the right G-action or s u b s p a c e s which are irreducible with respect to the G x G action.

We will need

LEMMA 6 . 1 . // V is a G x G

irreducible

subspace,

k

(6.1)

V = 0

W,

k = dim W

A SZEGO-TYPE THEOREM FOR SYMMETRIC SPACES

where

W i s G-irreducible.

Moreover,

every irreducible

75

representation

of

G occurs as such a W .

Proof.

See [8].

We will define on G a commutative algebra of zeroth order pseudodifferential operators a s follows. F i r s t define C : C°°(G) -> C°°(G) to b e the operator which on each G x G is equal to |j8 + p | Identity, ß

—irreducible s u b s p a c e , V , of L (G)

being the maximal weight of the représenta

tion of G on W in (6.1). By Lemma 5.2, C i s a positive, self-adjoint bi-invariant pseudodifferential operator of order one with symbol

(6.2)

o(o(x,o = ia

at all points ( x , ( ) f T

X.

Let I) be a Cartan subalgebra of Q . Given f e C°°(G) and H e Ï), let L H f

be the L i e derivative of f with respect to the left invariant vec-

tor field represented by H . L e t D H b e the zeroth order pseudodifferential operator

(l/V r î)L H C- 1 . The D H ' s

are self-adjoint and commute among themselves; so they gener-

a t e a commutative *-algebra, K , of zeroth order left invariant pseudodifferential operators. If H j j ' - s H j . are a b a s i s of I), then D H

,'",DH

are a s e t of generators of K . Moreover the map (6.3)

SI)-K,

p ( f ) -> p(D H )

from the ring, SI), of polynomial functions on I)* to K is a ring isomorphism. L e t us compute the symbol of p(D H ) for a given p e SÏ). Since p(D H ) is left-invariant, it is enough to compute the symbol at ( T ^ , the unit sphere in T . Identify

(T )j

with the unit sphere, q1 , in q ,

and let r: g* -> Ï)* be the t r a n s p o s e of the inclusion map of Ï) into Q .

76

VICTOR GUILLEMIN

LEMMA 6.2.

The symbol of p(D H ) i s the restriction

back under r of the polynomial

Proof.

function,

p,

to q1

of the pull-

on Ï) .

It is enough to check this for the generators, D H = ( 1 / V - 1 ) L H C~ i i

of the ring K .

However, a(C~

) = a(C) = 1 on Q^ by (6.2) and

a ( ( l / v ^ ï ) L H . ) ( e , £ ) - < H i , ^ > = pull-back of ej{.

So, for D R

the a s s e r -

tion is true.

Q.E.D.

Let W be a vector s p a c e on which G a c t s irreducibly. 6.1 there e x i s t s a s u b s p a c e , V , of L (G) on which G x G

By Lemma a c t s irreduci-

bly such that W s i t s in V a s a summand of type (6.1). L e t n„ orthogonal projection of L (G) onto V .

be the

Each element, p ( D H ) , of K

preserves V and preserves the direct sum (6.1). It also preserves each weight space, W ^ , of W and on this weight s p a c e is equal to

p

(^i) , d e " , i , y

ß ( s ) e Ï)* being the weight a s s o c i a t e d with W ( s ) .

^

Thus

trace 7r v p(D H )7T v

j

^

/ )3(s)

dimV

dim W ^

\|j8 + p|

J

the sum taken over all the weights of W , each weight counted with appropriate multiplicity. Let Wp W2, ••• be a s e q u e n c e of irreducible representations of G with N^ = dim W^ tending to infinity.

L e t Vj, V 2 , ••• be a corresponding

sequence of Gx G-irreducible subrepresentations of L (G). L e t ß-

be

the maximal weight of \A , and assume ß1/\ß-^

By

tends to a limit, y .

Theorem 3

(6.5)

lim trace nv p(DH)77-v Î = :n--L dim V4

~ a(p(D H ))d M . ^~H"^y

|

°y

A SZEGO-TYPE THEOREM FOR SYMMETRIC SPACES

where 0

i s the G x G - o r b i t o f

y in T G and /x

Gx G-invariant probability measure on 0

77

the unique

. To evaluate the right hand

s i d e of (6.5) we need

LEMMA 6.3. Let

0'

of G and let

be the unique

[L

/ / f i s a continuous striction

to T * = 8*,

be the orbit of y in

G-left-invariant

under the co-adjoint

probability

function

action

measure on O' .

on T G - 0 and f its re-

then

|

fd/zv = J

o

fd^ .

~o'

y

Proof.

G-invariant

J '^-J

(6.6)

q

y

The ring of continuous G-left-invariant functions on T G - 0 is

isomorphic with the ring of continuous functions on T hand s i d e of (6.6) defines some measure on q - 0 .

- 0 , so the left

T h i s measure is

obviously a probability measure and is obviously supported on CT . Q.E.D. Combining (6.4), (6.5), (6.6) and the formula for cr(p(D H )) in Lemma 6.2, we get

lim

dï^2,pfeT7î)= ^i^p\\ßl+p\) = JI , '*PS =
for all polynomial functions,

p , on Ï)*. Since the polynomial functions

are d e n s e in the continuous functions, this concludes the proof of Theorem 1.

BIBLIOGRAPHY [1] L. Boutet de Monvel and V. Guillemin, "Spectral theory for Toeplitz o p e r a t o r s / ' (to appear). [2] S. Helgason, Differential Geometry and Symmetric P r e s s , New York (1962).

Spaces,

Academic

78

VICTOR GUILLEMIN

[3] S. Helgason, " A duality for symmetric s p a c e s with application to group r e p r e s e n t a t i o n s / ' Advances in Math. 5, 1-154 (1970). [4] J. J. Kohn and L. Nirenberg, "An algebra of pseudodifferential o p e r a t o r s / ' Comm. P u r e Appl. Math., 18, 269-305 (1965). [5] B. Kostant, " O n convexity, the Weyl group and the Iwasawa decomp o s i t i o n / ' Ann. Sei. E c . Norm. Sup. 6, 413-455 (1973). [6] G. Schwartz, "Smooth functions invariant under the action of a compact group/ ' Topology, 14, 63-68 (1975). [7] H. Widom, " E i g e n v a l u e distribution theorems in certain homogeneous s p a c e s / ' (to appear). [8] H. Weyl, The Classical New J e r s e y (1956).

Groups, Princeton University P r e s s , Princeton,

SOME MICRO-LOCAL ASPECTS OF ANALYSIS ON COMPACT SYMMETRIC SPACES Victor Guillemin §1. L e t X be a compact Riemannian manifold, A the Laplace-Beltrami operator on X and y C X a periodic geodesic. It is part of the folklore of spectral theory that the spectrum of A contains a sequence of eigens p a c e s which are in some s e n s e " c o n c e n t r a t e d " on y . [3] and [6] one can find r e s u l t s about "quasi-modes,"

For instance in

functions which are

asymptotically eigenfunctions of A , concentrated on y . A result of a more p r e c i s e nature was discovered by us a couple of years ago: Let X = S , the standard unit sphere in R 3 , let R^ : S 2 -> S about the z-axis and let y C S

be rotation

be the geodesic lying in the x, y plane.

The eigenvalues of A are the integers, k ( k + l ) , k = 0, 1, 2, •••, the k-th eigenvalue occurring with multiplicity 2 k + l . there is a normalized eigenfunction, with the property that R^<£ k = e

lkö

cp^,

In the k-th

eigenspace

unique up to constant multiple,

<£ k for all de(Q,2n\.

L e t n be the

orthogonal projection of L (S ) onto the s p a c e spanned by the <£u's. Then n is pseudolocal and h a s i t s singular support concentrated on y ; so the <^k's do, in fact, live on y micro-locally. In this paper we will extend the result we have just described to an arbitrary compact symmetric s p a c e , X . In our generalization, the Laplace operator gets replaced by the whole ring of invariant differential operators

© 1979 Princeton University Press Seminar on Micro-Local Analysis 0-691-08228-6/79/00 0079-33 $01.65/1 (cloth) 0-691-08232-4/79/00 0079-33 $01.6 5/1 (paperback) For copying information, see copyright page 79

80

VICTOR GUILLEMIN

on X and geodesies by flat, totally geodesic submanifolds.

In order to

s t a t e our result we will need a little notation. L e t G be a compact, semi-simple L i e group, let K b e a c l o s e d subgroup of G and let X be the homogeneous s p a c e , G / K . and K.

Let

q and f be the L i e algebras of G

Then q = f ©p

(Cartan decomposition)

where p i s the orthogonal complement of Î in g with r e s p e c t to the Killing form. X is a symmetric s p a c e if [p,$] C f . From now on we will assume this to be the c a s e . Let a be a maximal s u b s p a c e of p with the property, [a, a] = 0 , and let \) be a Cartan subalgebra of 0 of the form {) = T®a with T C Ï . A function,

<£ , on X is called a conical

function

if it is the maximal weight vector in some irreducible sub-representation of L (X). (Note that the maximal weight vector in each irreducible subrepresentation of L (X) is fixed, up to scalar multiple, by the choice of Ï).) The irreducible s u b s p a c e s of L (X), and hence the conical functions, are indexed by the points of a

fl £

, a

being the positive Weyl cham-

ber in a and £ C a a certain co-compact lattice. We will denote by

0o

the conical function indexed by the point, ß , in a

be

the closed s u b s p a c e of L orthogonal projection of L

H£

. Let H

spanned by the ß's , and let n b e the onto H . T h i s is the obvious analogue of

the projection operator described above in the c a s e X = S . In §3 we show that for each open sub-simplex, S, of the positive Weyl chamber, a

, there corresponds a s u b s e t ,

S = Int a

X s , of the singular support of n . For

, X s is a flat totally geodesic submanifold of X and for

S C da, , X s is an orbit of the parabolic subgroup, Gg , of G

associated

with S. The wave-front s e t of n has an analogous decomposition; to each subsimplex,

S , corresponds a component, S s = X g x S , of the wave-

front s e t lying above Xg in the cotangent bundle of X. We will denote by S Q the component of the wave-front s e t a s s o c i a t e d with S = Int a . In §5 we show that on 2

, n i s pseudolocal, in fact, even a pseudo-

differential operator of Hermite type. The proof of this requires r e s u l t s

81

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

from the paper [2] of Boutet de Monvel and Sjostrand which we briefly review in §4. In §8 we d e s c r i b e , without proof, similar r e s u l t s for the lower dimensional s u b s i m p l i c e s of the positive Weyl chamber. While we were writing this paper, we learned from J o e Wolf about some r e s u l t s of h i s student, W. L i c h t e n s t e i n , on the asymptotic properties of R , ß e Int a , a s

|/31 tends to infinity.

We noticed that t h e s e results

(or at l e a s t r e s u l t s very much like them) could be derived easily from the micro-local properties of n described above. We indicate how this is done in §6, and in §7 s u g g e s t a way of interpreting t h e s e results microlocally. T h i s paper would never have got written without the a s s i s t a n c e of Sigurdur Helgason, David Vogan and Gregory Zuckerman. I would like to e x p r e s s to them my heart-felt gratitude.

§2. The defining

equations

L e t g = ï© V - ï P ^

of H e

the non-compact form of the L i e algebra,

L e t n be the maximal nilpotent subalgebra of g #

position, Q = î © \ / - ï û © n .

#

valued) vector field,

v', on X .

(2.1)

in the Iwasawa decom-

Let m be the centralizer of a in

Since G a c t s on X each element, v , of g

q.

k.

determines a (complex-

Let

D v : C°°(X) -> C°°(X)

be the differential operator, f -> 1/V-Ï v'f.

Consider the system of

equations: (2.2)

Dvf = 0 ,

v€Ti

and consider also the system of equations obtained by adding to (2.2) the supplementary equations (2.2)'

D f = 0,

V6m .

82

VICTOR GUILLEMIN

PROPOSITION 2 . 1 . The space of all smooth solutions with the space of all smooth solutions L, -closure

Proof.

of (2.2) i s

identical

of (2.2) plus (2.2)', and the

of this space is H .

It i s enough to check that (2.2) and (2.2) plus (2.2)' have the same

solutions in each irreducible s u b s p a c e , V , of L ( X ) . Since V s i t s inside of L (X) , it contains a K-fixed vector by the Frobenius reciprocity theorem; and it i s well known ( s e e [12]) that if a s p a c e on which G a c t s irreducibly contains a K-fixed vector, i t s maximal weight vector i s , up to a constant multiple, the unique solution of (2.2), and is also a solution of (2.2)'.

Q.E.D.

P R O P O S I T I O N 2.2. n

satisfies

(2.3)

D V TT = TTDV

for all v e û .

Proof.

It i s enough to check (2.3) on each irreducible s u b s p a c e ,

V , of

L ; however (2.3) holds identically on the maximal weight s p a c e of V and is zero on the other weight s p a c e s .

Q.E.D.

Under the action of a d ( a ) , n breaks up into a direct sum of onedimensional s u b s p a c e s , n ^ , i = l , - - - , N

where N i s the dimension of n .

Moreover, there e x i s t s an element, a- e a* such that if v^ i s a non-zero vector in n(2.4)

[H, V i ] = \ T Ï

öi(H) Vi

for all

Hea .

The a ^ ' s are called the restricted positive roots of G . L e t a subset of û on which all the a - ' s are > 0 , and let ( a ) of a

be the

be the image

in a* under the identification of a with a* given by t h e Killing

form. Consider the operators

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

83

A v = D V (A + 1)- 1 / 2

(2.5) for wed,

A being the Laplace-Beltrami operator on X . T h e operators,

(2.5), are bounded, self-adjoint and commute with each other. It i s p o s s i b l e to c h o o s e a b a s i s ( v j , •••, v f ) of a with the property that

Q

+ ={2CiVi'Ci-°}*

L e t f be a smooth function on a

and let f be the function on R r

obtained by composing f with the mapping R r -> a*,

(clt-~,cx)

-> (c1v1 + — + cTvT) .

T h e spectral theorem allows us to form the operator, f(Aj, •••, A f ) a s an operator in the uniform closure of the ring of operators generated by Alf'-fAr.

Set

f(A)=f(A1,...,Ar).

LEMMA 2 . 3 . f(A) / s a zeroth order pseudodifferential its symbol

Moreover,

is:

(2.6)

Proof.

operator.

a(f(A)) = f(a (A x ) , - . , a ( A f ) ) .

See [5].

PROPOSITION 2.4. F o r all

f e C°°(a*) with support in the

complement

of ( a + ) * . (2.2)"

Proof.

f(AV = 77f(A) = 0 . L e t V be an irreducible subrepresentation of L 2 ( X ) , yf^\ ß the

maximal weight of the representation of G on V and v a non-zero vector in the maximal weight s p a c e . Then (A + l ) _ / 2 v = cv with c > 0 and f(A)v = f(c/3)v.

Since ß

is in (a ) * , the right hand s i d e of this equa-

tion is zero for all f with support in the complement of a*.

84

VICTOR GUILLEMIN

§3. The characteristic

variety

of the defining

equations

of H

Let x be an arbitrary point of X represented by the c o s e t , G/K.

gK, of

We recall that the infinitesmal action of g on X a s s o c i a t e s to

each veQ

a vector field, v ' , on X. The map

(3.1)

8

sending v onto v'(x) i s surjective. isotropy group at x .

Its kernel is the L i e algebra of the

Since the isotropy group i s g K g -

, the kernel of

(3.1) is (Adg)f . T h e cokernel can, therefore, be canonically identified with (Adg)p . In other words there are canonical identifications (3.2)

*x

—

Tx

a

(Adg)p

the identification of T * with T x being by means of the inner product on T

coming from the Killing form on (Adg)p . (Note that, s i n c e AdK

maps $ into itself, (adg)p depends only on x = gK , not on the choice of g . ) Given x = gK, (3.3)

let 2 X - (Adg)p n a + - 0 .

B e c a u s e of (3.2) we can think of 2 subset of

a,

both a s a s u b s e t of T x and a s a

L e t 2 be the set-theoretic union of all the 2 x ' s .

Then

we have canonical imbeddings T*X (3.4) Xxa, Let us identify

2

with i t s image in T X .

PROPOSITION 3 . 1 . 2 equations,

is the characteristic

(2.2) plus (2.2)' plus (2.2)".

variety

of the system

of

85

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

Proof.

Let g

b e the non-compact form of g . It is well known that

\ / ^ l û is orthogonal to n in the Iwasawa decomposition: g # = k © \ / ^ î û © n ; s o a and n are orthogonal in g <8> C . Since a i s in p and m i s in Ï ,

a and m are orthogonal in g . A simple dimension

count now shows that the orthogonal complement of a in g is the s p a c e

m©gn(n©n)

(gHa 1 ? \/^T(v-v) for all v
Therefore, by (3.1) and (3.2), the characteristic variety of the system of e q u a t i o n s , (2.2) plus (2.2)', at x = gK i s (Adg)p PI a .

If we throw in the

additional equations, (2.2)", then by (2.6), this gets cut down to (Adg)t>Da+.

Q.E.D.

T h e most convenient way of d i s c u s s i n g the structure of 2 i s a s a s u b s e t of X x a , . Given a e a^ - 0 , let G . be the centralizer of a in G and let X Q b e the orbit of G Q in X containing the identity coset.

PROPOSITION 3.2. 2 /s the union, over all

a e a + - 0 of the

sets

Xaxlai. Proof.

L e t x = gK and let a Q b e in the intersection of a

with

(Adg)£ . T h e map K x Q+ ^ p sending (k, a) onto (Ad k)a i s surjective, ( s e e [7], p. 381); s o we can choose g in the c o s e t gK ; so that aQ e (Ad g) a . Consider now a Q and (Adg)~ a Q a s elements of the Cartan subalgebra, \) = r®a, Since a

of g .

i s the intersection of a with the positive Weyl chamber,

Ï) ,

of I), both a Q and (Ad g)~ aQ lie in l) + . But conjugate elements of Ï) lying in the same Weyl chamber are equal, ( s e e , for instance [10], p. 61); so g c e n t r a l i z e s

aQ .

Given two elements, a1

Q.E.D. and a 2 , of a , we will say that a1

is

86

VICTOR GUILLEMIN

equivalent to a2(al

~

a

e

2^ ^ ^

set

°^

roots

which vanish on a^

is

identical with the set of roots which vanish on a 2 . The relation, a

l ^

a

2 ' *s

eas

^y

seen

to be an equivalence relation, and i t s equiva-

lence c l a s s e s the open subsimplices of a

. (We will regard Int a

as

being, itself, such a subsimplex.)

LEMMA 3.3. // a1 ~ a 2 , then

G Q = GQ

ancf vice versa.

Proo/. See [7], page 249. If S is an open subsimplex of a group, G_a

0

, we will define G g to be the

, a n being an arbitrary element of S. By the lemma, this

definition does not depend on a Q . We will let X g be the orbit of G g

in

X containing the identity coset. From Proposition 3.2, we obtain PROPOSITION 3.4. 2 simplices,

S, of

a

/ s equal to the disjoint , of the sets,

union over all open

S s = X g x S, in X x a

sub-

.

From the results above we can already make some non-trivial conclusions about the projection operator, n : L (X) -> H (X). We recall that if X and Y are manifolds and A : C^(X) -> C~°°(Y) a continuous linear operator, there e x i s t s a distributional function, called the Schwartz

eA(x,y),

on X x Y ,

kernel of A , such that

e.(x,y)f(y)dy • J «A«.

(Af)(x) .

The wave-front

set of A is the s e t of points, (x,
such that (x,
If A is regular then it maps C^(X) into C°°(Y),

maps

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

87

C Q ° ° ( X ) into C °°(Y), and h a s a transpose with the same properties. (See [9], §2.4.) Consider now the operator, n,

above. From the results of §2, we

can immediately conclude that the wave-front s e t of n is contained in 2x2;

however, we can conclude a little bit more b e c a u s e of Proposition

2.2. L e t p : 2 ^ Q + be the projection (x, a) -> a and let 2

be the

fiber product of this mapping with itself.

PROPOSITION 3.5. The wave-front

Proof.

Let (x,
7j are in a

set of n is contained

be in the wave-front s e t of n.

in 2

.

By (3.3),
; and, by Proposition 2.2, (
( , ) being the Killing form. Hence é; = 77.

COROLLARY. on the set

The operator,

n,

Q.E.D.

is regular and is a smoothing

operator

X - U Xg .

REMARK. We will s e e later on that the wave-front set of n i s , in fact, considerably smaller than the s e t , 2 NOTATION. We will denote by 2 Clearly (3.5) 2

=U2

. the fiber product with itself of

, union over the subsimplices of

§4. Some facts about Szego

2 g ->S

a .

kernels

To obtain more p r e c i s e micro-local information about n,

we will re-

quire some general facts about reproducing kernels defined by degenerate elliptic equations.

The r e s u l t s we are about to describe are due to Boutet

de Monvel and Sjostrand and can be found in §2 of the paper [2]. L e t X be a smooth manifold and let D j , • • • , D N

be a collection of

first order pseudodifferential operators on X . We will say that the system of equations (4.1)

Dxf = 0 , . . . , D N f = 0

VICTOR GUILLEMIN

88 i s in involution

if there exist zeroth order pseudodifferential operators,

QK , such that

[D i(Dj ] = 2 Q ^

k

for all 1 < i, j , k < N . L e t a^ be the symbol of D i , and let 2 be the s e t of points in T*X - 0 satisfying:

^ ( x , £ ) = 0, •••, <7N(x, £ ) = 0 . At

each point (x,
(4.2)

l/v^îla^âjKx,^).

The results of Boutet-Sjostrand concern the micro-local behavior of the system (4.1) at points where the Levi form is positive definite.

It is a

simple exercise in symplectic geometry to show that S i s a manifold of codimension- 2N at such points. It is a l s o easy to s e e that 2

is sym-

plectic at such points: the restriction to 2 of t h e symplectic form on T*X is non-degenerate. To describe t h e results of [2] we need to introduce a little terminology Let X be a d i f f e r e n t i a t e manifold and A.

and A 2 continuous linear

operators on C~°°(X). Given an open conic s u b s e t , £ , of T*X - 0 we will say that A. and A 2 a r e equivalent every generalized function,

on C (A. ~ A 2 on L ) if for

u , with wave-front s e t contained in (2

(A1 - A 2 )u i s smooth.

THEOREM 4 . 1 . Let

(x, £ ) be a point in 2 at which the Levi form (4.2)

is positive

Then there exists

definite.

(x,
(i)

77 ~ ^

(ii)

I ~ 77+^LiDi

a conic neighborhood,

n, L j , •••, L N

(2, of

suchthat

£

on £

(iii) D ^ ~ 0 on C for all i . Moreover, Proof.

n is uniquely

determined

See [2], Theorem 2.14.

up to equivalence

by these

properties

89

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

A companion theorem to Theorem 4.1 describes more precisely the nature of the operator, n. t e manifold,

Suppose, in general, we are given a differentia-

X and a conic symplectic submanifold,

2,

of T X - 0 .

In [1] Boutet a s s o c i a t e s with the pair (X, 2 ) a natural c l a s s of symbols K k (X, 2 ) which are of type S k ^

on all of T*X - 0 and are of type

_

S °° outside every conic neighborhood of 2 . T h e s e symbols are called Hermite symbols and the s p a c e of operators a s s o c i a t e d with them: O P K k ( X , 2 ) , are called Hermite operators.

Boutet proves in [1] the

following facts about t h e s e operators. 1. A e OPK => WF(A) i s contained in the diagonal in 2. If A e OPK

and Q is an ordinary pseudodifferential operator of

order £ then AQ and QA are contained in 3. If A e O P K k

2x2.

opK k + £ .

and B e OPK £ then AB e O P K k + £ .

4. If A e O P K k , A t e OPK k . In [4] it is shown that Hermite operators have intrinsically defined leading symbols. T h e theory of t h e s e symbols involves the metaplectic group and the " symplectic s p i n o r s " of Kos tant. An example of a Hermite operator which "occurs

in n a t u r e ' ' is the following.

pseudo-convex domain in C

n

and let H

Let X be a strictly

be a L -closure in L 2 (dX) of

the s p a c e of C°° functions on dX which are restrictions of holomorphic functions on X . Associated with the contact structure on dX is a conic symplectic submanifold,

2 , of T dX - 0 . In [2] Boutet and Sjostrand

show that the Szego projector, n : L (dX) -> H (dX),

is a Hermite operator

in fact they show that n belongs to OPK (dX, 2 ) . More generally if q e C°°(dX) the Toeplitz operator f € H2

-> 77 qf

belongs to OPK ( d X , 2 ) . Getting back to the system of equations (4.1) Boutet and Sjostrand prove

THEOREM 4.2. The projector OPK°(X,2).

n described

in Theorem 4.1 is in

90

VICTOR GUILLEMIN

§5. Micro-local

properties

of n in the interior of the positive

Weyl

chamber

We recall that the nilpotent algebra, n , can be decomposed into a direct sum of one-dimensional p i e c e s

n

= Sni

such that n- is Ad(û)-invariant and (5.1)

[ H , V i ] = yfTai(U)vi

for all

H 6a

V: being any non-zero vector in n- . (See (2.4).) To each a- € a corresponds a unique vector H | 6 û (H^ H) , - (

such that for all H e a , a^(H) =

, ) being the Killing form.

LEMMA 5.1. // v^ is a non-zero vector in r i j , then with

there

l/\/-\ [v^, v^] = c^H^

ci > 0 .

Proof. See [7], page 246.

COROLLARY. One can choose a basis holds and, in

of n such that (5.1)

addition,

(5.2) where

v x , ••*, v N

l / V ^ I t v ^ V j ] = U{ (U[t H) = a{(U)

for all

H 6 a.

Let P be the differential operator: N

P =V ^

Dt D . v

i

v

i

i=l

PROPOSITION 5.2.

There exists

f e C°°(X) is perpendicular

a constant,

C > 0,

to the kernel of n , then

such that if (Pf, f ) > C ||f \\\ .

91

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

Proof.

By Lemma 5 . 1 , v^, v^ and yj^î H^ satisfy the bracket relations [ v ^ v M H ^ a ^ ,

where a{ = (H if Hj) > 0 .

[v^^TÏH^-a^,

Set v + = l / > / â j V i , +

z= l/a-\/-ÎH.

Then v ,

(5.3)

[v+,z] = v+,

i.e. v + ,

[ v ^ ] = y/^î^

,

v" = 1/Vâj Vj and

v~ and z satisfy the bracket relations [v"fz] = - v - f

[v+,v-] = z ;

v~ and z are the standard b a s i s for sl(2, R ) . For the follow-

ing lemma, s e e [11].

LEMMA 5.3. Let module.

V^ m ' bean

m+l-dimensional

irreducible

sl(2, R)

Then m

y(m)

=

^

y(m)

i=0

where

z is equal to (m-2i) times the identity m)

equal to i ( m - i + l ) I on v[

on V^

and

v~v +

is

.

Let V be any irreducible s u b s p a c e of L (X). Under the action of a d ( û ) , V decomposes into a direct sum of one-dimensional weight s p a c e s

2 v ßß -

\ ' max © * ^

V

v

being the maximal weight s p a c e . If f is a non-zero element of Vp ,

then for some V| , D (5.3) we get

i

f ^ 0.

Applying the lemma to the rescaled vectors

(DtvDv.f,f)=

||Dv.f||2>ai||f||2

For the rest of the v - ' s we have (D^.D f , f ) = J J

||Dvf||2>0; j

92

VICTOR GUILLEMIN

so with C = min a^ we get (Pf,f)>

C||f|| 2

for all f in the orthogonal complement of V m a x

in V .

Since L (X) is

a direct sum of its irreducible s u b s p a c e s this proves Proposition 5.2. Q.E.D. Let A be the standard Laplace-Beltrami operator on X. For every irreducible s u b s p a c e , V of L (X), A maps V into V and, in fact, is equal to a constant multiple of the identity on V . Hence A commutes with P .

Let Q be the unique operator which is equal to zero on the

range of n and equal to P _

on the Kernel of n. 9

By Proposition 5.2,

9

Q is a bounded operator from L

to L .

PROPOSITION 5.4. Q is a bounded operator from H s and is regular in the sense

Proof. to H s .

so does Q . However

(A+l)s/

isomorphically onto H s ; so Q is a bounded operator from Suppose (x,f, y, 77) is in the wave-front s e t of Q .

QA, <7(A)(x f £) = <7(A)(y f i/) f

i.e.

|£| = |*/| . In particular if £ = 0 , Q.E.D.

Define the operator L i : C°°(X) -> C°°(X) to be the operator: L^f = QD

In addition, of course, n (5.5)

Hs

Since AQ =

77 = 0 and vice-versa.

(5.4)

s

of §3.

Since P commutes with A ,

maps L

to H s for all

f.

Then

Identity = n + ^

L- is regular and s a t i s f i e s

LiDy .

satisfies TT = TT2 = TT1

93

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

and (5.6)

DV>7T = 0

Therefore,

for

i=l,...,N.

n s a t i s f i e s the conditions (i), (ii) and (iii) of Theorem (4.1);

(in fact, it s a t i s f i e s t h e s e conditions globally not just micro-locally). We will now u s e the uniqueness assertion in Theorem 4.1 to draw some conclusions about the structure of n in the interior of the positive Weyl chamber. F i r s t , however, we will need some notation. Let S Q = Interior

û, , let G^ = GbQ + ° o

and let X^ = XQb . Let A be the connected ° o

L i e subgroup of G having a

for its L i e algebra. The group, A ,

compact ([7], p. 210) and abelian. Let M be its centralizer in LEMMA 5.5. G = AM and identity

Proof.

is

K.

XQ /s the orbit of A in X containing

the

coset.

For the first assertion, s e e [7], Chapter VII.

If o is the point in X representing the identity c o s e t , then M is contained in the isotropy group K , of o ;

so X = AM • o = A • o . Q.E.D.

The group A PI K is abelian and is also discrete s i n c e ûfl Ï = \0\. Therefore

x 0 = A/A n K is a compact abelian group of the same dimension a s A . One can show that XQ , viewed a s a submanifold of X , is flat and totally geodesic. Moreover, every flat, totally geodesic submanifold of X of the same dimension a s A is of the form, gX Q , for some g e G. (For the proof of t h e s e a s s e r t i o n s s e e [7], page 210.) Consider now the open subset, 20=XQxS0,

of 2 .

We will prove:

THEOREM 5.6. / / ( x , f ) e 2 (5.7) has positive

the system

Dv.f=0f definite

of

equations

i=lf...fN

Levi form at (x,
94

VICTOR GUILLEMIN

Proof.

The Levi form at (x,
(5.8)

lA/^KUv^Vj]),

If i ^ j , [v^,"v-] is in n © Ï Ï ;

1 < i,j < N .

s o , with f

in û , (5.8) is zero. If i = j ,

then (5.8) becomes just (f, H^) by Lemma 5 . 1 . Since £ e Int a+ , (5.8) i s positive when i = j . T h i s shows that (5.8) is a diagonal matrix with positive entries along the diagonal.

COROLLARY 1. The characteristic (5.7) contains

Proof.

S Q as an open

Q.E.D.

variety

of the system

of

equations

subset.

Since the Levi form is non-degenerate at (x,
variety of the system (5.7) i s locally a submanifold of dimension equal to twice the dimension of X minus twice the dimension of n . It is easy to check that this is also the dimension of 2

COROLLARY 2. Let

.

( x , f ) and (y, 77) be elements

(x,f, y, 77) e WF(77-) and either

of 2 .

//

(x,
(x,f) =

(y» 1) •

Proof.

Suppose ( x , f ) f X Q . By Proposition 3.5, f = 77, s o (y,r/) is

also in 2

. It remains to show that x = y . If this were not the c a s e ,

then by Proposition 4.1 we could find a conic neighborhood, &1 , of (x,f) (i)

not containing (y, 77) and operators, n', n ~ (TO2 -

L/1,*-*,L/N

suchthat

00*

(ii) I ~ ^ + 2 ^ ^ and (iii) Dn\

~ 0,

i= 1 , - - , N

on C j . Let C 2 be a conic neighborhood of (y,77) not intersecting &1 . By (5.4), n ~ - S D ^ . L i on ^ x ^ .

In addition, 7T'D[ ~ ( D ^ T T ' ) * - 0

95

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

on &1 . Therefore, we conclude TT ~

on C.xC2

(jr'+ S L ^ D y ) TT ~

TT'TT ~*> TT*

. T h i s proves that TT is smoothing on ( ^ x(2 2 which is a

contradiction.

Q.E.D.

COROLLARY 3. On S Q , n is micro-locally

in

OPK0(X,SQ).

We will next show that we can modify TT a little on the boundaries of the Weyl chamber so a s to get an operator which is globally 0

OPK (X,2O). with

in

To start with, we will u s e the Killing form to identity

a* . Let f be a function

on

a

a+

which is smooth except at the

origin, homogeneous of degree zero, and vanishes on a neighborhood of dû

- 0 . If V o is an irreducible s u b s p a c e of L (X) and ß

the maxi-

mal weight of the representation of G on V ß , then ß e û+ . Define an operator TT^ : L 2 (X) -> L 2 (X) as follows:

'ß

1) If V Ö is a non-zero maximal weight vector in V = Vg

set

2) If V is orthogonal to the maximal weight vector in V set 7Tf(V)=0.

Together, 1) and 2) define n^ unambiguously on all of L (X).

This

operator is a kind of smoothed out version of TT in which the pathologies occurring at the boundary of

THEOREM

Proof.

û

+

have been eliminated.

5.7. ni belongs to OPK°(X,S Q ).

Let f(A) be the operator defined at the end of section 2. (See

Lemma 2.3.) By Lemma 2.3, f(A) is an ordinary pseudodifferential operator of order zero. We will show that the wave-front s e t of f(A)7r is concentrated on 2

. To s e e this, let g be any homogeneous function order

96

VICTOR GUILLEMIN

zero on û which is smooth except at the origin. If f and g have nonoverlapping supports (5.9)

f(A)g(A) = 0 .

Let ( x , f ) be a point of S - S Q .

We can choose g such that g(f ) ^ 0

and g and f have non-overlapping supports. By 2.6 g is elliptic at (x,
. From Corollary 3 of

Theorem 5.6 we conclude that f(A) 77 e OPK (X, 2 ) . We will now show that 77f = f(A) 77. To s e e this let Vg be an irreducible subrepresentation of L (X) with maximal weight ß

and maximal weight vector v n .

(A-+ 1)~ VU = C V D . Then, by definition of

f(A)nvß

Let

f(A),

= f(A)Vjg = fCcjS)^.

Since f is homogeneous of degree zero the term on the right is

f(/3)vo

or TTcVß. Both 77r and f(A) 77 are zero on the orthogonal complement of V/D in V U ; so they are identical on V o .

V D being arbitrary, they must

be identical on L (X).

§6. Some asymptotic properties positive Weyl chamber

of conical

functions

on the interior of the

Let us introduce the following notation. If V is an irreducible subrepresentation of L (X) with maximal weight, ß , let v o be its maximal weight vector. We will normalize v n so that its L

norm (as an element

2

of L ( X ) ) is one. It is well known that every irreducible representation of G which occurs as a subrepresentation of L (X) occurs with multiplicity one; so v g is determined completely by ß

up to multiplicative

constant of modulus one. To distinguish between VD regarded a s a vector and VD regarded as a function on X we will denote the function which v o represents by cj>o. ß's are the conical Let ffl be a conic s u b s e t of Int û properly contained in Int a

functions

whose closure in

on

X.

û - 0 is

. We will show that the conical functions

97

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

have the following asymptotic properties. (The first two of these propert i e s were discovered by W. L i c h t e n s t e i n . )

PROPOSITION

6.1. Let

q be in C~(X-X Q ).

Then

J q\ß\2x = 0(101- k )

(6.1)

X

for all

ß 6 ffi and

PROPOSITION

k> 0.

6.2. Let q be in C°°(X). Then

(6.2)

qdx o+ 0(|/S|- 1 )

J ql^^dx = f X

X o

for all

ß e W , dx

COROLLARY. S ÄY

, as

o

ß

being normalized

The function,

tends to infinity

Haar

measure.

\ß\ converges

to the delta

function,

in ffl .

Proposition 6.2 can be generalized:

PROPOSITION 6.3. Let differential

(6.3)

Q : L 2 (X) -> L 2 (X) be a zeroth order

operator with leading symbol,

J

cr(Q). Then for all

pseudoß e ffi

a(Q)(x,iS)dx+ OOiSr1) .

Q0j80j8(x)dx = f o

We will first prove Proposition 6 . 1 . Let f be a smooth function on û + - 0 which is homogeneous of degree zero, supported in Int a

and

equal to one on CO . L e t nr be the operator described in Theorem 5.7.

98

VICTOR GUILLEMIN

Let q be a smooth function on X with support in X - X 2

2

M : L (X) -> L (X) be the operator "multiplication by q . " 5.7, M ne is smoothing; s o , for all k ,

K

M 7rA

particular, bounded as an operator from L

and let By Theorem

is smoothing and, in

to L . L e t pea

be half

the sum of the restricted roots. Then, by [7],

(6.4)

p\2-\p\2)ß

A 0 0 = (\ß +

for all ß . Since n-f <£/Q = 4>ß when ß e ffi we conclude from the L 2 -boundedness of M 7rA :

C > <M q fr f A k ^ /8 ,^ /8 > L2 = (l^ + p| 2 -|p| 2 ) k j q | ^ | 2 d x when ß e ffi . This proves Proposition 6 . 1 . To prove Proposition 6.2 we first note that if a e A , q e C°°(X), and q

is the translate of q by a ;

i.e. q a (x) = q(ax) , then for all ß ,

L2 = s i n c e |<£o| 2 (ax) = \o\2(x).

<

^ß>(t>ß>^2

Therefore, if we s e t

,.(x) = J qqa(x« d a ,

x q1 ava v .( ) =

a

' A

da being normalized Haar measure on A , «lß.ß>^2 = Let c = f

q d x Q . Then q


av^iS^iS>L2 '

is equal to c on X ; so if we set

o q

o

= q

(6.5)

av "

c

'

we

have

<<\ß,ß> 2 = c + <%ct»ß>ct>ß>

2

99

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

with qQ vanishing on X . Now nM

nç is a zeroth order Hermite opera-

tor whose leading symbol v a n i s h e s , s i n c e q

v a n i s h e s on XQ ; so it

can be written in the form nQn where Q is a standard pseudodifferential operator of order - 1 . The second term in (6.5) can be rewritten: (l^+p|2-|pl2r1/2

(6.6)

2

.

The second factor is bounded in ß ; so (6.6) is of order 0(|/3|

) . This

proves Proposition 6.2. The proof of Proposition 6.3 i s similar. Let us denote by T a : X -> X the mapping x -> a x . If Q is a pseudodifferential operator of order zero, L 2 =


aQT*a(^iS^iS>L2 '

so it is enough to prove (6.3) for the operator,

T*QT* da ;

/ i.e. it is enough to prove the theorem for operators which commute with Ta

for all a 6 A . However, every such operator can be written in the

form: f(A) + Q 0 where f is a smooth homogeneous function of degree zero on a - 0 , i s defined a s at the end of section 2, and Q on 2 Q .

Now

<

(

(

>

Q0 t>ß> t>ß

2

=

h a s a symbol which vanishes

0(|/3|~ ) by the same reasoning as above,

and (6.6)

f(A)

ß,ß>

2

= f(j8).

Combining (6.6) with Lemma 2.3, we get (6.3).

100

VICTOR GUILLEMIN

§7. The restrictions

of conical

functions

to X Q

We recall that X Q is the orbit of A containing the identity c o s e t , o € X . Therefore, there i s a mapping:

(7.1)

K: Û

-> X 0

mapping a 6 û onto (exp a) • o . The preimage, K~ (O) , c o n s i s t s of all a e û such that exp a e K . L e t us denote this preimage by £ . It is easy to s e e that £

is a co-compact lattice in û . Given a conical func-

tion, 0 ö , the restriction of c/>o to X Q transforms under the action of A like the group character e ^ X

must vanish, or ß

P^ioë a) •

SOf

either
must be in the dual l a t t i c e , £

. It turns out that

only the second alternative can occur. For the following, s e e [8],

PROPOSITION 7 . 1 . The conical with the points in £

H Q

there exists

function,

a conical

(7.2) with

functions

are in 1-1

More specifically, o , such

ß\X0 =

for each

correspondence ß e£

H û ,

that

cßeM'*>

co £ 0 .

We will investigate the restriction mapping, (7.2), from the micro-local point of view. Let f be a smooth function on

a, - 0 which is homogene-

ous of degree zero and is supported in Int û . In §5 we a s s o c i a t e d with f the operator nf e O P H ° ( X , S Q ) .

We will now show that this is closely

related to the operator (7.3)

P f : L 2 ( X Q ) ^ L2(X0)

defined by setting Pf eo = f(/3) e g for all ß e £

. (Here we have s e t

eo(x) = e^~ (PfX)^ Note that Pf is a pseudodifferential operator of order zero and that it commutes with the action of A on X .

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

LEMMA 7.2. Let

i xv

o

be the inclusion

(7.4)

Proof.

i* n{=

mapping of X_ into

°

101 X.

Then

P f i* „ .

T h i s is an immediate consequence of (7.2).

A simple argument, b a s e d on wave-front s e t considerations, shows that i Y TT is regular; so its transpose is well-defined as an operator from o C°°(X 0 ) to C°°(X). We will denote this transpose by W and the transpose of i x TTr by Wr. From (7.4) we get

(7.5)

Wf = WPf .

We will show in a moment that (7.5) can be considerably generalized.

LEMMA 7 . 3 . W Wf i s a pseudodifferential

operator of order -N ,

N = dim X - dim X . The symbol of this operator is non-vanishing ever the symbol of Pr is

where wher-

non-vanishing.

The proof of this lemma involves standard composition formulas for Hermite operators, for which we refer to [1]. REMARKS:

1) The operator,

W Wf, commutes with the action of A on X , so

it is a convolution operator. 2) Though we won't need this in what follows, an explicit formula can be given for the total symbol of W Wf,

(7.6)

namely

^total(wtwf)03) = ' ^ 4 fOS) w(jS)2

where c is the Harish-Chandra c-function and w is the polynomial occurring on the right hand s i d e of the Weyl character formula.

102

VICTOR GUILLEMIN

By Lemma 7.3 we can find an invertible pseudodifferential operator, C , on X

which is positive, self-adjoint, elliptic of order - N / 2 ,

A-invariant and s a t i s f i e s W*Wf = C 2 P f = C P f C .

(7.7) We will define

U = WCT1

(7.8)

U f = W f C~ 1 .

and

Then, by definition, U t U f = Pf .

(7.9) Let d iÄv X

o

: TX

°

-> TX be the derivative of the inclusion mapping of

into X. The Reimannian inner products on X and X Q give us

identifications TX so, from d i Y

o

(7.10)

s

T*X

and

TX Q

s

T*X Q ;

we get a mapping j : T * X 0 -> T*X .

T h i s mapping is symplectic in the weak s e n s e that the pull-back of the canonical one-form on T X is the canonical one-form on T X .

PROPOSITION 7.4. Uf is a Hermite-Fourier zero, associated

with the symplectic

Integral

operator of order

mapping (7.10).

At the end of §4, we attempted to explain, in terms the ordinary layman can understand, the theory of Hermite pseudodifferential operators.

For

the theory of Hermite Fourier Integral Operators and their composition properties, of which the above result is an immediate c o n s e q u e n c e , we refer to [1]. Let ffi be a convex, conic subset of

a

on

which f = 1 . Then, for

103

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

Pf e g = e g ; s o Uf maps the s u b s p a c e spanned by ea , ß e (iu , isometrically into H 2 ( X ) . (Recall that e g ( x ) = e ^

, x )

. )

From this

and the A-invariance of Ur we conclude:

LEMMA 7.5. For all

ßelft,

Ufeo-0o.

The following i s a sharpening of Proposition 6.3.

PROPOSITION 7.6. Let Q be a pseudodifferential there exists

a pseudodifferential

(7.11)

operator,

operator on X . Then

Q , on X such

that

GrQiOUf = U f Q 0 .

Moreover, on the subset

of T X where

the symbol of Q are related (7.12)

cr(Pr) ^ 0 ,

the symbol of Q and

by: a(QQ) = j V ( Q ) ,

j being the mapping (7.10).

Proof.

Let f1 be a smooth function, homogeneous of degree zero on

û - 0 with support in Int Q

and with the property f1 = 1 on support f,

Set Q 0 = IjtQU^ .

It is easy to check that this has the desired properties.

COROLLARY. / / ß and ß' arein ffi then (7.13)

Jw>ßtßdx=j

Q0eßeß,dxQ.

104

VICTOR GUILLEMIN

Proof. The integral on the left is < TTQTTC/> n, c/> o> >

. If ß

2

and ß'

are

in (S , we have ß,4>ß'> = <77-Q77-Uf e ß , U f e ß ' > = =

= <%eß>eß'> > which is the integral on the right in 7.13.

§8. Micro-local

properties

of TT on the walls of the positive

Weyl

chamber

We will describe briefly how TT b e h a v e s on the other p i e c e s of its characteristic variety, namely on the s e t s , S s , S ranging over the subsimplices of dû

. To begin with we will introduce a partial ordering

among the subsimplices of

û + by s e t t i n g S < T if S C dT . Given a 9

9

subsimplex, S, of û , let Hj; be the L closure of the s p a c e spanned by the conical functions, 0 D , ß an element of the closure of S in û . Let n

be the orthogonal projection of L 2

THEOREM 8.1. The wave-front all

S'^S,

of the sets

2^.

set of TT Moreover

onto H 22 . S'

IS contained S

TT ~ TT on S

in the union, A

-

\J

over

2^.

S 7 >S

The proof of this will be given elsewhere. (For the definition of S s , s e e (3.5).) We recall that if a^ and a« f S in G is equal to the centralizer, Gs

to be equal to G a ,

be the L i e algebra of G s (8.1)

a

Ga

then the centralizer, , of a .

in G.

G a , of a^

In §3 we defined

being a representative element of S. L e t

gs

and let f) s be the following s u b s p a c e of g s .

$„ = i v 6 ö , [ v , a ] = 0,(v,a) = 0 , V a 6 S i .

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

LEMMA 8.2. f) s /s a Lie subalgebra

Proof.

105

of g s .

Given elements Vj and v 2 of I) s , [[v1, v 2 ] , a] = 0 for all

a 6 S by J a c o b i ' s identity.

Also we have:

( [ v 1 , v 2 ] , a ) - tKadCtvj, v 2 ] ) a d a ) = trCadvj adv 2 ada) - tr(adv 2 advj ada) = trCadvj ada a d v 2 ) - tr(adv 2 advj ada) = 0 . (In the second-to-last line we used the fact that adv 2

and ada commute.) Q.E.D.

Let H s be the connected L i e subgroup of Gg a s s o c i a t e d with

ï) s . It

is not hard to s e e that H g is closed in G g . In addition we have:

LEMMA 8.3. H s i s a normal subgroup of G g and the quotient G s / H s , /s

Proof.

group,

abelian.

Let ûg be the abelian subalgebra of

û spanned by la, a e S i .

It is clear from (8.1) that (8.2)

8S = o s ® ï ) s .

Since I ) s is in the centralizer of ûg , it is a subideal of g s . This shows that Hg is a normal subgroup of Gg . By (8.2) the L i e algebra of Gg/Hg

is ûg ; so Gg/FL

is abelian. Note that G , hence Gg is

connected.

G.E.D.

PROPOSITION 8.4. / / ß function, Proof.

cf>o, is

is in the closure

of S in a+ , the

conical

U^-invariant.

We decompose n into one-dimensional root s p a c e s , tt: , associated

VICTOR GUILLEMIN

106 with the roots, a^,

and choose v ^ e î t ^ ,

Vjfîtj

and H i = \/yf^\ [v^ v ^

a s in the corollary to Lemma 5.1. Let us number the roots, a j ,

i=l,---,N

so that the first N' roots are precisely the roots which vanish on S. is clear that H-,

f) s ®C

i s spanned by m

It

and by the vectors v-, VJ and

1 < i < N'. Let V be the irreducible s u b s p a c e of L (X) in which

(f>o s i t s and let VD be the maximal weight vector in V : the abstract element in V which is represented concretely on X by the function, (f>a(X).

Let p be the representation of

8®C on V .

2.1, p(m)v/D = 0 and by definition p(Vj)vo = 0 . pCH^vo = ß(U^)Wß = 0 s i n c e ß

By Proposition

Moreover, for i < N',

is perpendicular to H i by Lemma 5.1.

To show that p(v-)vö = 0 for i < N' we need

v+,

LEMMA 8.5. Let let

V

v~ and

be an irreducible

z be the standard

sL(2, R)-module.

zero vector which is annihilated dimensional

and

by both

s L ( 2 , R ) acts trivially

basis of sL(2, R) and

Suppose v

+

and

VQ contains

z.

Then

V

a nonis one-

on it.

Proof. T h i s i s a corollary of Lemma 5.3 of §5. Now v-,"v- and \ / - ï H^ span a subalgebra of

Q®C isomorphic to

sL(2, R ) . Let us decompose V into a direct sum of irreducible sL(2, R) modules

v =2v(k) and let vß = ^

v

ß

}

with

v

ß}

€

^

:

Since

P ( v i ) v ß } = P( H i) v jg° - 0 ,

we conclude from the lemma that either v ^ = 0 or v ^ by ~v- . T h i s shows that vn

is annihilated by "v- and concludes the

proof of Proposition 8.4.

Q.E.D.

We recall that X g is the orbit of G g coset. Let K s = G g H K.

is annihilated

We can identify

space, G s / K s . By (8.1) f f l g s C l ) s

in X containing the identity X g with the homogeneous

so we have

107

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

LEMMA 8.6.

The connected

COROLLARY. H S K S dimension

as

component

is a closed

of K g /s contained

in H g .

normal subgroup of Gg of the same

H~ and the map G

is a finite covering

S / H S -*

G

S/HSKS

map.

We will denote by Yg the quotient s p a c e , Gg/HgKg . By Lemma 8.3, Yg is a compact abelian group, i.e. a torus. Since Xg = Gg/Kg we have a fiber mapping, (8.3)

T: X S

We recall from §5 that

û contains a natural lattice, £ . As in (8.2) let

ûg be the subalgebra of

LEMMA 8.7. £_~ g by £~ ^ s i s identical

-> Ys .

û spanned by i a , a e û g ! and let

is da ^U-^UlU[JctL,l co-compact IÙ with

lattice IctllKJC

in III

Q~s U

and ctllU

Sg^ugflx.

the L[UUllGllL quotient l/itr

of Ul

aU9^

Yg Ys .

We will omit the proof. Let £ g 1

be the dual lattice to £ g

3

tion, e ^ ' ^ , on

û

in ûg . Given ß e £ g

i s ^ - p e r i o d i c ; so it defines a function,

the funce o , on

Yg . Conversely, every exponential function on Yg is of this form. We can now describe quite precisely the micro-local structure of the projector,

n . To get a global result we will do what we did in §5 and

kill off the contribution to n

coming from the lattice points on dS . Let

f be a smooth function on S which is homogeneous of degree zero and h a s support contained in the complement of a neighborhood of d S . If V is an irreducible subrepresentation of L (X) with maximal weight ß e S , S

S

we define n^ on V by setting, nç v o = f(/3)vo , v o being the maximal weight vector, and nc = 0 on the orthogonal complement of v o in V .

108

VICTOR GUILLEMIN g

This defines nç on all irreducible s u b s p a c e s of the type above. On all other irreducible s u b s p a c e s of L (X) we s e t n^ = 0 . S

THEOREM 8.8. The wave-front

set of nc is the set of all points

of the

form i ( x , f , y , f ) , f * support f,

x,yfXs,

r(x) - r(y)} ,

T being the mapping (8.3).

We will denote by Sg (8.4)

the s e t

i ( x , f , y , f ), £ 6 S, x, y e X g f r(x) = r(y)| . g

Theorem 8.8 s a y s that for all f the wave-front s e t of n^ i s contained in Sg . Note that 2j? is usually much smaller than the s e t , Sg , of Proposition 8 . 1 . In the paragraph below we will u s e the following convenient notational convention. We will denote by - T X the cotangent bundle of X , but with the reverse of its usual symplectic structure; i.e. with the symplectic structure given by the one-form, ^

- f - dx^.

LEMMA 8.9. 2J? i s an isotropic

submanifold

product symplectic

to be zero on S s .

form restricts

of (T*X) x (-T*X) ; i.e. the

Proof. LTR. The following i s an analogue of Theorem 5.7.

THEOREM 8.10. n* belongs

to OPK°(Xx X, 2j?) .

For notation, s e e [1]. Let f be a smooth homogeneous function on S which vanishes near * S

dS as above and, for each ß e £~ H S set

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

\JSfeß = {(ß)d>ß ,

(8.5) eo

109

being the exponential function

e 1 ^ ' ^ and c/>ß the conical function

indexed by ß . By (8.5) we can extend Uf to all of L 2 ( Y S ) . As in (6.5) let (8.6)

j : T*X S -> T*X

be the symplectic imbedding given by the Riemannian metrices on X s and X and let A s be tthe isotropic submanifold of (T Y g ) x (-T X g ) c o n s i s t i n g of the points (8.7)

(y,77,x,f),

X6XS,

y = r(x) , £ = j ( d r x ) V

The following i s an analogue of Proposition 7.4.

T H E O R E M 8.11.

Uf

is in

OPK°(Ysx X , A S ) .

The proof of the Theorems 8.8, 8.10 and 8.11 will be given elsewhere. We will, however, give a few indications here about how the proofs go. Consider the system of equations (8.8)

Dv.u = 0 ,

i=l,..-,N,

on X the v - ' s being a s in Lemma 6 . 1 . We recall from §3 that S g i s one component of the c h a r a c t e r i s t i c variety of this system. On this component the Levi form i s a diagonal matrix with non-negative entries along the diagonal a s before; however, precisely

N ' entries

along the diagonal

are zero, namely the first N ' diagonal entries, s i n c e l / v £ ï (^[vj.Vi]) = ( £ H i ) = 0 when f e S and i < N ' , H^ being perpendicular to S by Lemma 5.1. To prove Theorem 8.10 (from which the other theorems follow by functorial nonsense) one h a s to resort to analogues of the Boutet-Sjostrand results

110

VICTOR GUILLEMIN

described in §4 for systems with positive semi-definite Levi forms of constant

nullity.

We will d i s c u s s the micro-local theory of such systems

in a forthcoming article. We conclude with a couple of results on the asymptotic behavior of the conical functions, n , ß e S as ß

THEOREM 8.12. Let û - 0 /s disjoint

w be a conic open subset

from dS , and let

tial operator on X with symbol

f

of S whose closure

Q be a zeroth order

cr(Q).

in

pseudodifferen-

Then

a(Q)(x,iS)dx+0(|iS|-1)

Qcf>ß4>ßdx = f

X

tends to infinity in Int S .

Xs

for all ß effi.

COROLLARY. AS ß function

tends to infinity

in ffi, \4>ß\2

tends to the delta

of X g .

BIBLIOGRAPHY [1] L. Boutet de Monvel, "Hypoelliptic operators with double characteri s t i c s and related pseudodifferential o p e r a t o r s / ' Comm. P u r e Appl. Math. 27(1974), 585-639. [2] L. Boutet de Monvel and J. Sjostrand, "Sur la singularité des noyaux de Bergman et de S z e g o , " Astérisque 134-35(1976), 123-164. [3] Y. Colin de Verdiere, "Quasi-modes sur les variétés r i e m a n n i e n n e s , , , Inventiones Math, (to appear). [4] V. Guillemin, "Symplectic spinors and partial differential e q u a t i o n s / ' Colloque international de géométrie symplectique, Aix (June 1974) C.N.R.S. [5] V. Guillemin and S. Sternberg, "On the spectra of commuting pseudodifferential operators: recent work of Kac-Spencer, Weinstein and o t h e r s / ' Utah conference on a n a l y s i s , (February 1977) (to appear). [6] V. Guillemin and A. Weinstein, " E i g e n v a l u e s a s s o c i a t e d with a closed g e o d e s i c / ' Bulletin of the AMS, Vol. 82, no. 1 (Jan. 1976), 92-94.

SOME MICRO-LOCAL ASPECTS OF ANALYSIS

and symmetric

m

[7]

S. Helgason, Differential geometry P r e s s , New York (1962).

[8]

, Analysis of Lie groups and homogeneous spaces, Regional conference s e r i e s in mathematics, no. 14, AMS, Providence (1972).

[9]

L. Hormander, " F o u r i e r Integral operators I , " Acta Math. 127(1971), 79-183.

[10] J. Humphreys, Introduction to Lie algebras Springer Verlag, New York (1972). [11] N. J a c o b s o n , Lie algebras, [12] G. Warner, Harmonic analysis Verlag, New York (1972).

spaces,

Academic

and representation

theory,

Interscience, New York (1962). on semi-simple

Lie groups,

Springer

ON HOLONOMIC SYSTEMS WITH REGULAR SINGULARITIES Masaki Kashiwara

and Takahiro Kawai

A holonomic system, i.e., a left coherent ë-Module (or ©-Module)^' whose c h a r a c t e r i s t i c variety is Lagrangian, s h a r e s the finiteness theorem with ordinary differential equations, namely, all the cohomology groups a s s o c i a t e d with its solution sheaf are finite dimensional ([4], [8]). Hence the study of a holonomic system will, in principle, give us almost complete information concerning the functions which satisfy the system, as in the one-dimensional c a s e . When we try to work out such a program, concentrating our attention on holonomic s y s t e m s " w i t h regular s i n g u l a r i t i e s " will be a natural choice. The purpose of this report is to summprize the article of Kashiwara-Kawai [10], which e s t a b l i s h e s a solid b a s i s for the theory and, at the same time, clarifies the role that holonomic systems with regular singularities play among general holonomic s y s t e m s .

(See

Theorem 1 below for the p r e c i s e statement.) Before beginning the d i s c u s s i o n s , we mention that our work is closely related to the work of Nilsson [17], Leray [13], and Deligne [2]. Especially our argument makes e s s e n t i a l u s e of the results of Deligne [2]. A very Supported in part by NSF grant MCS77-18723 '<& (resp. JJ) d e n o t e s the sheaf of micro-differential (resp. linear differential) operators of finite order.

© 1979 P r i n c e t o n University P r e s s Seminar on Micro-Local Analysis 0-691-08228-6/79/00 0113-09 $ 0 0 . 5 0 / 1 (cloth) 0-691-08232-4/79/00 0113-09 $ 0 0 . 5 0 / 1 (paperback) For copying information, s e e copyright page

113

114

MASAKI KASHIWARA AND TAKAHIRO KAWAI

interesting paper of Ramis [18] is also closely related to a part of our results/ 2' In this report we u s e the same notions and notations as in [10]. See also [11], [19]. We first recall some notions needed to define s y s t e m s with regular singularities ([11]). We denote by ë x ( m )

the sheaf of micro-differential operators of order

at most m . Let V be a homogeneous involutory subvariety (possibly with singularities) of T X and I v on T*X which vanish on V . i P 6 Ê x ( l ) ; a 1 ( P ) 6 l v ! and by ë $v . Using the sheaf ë

v

the sheaf of holomorphic functions

Then we denote by $ v y

the sheaf

the sub-Algebra of ë

x

generated by

, we define the notion of ë x - M o d u l e with regu-

lar singularities along V as follows: DEFINITION 1. Let )H be a coherent ë x - M o d u l e defined on 12 C T * X . We say that IK h a s regular singularities along V if one of the following three equivalent conditions i s satisfied: (i) For any point p in Î Î , there e x i s t s a neighborhood U of p and an ë v - s u b - M o d u l e %.Q of %. defined on U which is coherent over

ë(0)

and which generates %. a s ë x - M o d u l e , i.e., % = ë x 5K 0 * (ii) For any coherent e(0)-sub-Module £ set of Q , ë y £

of %. defined on an open

is coherent over ë ( 0 ) .

(iii) Let ÎÏ be an ë v - s u b - M o d u l e of %. that i s defined on an open s e t of Q and that is locally of finite type over ë ated by finitely many s e c t i o n s of ë

v

v

, i.e., locally gener-

. Then ÎÏ is coherent over ë ( 0 ) .

DEFINITION 2. For an ë x - M o d u l e Jtl and an involutory variety V which contains Supp %., IR(5H, V) denotes the s e t ip; 5K does not have regular singularities along V on any neighborhood of p Î.

^ ^Even though the definition of " r e g u l a r s i n g u l a r i t i e s " of Ramis [ l 8 ] is different from ours, an analytic characterization ([lO], Chap. VI) of holonomic JJ-Modules with regular singularities shows that they are actually the same.

ON HOLONOMIC SYSTEMS WITH REGULAR SINGULARITIES

115

DEFINITION 3. A holonomic ë x - M o d u l e 5R is said to be with R.S. if and only if Supp % R IR(5H, Supp ÎIÏ) i s nowhere dense in Supp Jtl. REMARK 1. Note that we have defined the notion of a holonomic system with R.S. by a property of the system at the generic points of its characteristic variety. However, we can eventually prove ([10], Chap. V) that %. h a s regular singularities along V for any involutory s e t V containing Supp M if !m i s a holonomic system with R.S. DEFINITION 4. For a holonomic ë x - M o d u l e Jtl we define a subsheaf j Rreg of 5n°° = g°°<8)5n by defining the presheaf {JlL„(U)} by JRi*~& rM r _ a (U) = reg i s € 5H°°(U) ; for any point p in U there e x i s t s an Ideal § C g near p s u c h t h a t is = 0 and that fb/i

defined

is with R . S . ! .

REMARK 2. Making u s e of the detailed a n a l y s i s on the structure of %. at the generic points of Supp JÎI, we can prove ([10], Chap. I, §3) that 5K ree is a holonomic g x - M o d u l e with R.S. Now, one of the most important r e s u l t s of [10] is the following:

THEOREM 1. For any holonomic

^-Module

)R , ë°° ® )H r M = ë°° ® ÎR holds

g

reg

In view of Remark 2, this theorem a s s e r t s that any holonomic

g g-Module

can be transformed into a holonomic g-Module with R.S. by microdifferential operators of infinite order. As far a s we know, such a clear result h a s not been known even for ordinary differential equations, even though several transformations are employed in analyzing equations with irregular singularities (by Birkhoff, Hukuhara, Turrittin, •••)• The proof of this theorem is achieved by constructing sufficiently many multi-valued analytic solutions of the system Jil s o that we can imbed 5lî into a 5)-Module. Here we e s s e n t i a l l y u s e two r e s u l t s of Deligne [2]: The first is the result to the effect that for any multi-valued analytic function cf> with finite determination we can find a Nilsson c l a s s function I/J with

116

MASAKI KASHIWARA AND TAKAHIRO KAWAI

the same monodromy structure a s 0 .

As a matter of fact, what we need

is a mere sophisticated version of this result which makes u s e of linear differential operators of infinite order. It follows from " R e c o n s t r u c t i o n Theorem," i.e., a theorem which e s t a b l i s h e s the exact correspondence between the category of holonomic ® x -Modules and the category of constructible s h e a v e s on X. See [10], Chap. I, §4 and Chap. II, §2 for det a i l s . The second is the result to the effect that a Nilsson c l a s s function satisfies a holonomic system of linear differential equations. (See also Kashiwara [6].) The complete proof of Theorem 1 is too long and complicated to reproduce here and we refer the reader to [10] for it. A prototype of the argument can be found in Kashiwara-Kawai [9]. See a l s o KashiwaraKawai [7], [8] and Bony-Schapira [1]. A recent result of KashiwaraSchapira [12] on micro-hyperbolic s y s t e m s is effectively used in the proof. In the course of the proof of Theorem 1 we find the following Theorem 2, which is a l s o very interesting and important.

THEOREM 2. Let

M be a holonomic

Supp )K is in a generic in a neighborhood and satisfies

the

position

of p .. Then Th

near %.

Q>x~Module with p , namely, is a Î L

R.S. Assume

Supp JtiCi n~ n(p) = C x p^

, .-module^

' of finite

following:

ÎIL

V1®

1 n

if

if

qen

n(p) - T x X - L p

REMARK 3. We conjecture that this result holds for any holonomic ë x - M o d u l e with R.S.

(3) c x denotes the multiplicative group of non-zero complex numbers and C p means i ( x , c ^ ) ; c f C X | with p = (x, £ ) . v 'n denotes the c a n o n i c a l projection from T X to X . X

that

type

ON HOLONOMIC SYSTEMS WITH REGULAR SINGULARITIES

\yj

Using t h e s e r e s u l t s we can e s t a b l i s h the fact that the family of holonomic s y s t e m s with R.S. is closed under integration procedure and restriction procedure. More precisely we have the following results.

(See

[10], Chap. V. See a l s o [19], Chap. Ill, §3.5 and §4.2 and [5], §4 for related topics.)

THEOREM 3. Let tb^-Module

with

cf> : Y -> X be a holomorphic

R.S. defined

on an open set

be an open set in T*Y - T*Y finite. with

such that a T

(Supp jR)np (W) -> W is

on W. Here and in the sequel

projection

Let W

_1

is a holonomic

R.S. defined

holonomic

U in T * X - T ^ X . 1

Then

the canonical

map and % a

5) (resp.

from Y x T X to T X (resp.

ë y -Module p)

denotes

T Y).

X

THEOREM 4. Let

X be a holomorphic

map,

T X - T X X and

W an open set in T Y - T y Y . Let

right

&Y~Module

defined

with

R.S. Assume

Then

furthermore

(fa^A = to^ip

R.S. defined

on W. Assume

Jl

on U .

®

that p~ ^Y->X^

U an open set in 51 be a

coherent

that 5l is a holonomic Supp Î1 PI zf

2S a r

system

(U) -> U i s

n

*& * holonomic

finite.

tb^-Module

with

P~ &Y

If we assume in addition that 5H i s a ® x -Module, we can generalize t h e s e results as follows:

THEOREM 5. Let ^)x-Module 3)Y-Module

with with

0 : Y -> X 6e a holomorphic

R.S. Then 0*)R = C y

with

holonomic

$ „-Module

f )R

is a

holonomic

holonomic

t 0x

<£> : Y -> X 6e a projective

§Y-Module

_1

-1

R.S.

THEOREM 6. Let

®

map and % a

R.S. Then, /or any k , with

R.S.

map and %. a

k t f 5R == R ^ Ö ^ y

holonomic

L
118

M A S A K I K A S H I W A R A A N D T A K A H I R O KAWAI

REMARK 4. Since Theorem 6 gives information on the c h a r a c t e r i s t i c k

variety of f %., it will be useful in manipulating Nilsson c l a s s functions. Next we shall d i s c u s s how we can ' a n a l y t i c a l l y ' characterize holonomic systems with regular s i n g u l a r i t i e s . The b a s i c properties of holonomic systems with R.S. stated so far are effectively used for this purpose ([10], Chap. V and Chap. VI). Let us first recall the most important characteristic property of the ordinary differential equations with regular s i n g u l a r i t i e s , namely, the validity of the comparison theorem of the following type.

THEOREM 7 (Malgrange [15]). Let SS(?R) = Tj*0jC U T*C . Then

% be ® C / 3 ) C P

(P e 3)Q Q).

Ml has regular singularities

Assume

at 0 if and

only if

(i)

ë-tip <jn,Oc)0 = ëxtjjj

holds for j = 0, 1 . Here Oç ideal of Oç

0

.

Q

= lim Oç

0/nt

<JH,ÖCJ0)

, where

m

is the

maximal

k

It is noteworthy that such a b a s i c result had not been obtained apparently before Malgrange [15]. Probably this is due to the fact that the characterization of regular singularities requires not only the study of the 0-th cohomology group but also that of the first cohomology group,

^ while

s p e c i a l i s t s in the theory of ordinary differential equations had rarely considered higher order cohomology groups before Deligne [2].

In order to i l l u s t r a t e this, we consider the equation with P = x D — a(a?^0). Clearly m is^not with regular singularity at the origin. However, holds—actually, both hand s i d e s are zero! ft) See a l s o Gerard-Sibuya [3] and Majima |_14J for very i n t e r e s t i n g related r e s u l t s on a c l a s s of Pfaff s y s t e m s .

ON HOLONOMIC SYSTEMS WITH REGULAR SINGULARITIES

119

The holonomic system with R.S. in our s e n s e shares such comparison theorems with ordinary differential equations with regular singularities a s follows:

THEOREM 8. Let

holds for any

Jti and 7l be holonomic

with

R.S.

Then

j.

THEOREM 9. Let

Jtl be a holonomic

j and any analytic

j-th algebraic

relative

3)x~Module

with

[Yl

®x for any

^-Modules

subset

cohomology

m

lim fbtdh (0 x /4 ,!)]I), where m X

$Y

R.S.

Then

\ Y of X. Here

K r Y ] ( ! l ) denotes

group of %. supported is the defining

by Y ,

Ideal of

namely,

Y.

REMARK 5. A s p e c i a l c a s e of Theorem 9 where JH = 0 X

was proved by

Mebkhout [16]. Ramis [18] defines the notion of a fuchsian holonomic ©-Module by using this property as its c h a r a c t e r i s t i c property. As a corollary of Theorem 8 we obtain the following Theorem 10, which justifies our u s a g e of the terminology " w i t h

R.S."

THEOREM 10. Let

with

holds for any v A, X

3)x~Module

R.S.

Then

of M .

Here

ê ^ J s (5n,0 x ) x - ê x t ^ (JR,ÔXfX)

(2)

0Y

Jti be a holonomic

j and any

= lim 0 Y ^

v/ A., X

m

x in the domain of definition

, where

the

m is the maxima 1 ideal of 0 V ... A., X

120

MASAKI KASHIWARA AND TAKAHIRO KAWAI

Furthermore we can prove that the validity of (2) is actually a characteristic property of holonomic ® x -Module with R.S., namely, we have the following Theorem 11 as a generalization of Theorem 7.

THEOREM 11. Let Then

% be a holonomic

§x-Module.

Assume

that (2)

holds.

JR is with R.S.

REFERENCES [I]

Bony, J. M. and Schapira, P . Propagation d e s singularités analytiques pour les solutions d e s équations aux dérivées p a r t i e l l e s . Ann. Inst. Fourier 26, 81-140(1976).

[2]

Deligne, P . Equations Différentielles à P o i n t s Singuliers Réguliers. Lecture Notes in Math. No. 163, Berlin-Heidelberg-New York, Springer, 1970.

[3]

Gérard, R. et Sibuya, Y. Etude de c e r t a i n s s y s t è m e s de Pfaff au voisinage d ' u n e singularité. C. R. Acad. Se. P a r i s 284, 57-60(1977).

[4]

Kashiwara, M. On the maximally overdetermined system of linear differential equations, I. Publ. RIMS, Kyoto Univ. 10, 563-579(1975).

[5]

B-functions and holonomic s y s t e m s . Inventiones math. 38, 33-53(1976).

[6]

On holonomic s y s t e m s of linear differential equations II. To appear in Inventiones math.

[7]

Kashiwara, M. and Kawai, T. Micro-hyperbolic pseudo-differential operators I. J. Math. Soc. Japan 27, 359-404(1975).

[8]

F i n i t e n e s s theorem for holonomic s y s t e m s of microdifferential equations. P r o c . Japan Acad. 52, 341-343(1976).

[9]

Holonomic character and local monodromy structure of Feynman integrals. Commun, math. P h y s . 54, 121-134(1977).

[10]

On holonomic s y s t e m s of micro-differential equations III. Systems with regular singularities. To appear.

[ I I ] Kashiwara, M. and Oshima, T. Systems of differential equations with regular singularities and their boundary value problems. Ann. of Math. 106, 145-200(1977). [12] Kashiwara, M. and Schapira, P . Micro-hyperbolic s y s t e m s . To appear in Acta Math. [13] Leray, J . Un complément au théorème de N. Nilsson sur l e s integrales de formes différentielles à support singulier algébrique. Bull. Soc. Math. F r a n c e 95, 313-374(1967).

ON HOLONOMIC SYSTEMS WITH REGULAR SINGULARITIES

121

[14] Majima, H. Remarques sur la théorie de development asymptotique en plusieurs variables I. Proc. Japan Acad. 54, Ser. A. 67-72(1978). [15] Malgrange, B. Sur les points singuliers d e s équations différentielles Enseignement Math. 20, 147-176(1974). Tl6] Mebkhout, Z . Local cohomology of analytic s p a c e s . Publ. RIMS, Kyoto Univ. 12, Suppl. 247-256(1977). [17] Nilsson, N. Some growth and ramification properties of certain integrals on algebraic manifolds. Ark. Mat. 5, 527-540(1963-65). [18] Ramis, J. P . Variations sur le theme " G A G A . " To appear. [19] Sato, M., Kawai, T. and Kashiwara, M. Microfunctions and pseudodifferential equations. Lecture Notes in Math. No. 287, BerlinHeidelberg-New York, Springer, 1973, pp. 265-529.

MICRO-LOCAL ANALYSIS OF FEYNMAN AMPLITUDES (Seminar Report on Linear P D E given in the fall of 1977) Masaki Kashiwara* and Takahiro Kawai* The purpose of this report is two-fold: On the one hand, we show that Feynman integrals are an interesting object for mathematicians to study, especially from the viewpoint of microlocal a n a l y s i s . On the other hand, we show that the micro-local analysis of Feynman integrals yields several physically interesting results on the analytic structure of Feynman integrals. Even though some results of this report can be extended to the S-matrix itself, we do not d i s c u s s it here. (See Kawai-Stapp [16], [17] for this topic.) Although we do not give any explanations why the study of analyticity properties of the S-matrix and Feynman integrals are physically interesting and important, we j u s t quote one s e n t e n c e from the celebrated and pioneering book of Chew [3], where he claims "Analyticity

a s a fundamental prin-

c i p l e in p h y s i c s " : "During the past ten years, n e v e r t h e l e s s , a feeling h a s been growing among many theoretical p h y s i c i s t s that the description of natural phenomena on subatomic level may be facilitated if analyticity is employed a s a primary rather than a derived c o n c e p t . " (Chew [3], p. 2, line 33-line 36.)

* Supported in part by National Science Foundation grant MCS77-18723.

© 1979 P r i n c e t o n University P r e s s Seminar on Micro-Local Analysis 0-691-08228-6/79/00 0123-15 $ 0 0 . 7 5 / 1 (cloth) 0-691-08232-4/79/00 0123-15 $ 0 0 . 7 5 / 1 (paperback) For copying information, s e e copyright page

123

124

MASAKI KASHIWARA AND TAKAHIRO KAWAI

For the d e t a i l s of the results d i s c u s s e d here, we refer the reader to Kashiwara-Kawai [9], [10], [11]; Kashiwara-Kawai-Oshima [13]; KashiwaraKawai-Stapp [14] and Sato-Miwa-Jimbo-Oshima [27]. See Nakanishi [20], Speer [28] and Kawai-Stapp [17] for notations related to Feynman integrals and diagrams and s e e Sato-Kawai-Kashiwara [26] for the b a s i c notions concerning microfunctions and holonomic (= maximally overdetermined) s y s tems of micro-differential (= pseudo-differential) equations.

See also

Nakanishi [20], Speer [28] and references cited there for the physical importance of Feynman integrals. F i r s t of all, we recall the definition of Feynman diagrams and Feynman integrals. DEFINITION 1. Feynman diagram D c o n s i s t s of finitely many points (called " v e r t i c e s " ) iV-!-_ 1 ...

' , finitely many one-dimensional segments

(called "internal l i n e s " ) iL£Î£ =1 ... (called "external l i n e s " ) i L ^ ! r = 1 ...

, and finitely many half lines

N n

.

Each of the end points Wj? and

Wj? of Lp and the end point of L^ must coincide with some vertex V-. We suppose Wj? ^ Wj? for all l S ' A four vector p f = (p r 0 , p f

1

,p f 2 , p f

3)

i s associated with each external line L ^ and a strictly positive^ ' constant mjT

is a s s o c i a t e d with each internal line Lp . We suppose that

each internal line and each external line are o r i e n t e d / ' The orientation is indicated by the arrow —•—. p Example of a Feynman diagram D:

' T h e r e are some c a s e s where we should omit t h e s e c o n d i t i o n s . However, we include t h e s e conditions in the definition of Feynman diagrams for simplicity. ' P h y s i c a l l y speaking, m^ r e p r e s e n t s the m a s s of relevant p a r t i c l e s .

MICRO-LOCAL ANALYSIS OF FEYNMAN AMPLITUDES

125

REMARK. A Feynman diagram represents diagrammatically the interaction of elementary p a r t i c l e s . See, e.g., Nakanishi [20] for more detailed physical explanations. DEFINITION 2. (Incidence number). If internal line Lj? s t a r t s from V(resp. ends at V-), the incidence number [j : £] is defined to be

-1

(resp. + 1 ) . In other c a s e s , [j : £] is defined to be zero. The incidence number [j : r] i s defined in the same way. DEFINITIONS.

A vertex Vj of D for which [ j : r ] = 0 holds for all r

is called internal vertex. Other vertices are called external. REMARK. For simplicity all diagrams considered below are supposed to be connected. DEFINITION 4. (Feynman rule)^ \

Feynman integral F D (p)

associated

with D is (formally) defined by

(1) FD(p) = F D ( Pl ,.,p n ), I tL-)fl tl 2 2 jJ(k r m e + VnO)

N

/ J ] d4k(? , l=1

£=1 3

where k 2 = k

2

I

^ - ^

Since F D (p)

H,v

'

" d e f i n e d " by (1) i s , in general, a divergent integral and

not well defined a s it s t a n d s , the so-called renormalization procedure is needed to make it well defined.

The recipe for renormalization given by

Bogoliubov-Parasiuk-Hepp is not convenient for our purposes; we use here

For simplicity we s u p p o s e all relevant p a r t i c l e s are s p i n l e s s . v

'In this report, we always u s e this Minkowsky metric to define four-vector k .

k

for a

126

MASAKI KASHIWARA AND TAKAHIRO KAWAI

the recipe given by Speer [28] (which is equivalent to that given by Bogoliubov-Parasiuk-Hepp a s i d e from finite renormalization). T h e recipe of Speer i s a s follows:^ ' F i r s t we consider the generalized Feynman integral F D (p; A.) defined as follows: (2)

FD(p;X) = F D ( p 1 , . . - , p n ; A 1 , . . . , A N ) N

[j:Qk^

ÏLAI±

N

tL—j

n

H

1=1

• /

^

1=1 This integral i s convergent for Re A» » 1 and meromorphically extended with respect to A(eC ) . Next we c h o o s e p o s i t i v e numbers Rj>((!= 1,---,N) so that (3)

0 < R x < R 2 < ••• < R N « 1

and £-1

(4)

R* > 2

R

k

k=l

hold. We define Cp = ÎA e C; | A - 1 | = R^l. Then the renormalized integral F D (p) H fflN(FD(p; A)) i s given by

(5)

N! 2 j> w

'" j>

^a(l)

(A^-'-CVD^l-^N

<7(N)

where a i s a permutation of ( 1 , - - - , N ) .

Even though Speer's setting i s more general, we have chosen the most convenient recipe for our purpose.

MICRO-LOCAL ANALYSIS OF FEYNMAN AMPLITUDES

127

For our later d i s c u s s i o n s it is more convenient if we deal with integral (2) by compactifying the domain of integration, namely, we rewrite (2) as follows:

Ii 4

(6)

IPlS>

USr

r]

W\y.t'Uo\ II Wr + JLW^-'H Lj:tJ

n^HH^0^

(R ))N N

£=1 Here /x£ = 2A.£ + 4 # i j ; [j:£]^ O i - 5 = 2A.£ + 3 , (Kj?, C £) X <£< N is the homogeneous coordinate on P(R ) , namely, Kn/co^kn,

and

Û)(KÇ,CÇ)

3

is the volume element on P ( R 4 ) . (Sato-Miwa-Jimbo-Oshima [27], Jimbo[8]. Our starting point is to know how the singularity spectrum of F D (p; A.) is described.

As a matter of fact, it is described by the (positive

-a)

Landau-Nakanishi variety defined below. (Landau [18], Nakanishi [19] and unpublished article of Bjorken.)

The factor

II

c p is multiplied in the argument of 8

to make it

K';[j:n^o! n' well defined.

Note that

II

N II

en* = I I c g holds, because each

j=l \t;[yl'U0\ appears exactly twice in the left-hand side.

1=1

en

128

MASAKI KASHIWARA AND TAKAHIRO KAWAI

DEFINITIONS.

An 8n-real vector (p,u) = (pj,---»p , Uj, •••,u n ) i s said

to satisfy the positive -a

Landau-Nakanishi equations if it s a t i s f i e s the

following equations for some an € R + = ia(fR; a > 0 S , w j f R 4

u

r = ] £ [j:r]w.

and kg
(r=l,--,n)

(7.a)

j= l

n

2

N

[j-r]pr + ^ [ j : £ ] k ^ 0

r=l

(7)

(j=l.

,n')

(7.b)

(?=1,

,N)

(7.c)

<*=1,

,N)

(7.d)

£=1

n

^

[j:flwj = apkg

j=l

a£(k2rm2£)=0 The p o s i t i v e - a

Landau-Nakanishi variety £ ( D + ) i s defined to be the

s e t of points (p, yj-l u°o) e yj-l

S * R 4 n , where (p, u) is a solution of (7).

REMARK 1. Since F D (p) (resp. F D (p; A)) h a s the form S V V

tj:r]pr\fD(

j,r

(resp. 8 (^V [ j : r ] p r ) % ( p ; ^))»

we

often d i s c u s s the analyticity property

of f D (p) on M = < ^ [ j : r l p r = 0 ? C R 4 n . By the standard convention that (p, u) and (p', u') represent the same point in S M if and only if p = p ' and u - u ' = a e R , we may regard £ ( D + )

as a s u b s e t in \ / ^ î S*M . T h i s

convention will be often employed without explicit mentioning, if there is no fear of confusion.

The function

f D (p) (resp. f D (p; A)) is called a

Feynman amplitude (resp. a generalized Feynman amplitude). REMARK 2. We denote by <£(D) the variety defined by the equation (7) without the additional assumption that ao > 0 . The celebrated result of Landau and Nakanishi claims that the singularity of Feynman amplitude f D (p)

is described by p o s i t i v e - a

Landau-Nakanishi equations.

result can, actually, be formulated micro-locally as follows:

Their

MICRO-LOCAL ANALYSIS OF FEYNMAN AMPLITUDES

T H E O R E M 1.

S.S.F D (p) c

129

£(D+).

For the rigorous proof, s e e Chandler [1] and Sato-Miwa-Jimbo-Oshima [27]. See also Chandier-Stapp [2], Iagolnitzer-Stapp [5], Pham [22], Sato [25], Iagolnitzer [5], [6], Kawai-Stapp [16], [17] and references cited there for related topics. Concerning the analytic structure of F D ( p ) , Regge [23] and Sato [25] made the following important and intriguing conjecture. CONJECTURE.

F D (p) s a t i s f i e s a holonomic system of micro-differential

equations whose c h a r a c t e r i s t i c variety is contained in £ ( D )

, the com-

plexification of £ ( D ) . T h i s conjecture has been proved with a slight modification concerning its characteristic variety by Kashiwara-Kawai [9]. The study on N

^

(f£ + V 1 "! 0)

Yl

l

done by Kash iwara-Kawai [11] is e s s e n t i a l for the proof.

£=1 In order to s t a t e the result we recall the definition of extended Landau variety £ ( D ) .

T h i s variety appears naturally not only in mathematical

context but a l s o in physical context. (See Kashiwara-Kawai-Stapp [14].) DEFINITION 6. £ ( D ) = i(p, u) e T * C n ; cj

m)

and 4

m)

(£=l,...fN)

there e x i s t s a s e q u e n c e of s c a l a r s

and four-vectors

p< m ) , u ( f m) (r= 1, •••, n), k< m)

(£ = 1, •••,N) and w^ m ' (j = 1, •••, n') which s a t i s f i e s the following relation

(8)i.

130

MASAKI KASHIWARA AND TAKAHIRO KAWAI

P^ m ) - Pr

(r=l,-,n)

(8.a)

u r( m ) - u r

(r=l,-,n)

(8.b)

u<m> = j £ [j:r]- (m)

(r=l,-,n)

(8.c)

j=l

r=l

U:r]p(rm> + £ [j:QkJm> = 0 e=i

2

MW- m)

2

(8)

(j=l,-,n')

(8.d)

<e=i,-- , N )

(8.e)

aj m >(kp-m2)^0

(e=i,"- , N )

(8.f)

ci"1-* is bounded

<e=i,"- , N )

(8.g)

c<m)k<;m> is bounded

(?=i,- - , N )

(8.h)

(c< m \c< m V™>)^ 0

tf=i,- - , N )

(8.i)

+ aK( m ) k ( m )

t "•£

j=l

,(m)

Using £ ( D ) , the result of [9] is stated as follows.

THEOREM 2. Renormalized system contained

F eynman integral

5ltD of linear differential in the extended

equations

Landau variety

F D (p) satisfies

whose characteristic

a

holonomic variety

is

£(D).

Furthermore a recent result of Kashiwara-Kawai [12] shows that the system 5ltD is actually a system with regular singularities. T h i s property will turn out to be important when one tries to relate "holonomic system approach" to "monodromy structure a p p r o a c h . " (See Regge [24] and references cited there for related topics.)

MICRO-LOCAL ANALYSIS OF FEYNMAN AMPLITUDES

131

Even though a detailed study on the structure of 5ltD has not yet been fully done in general, several interesting p i e c e s of information on F D (p) can be drawn from the a n a l y s i s of !)HD if we assume some moderate conditions on D . (Kashiwara-Kawai [10], Kashiwara-Kawai-Oshima [13].) As an example, we determine the singularity structure of f D (p)

explicitly

at some physically important points. For the s a k e of simplicity, we consider for the moment a diagram which s a t i s f i e s the following conditions (9) and (10). (9)

At each vertex V- there e x i s t s exactly one external line that touches V- . T h e external line shall be indexed to be L e

and

supposed to be incoming. (10) The diagram D is simple in the s e n s e that for any pair of vertices (V\ > V: ) there e x i s t s at most one internal line (possibly no) that

h

h

joins V-J l Example

and V-J . 2

of a diagram

D satisfying

conditions

(9) and (10):

A (simple) daughter diagram D« of D with respect to L-

is, by

definition, a diagram obtained from D by deleting L.« and identifying the end points Wx and Wx of L x .

132

MASAKI KASHIWARA AND TAKAHIRO KAWAI

The diagram

D

D-. obtained

from D in the preceding

example:

r

We denote by £ Q ( D + ) the part of £ ( D + ) where all a^

0.

In our

+

c a s e we can verify that £ Q ( D ) and £ Q (D^) intersect normally along a codimension

1 submanifold.

Hence, if we assume in addition that

77(£ 0 (D + )) ^ ' and 77(£ 0 (D^)) are real hypersurfaces (i.e., with codimension 1) in M = j ^

U : r]p r = 0 [ , then we can introduce a local coordinate

system on M near p Q e 77(£ 0 (D + )) n 77(£ 0 (D+)) SO that 77(£ 0 (D + )) = {xfR4n-4;x1 = x2fx2>0}(**)and

77(£0(D+)) = i x e R 4 1 1 " 4 ; x x = 0 j . Then a

result of Kashiwara-Kawai-Oshima [13] a s s e r t s the following:

THEOREM 3. Under the assumptions following

on D so far stated,

system

holomoTphic functions hyper geometric

introduced

defined

near

above.

has the

p Q e 77(£ 0 (D + )) fl 77(£ 0 (D+))

form ( l l . A ) or (11.B) near the point

in the coordinate

f D (p)

Here

<^-(x) ( j = l , 2 , 3) a r e

p Q and F ( a , / 3 , y ; z) ^

;

i s the

function.

C a s e A: i / = - 2 n ' + ( 3 N / 2 ) + 3 / 2 d O , 1, 2, •••} .

7

(*) 7

77 denotes the canonical projection from y-1

^

*

SM

to M.

'The condition x« > 0 arises from the positivity of ap's . The part

Jx; x 1 = x 9 , x « < 0 | corresponds to the solutions of Landau-Nakanishi equations with a1<0 and a£ > 0 (£ ^ 1). ^ ^We consider the analytically continued function outside \z (C; z e R , z > 1|

MICRO-LOCAL ANALYSIS OF FEYNMAN AMPLITUDES

133

(11.A) ^ ( x ) f C d g f l ( X l - X 2f log( Xl -x2 + VTÏ 0) +

r(^ + i/2) V^

+ <^ 2 (x)f ^Ç^-

+

r(-^-l/2) ( yjn

2

(

1

+

^

or-Y, FL+1/2i

\ x 2 ( X l - X 2 f log(Xl-x2

^

0 fW-^l/2, \

x

lf 3/2 .

2 Y\

X1+y/-10jJ + vCÎ

0)

1,1/2;

%=-^\ xl+y/-10J!

+ <£3(x) . C a s e B: v- 1/2
x -x^x.^/MogCx, f V ^ 1 0 ) F -1/+1/2,1,3/2; x \ \ i

+

2=v / I Î 0-

172 + 2 9(x)[r(-l,)x„(x 2 x1-x2 2 + v c i 0)^+ (-i)""" y v^+i/2)!

x J - x ^ l o g O ^ + ^ ï 0)F(- l ,-l/2, 1, 1/2;

+

2 I d (x1 + V - 1 0 r /t-'/i f F a, 1,1/2;

^U-

^=—

a=~v-lh

+ 03(x) . See [13] Theorem 1.2, for the general formula which can be applied to generalized Feynman amplitude f D (p; A.). We note that the geometrical situation d i s c u s s e d in Theorem 3, i.e., the situation where two p o s i t i v e - a

Landau-Nakanishi varieties projected

to p-space are osculating along codimension

1 submanifold, is a l s o

MASAKI KASHIWARA AND TAKAHIRO KAWAI

134

crucially important in d i s c u s s i n g the relations among several Feynman amplitudes (e.g., relation between f D (p)

and fD (p)) —the so-called

hierarchical principle. See [4], [10], [21], [27], [30] and references cited there for this topic. We end this report by pointing out one interesting application of the theory of linear differential equations to the a n a l y s i s of Feynman integrals, which differs in its nature from what h a s been d i s c u s s e d so far. It i s a problem of asymptotic behavior of Feynman integrals. One of the simplest problems of this sort is the study on the behavior of F D (p; K

m

as

) ^

some m a s s e s m£ (£ e L Q C \ 1, •••, Ni) tend to zero. In

this c a s e , at l e a s t for a diagram D satisfying the condition (9), we can explicitly find equations with regular singularities along i(p,m); m£ = 0, £ € L Q ! (not holonomic) which F D (p; X; m) s a t i s f i e s (Theorem 4 below). Then we u s e the r e s u l t s of Kashiwara-Oshima [15] to obtain the asymptotic behavior of F D (p; À; m) a s mo t e n d s to zero (Theorem 5). T h i s provides us with an alternative approach to the study of zero mass singularities of Feynman amplitudes done by Speer-Westwater [30] and Speer [29].

This

topic shall be d i s c u s s e d in detail in a forthcoming paper of JimboKashiwara-Kawai-Oshima.

THEOREM 4. The generalized the following

differential

Feynman

equations

integral

F D (p; X; m)

with regular singularities

satisfies along

i(p,m); m£ = 0!

<12> [^%) 2 - m f(2tJ : « D Pj ) +2 (V2)^D m jF D = 0 0=1. ...,N). For simplicity let us consider the c a s e where one mo , say m1 , tends to zero. Then we find the following

'In order to emphasize that F~(p; A) depends a l s o on mo's , we u s e this notation here.

MICRO-LOCAL ANALYSIS OF FEYNMAN AMPLITUDES

THEOREM 5. Suppose FD,

- n ^ P ' ^'

m

') (

res

that X« is not an integer. P-

ing ( K ^ - c ^ m ^ + v C ï O ) m' = (m 2 , •••, n i N ) .

F D ' ( p ; h'> 1

by (K* + V ^ ï 0)~

integral obtained

x

exponent

= o^P»

2(2-^)

form (13) in a neighborhood

^lS*R4n+N;

by replac-

(resp. cJS 4 ^)). Here

^»

m

')

of [15]

with the an<

^ *hat

Definition character-

associated

i s given by

77 À 1 (À 1 + l ) F D ' ( p ; X; m ' ) . Furthermore,

following

Let us denote by

i(p,m); 114 = 0! associated

' 0 i s given by F D ,

with the characteristic e

))

tne

T/ien the boundary value (in the sense

4.8) of F D (p; X; m) along istic exponent^

m

135

F D (p; X; m) has the

of A+ = i(p, m; \/^T d m ^ )
114=0}:

lF

D(mi=0)Y(ml)

/ X.n^pi

9

2(2-X 1 )\

Note that the operators used in (13) are well-defined micro-differential operators defined near A+ .

REFERENCES [1] Chandler, C , Some physical region mass shell properties of renormalized Feynman integrals, Commun, math. Phys., 19(1970), 169-188. [2] Chandler, C. and H. Stapp, Macroscopic c a u s a l i t y conditions and properties of s c a t t e r i n g amplitudes, / . Math. Phys., 10 (1969), 826-859. ^ 'A characteristic exponent i s , by definition, a solution of the indicial equations associated with (12). (See [ l 5 j p. 174.) In our c a s e , the indicial equation is s(s+2(X1-2)) = 0 .

136

MASAKI KASHIWARA AND TAKAHIRO KAWAI

[3]

Chew, G., The Analytic

[4]

Eden, R., P . Landshoff, D. Olive and J. Polkinghorne, The S-matrix, Cambridge University P r e s s , 1966.

[5]

lagolnitzer, D., Analyticity property of scattering amplitudes: a review of some recent developments, Lecture Notes in Phys., 39, Springer-Verlag, Berlin-Heidelberg-New York, 1975, 1-21.

[6] Phys.,

S-matrix,

Benjamin, New York, 1966.

, The structure theorem in S-matrix theory, Commun, 41 (1975), 39-53.

Analytic

math.

[7]

lagolnitzer, D. and H. Stapp, Macroscopic c a u s a l i t y and physical region analyticity in S-matrix theory, Commun, math. Phys., 14(1969), 15-55.

[8]

Jimbo, M., A correction to "Holonomy structure of Landau singularit i e s and Feynman i n t e g r a l s , " Publ. RIMS, Kyoto Univ., 12, Suppl. (1977), 438-439.

[9]

Kashiwara, M. and T. Kawai, Holonomic s y s t e m s of linear differential equations and Feynman integrals, Publ. RIMS. Kyoto Univ., 12, Suppl. (1977), 131-140.

[10]

, Holonomic character and local monodromy structure of Feynman integrals, Commun, math. Phys., 54(1977), 121-134.

[11]

, On holonomic s y s t e m s for T T (îç + \/^î Publ. RIMS, Kyoto Univ. "

[12]

, On holonomic systems of micro-differential equations Illsystems with regular singularities, to appear.

0)

, to appear in

[13] Kashiwara, M., T. Kawai and T. Oshima, A study of Feynman integrals by micro-differential equations, Commun, math. Phys., 60(1978), 97-130. [14] Kashiwara, M., T. Kawai and H. Stapp, Micro-analytic structure of the S-matrix and related functions, Publ. RIMS, Kyoto Univ., 12, Suppl. (1977), 141-146. A full paper will appear in Commun, math. Phys. [15] Kashiwara, M. and T. Oshima, Systems of differential equations with regular singularities and their boundary value problems, Ann. of Math., 106(1977), 145-200. [16] Kawai, T. and H. Stapp, Micro-local study of the S-matrix singularity structure, Lecture Notes in Phys., 39, Springer-Verlag, BerlinHeidelberg-New York, 1975, 36-48. [17]

, Discontinuity formula and S a t o ' s conjecture, Publ. Kyoto Univ., 12, Suppl. (1977), 155-232.

RIMS,

[18] Landau, L. D., On analytic properties of vertex parts in quantum field theory, Nucl. Phys., 13 (1959), 181-192.

MICRO-LOCAL ANALYSIS OF FEYNMAN AMPLITUDES

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[19] Nakanishi, N., Ordinary and anomalous thresholds in perturbation theory, Prog. Theor. Phys., 22(1959), 128-144. [20]

, Graph Theory and Feynman Integrals, New York, 1971.

Gordon and Breach,

[21] Pham, F . , Singularités des p r o c e s s u s de diffusion multiple, Ann. Inst. H. Poincaré, 6A (1967), 89-204. [22]

, Microanalyticite de la matrice S, Lecture Notes in Math., 449, Springer-Verlag, Berlin-Heidelberg-New York, 1975, 83-101.

[23] Regge, T., Algebraic topology methods in the theory of Feynman relativistic amplitudes, Report of Battele Renctres., Benjamin, New York, 1968, 433-458. [24]

, Old problems and new hopes in S-matrix theory, Publ. RIMS, Kyoto Univ., 12, Suppl. (1977), 367-375.

[25] Sato, M., Recent development in hyperfunction theory and its application to p h y s i c s , Lecture Notes in Phys., 39,Springer-Verlag, BerlinHeidelberg-New York, 1975, 13-29, [26] Sato, M., T. Kawai and M. Kashiwara, Micro functions and pseudodifferential equations, Lecture Notes in Math., 287, Springer-Verlag, Berlin-Heidelberg-New York, 1973, 265-529. [27] Sato, M., T. Miwa, M. Jimbo and T. Oshima, Holonomy structure of Landau singularities and Feynman integrals, Publ. RIMS, Kyoto Univ., 12, Suppl. (1977), 387-438. [28] Speer, E. R., Generalized P r e s s , 1969. [29]

Feynman

Amplitudes,

Princeton University

, Mass singularities of generic Fevnman amplitudes, Ann. Inst. H. Poincaré, 26(1977), 87-105.

[30] Speer, E. R. and M. J. Westwater, Generic Feynman amplitudes, Ann. Inst. H. Poincaré, 14(1971), 1-55.

Library of Congress Cataloging in Publication Data

Guillemin, V 1937Seminar on micro-local analysis. (Annals of mathematics study ; no. 93) 1. Mathematical analysis—Addresses, essays, lectures. I. Kashlvra,ra5 Masaki, 19^7- joint author II. Kavai, Takahiro, joint author. III. Title. IV. Series: Annals of mathematics studies ; no. 93. QA300.5.G8U 515 78-70609 ISBN 0-691-08228-6 ISBN O-69I-O8232-U phk.