Proc. Indian Acad. Sd. (Math. Sci.), Vol. 90, Number 2, April 1981, pp. 95-102. (~ Printed in India.

A submersion principle and its applications

By

HARISH-CHANDRA

L Introduction

Let G be a real reductive group and n an irreducible admissible representation of G. Let | denote the character of n. We recall that | is a distribution on G defined by

| (f) = tr rt (f) (3" e C'* (G)).

It is well known that | is a locally summable function on G which is analytic on the set G' of regular elements. Fix 70 e G' and let F be the Cartan subgroup of G containing 70. Put

F ' = G ' N F a n d G r = tO x F ' x -1. mUG

The mapping (x, 7)I ~ "3C7 "Tg'''l of G X /'~t into G is everywhere submersive. Since | is invariant under all inner automorphisms of G, one proves easily ~ a t | defines a distribution 0 on F' . Let r be the algebra of all differential operators on G which commute with both left and right translations. Then | satisfies the differential equations

z | x(z)O

where Z is the infinitesimal character of n. We can transcribe these equations in terms of 0. In this way we obtain a system of differential equations for 0 on F' . It tarns out that this system is elliptic and therefore 0 is an analytic function on 1". Bat this implies that @ coincides with an analytic function on Gr.

We would like to prove a similar result in the p-adic case. Let s be a p-adic field and G the group of all O-rational points of a connected reductive O-group G [31. Then G, with its usual topology, is a locally compact, totally disconnected, animodular group. Let dx denote its Haar measure. We shall use the termino- logy of [3] without further comment.

Let n be an admissible irreducible representation of G. Then for every f e C** (G), the operator

g ( f ) = l f (x )~(x)dx G

95

96 Harish-Chandra

has finite rank. Put

O (f) = t r n (f).

Then O is a distribution on G. Let G' be the set of all points x e G where DG (x) 0 ([3], w 15). Then G' is an open, dense subset of G whose complement has measure zero. We intend to show that | coincides with a locally constant function F on G'. This means that

o 0 9 = i f (x )F(x)ax ,

for all f ~ C ~ (G').

In case char t2 = 0, this fact was first proved by Howe by making use of his Kirillov theory for p-adic groups. Moreover when char I2 = 0, it is known that O is a locally summable function on G [4, 5]. But these me~ods , which make extensive use of Lie algebras and the exponential mapping, do not seem to work in characteristic p. We shall therefore construct a proof on a totally different principle.

2. The submersion principle

Let P be a parabolic subgroup of G and x [-* x* tlle projection of G on G* = G/P

Theorem 1 (tile submersion principle). Fix 7 ~ G'. Then the mapping

x I-+ (x~ 'x-l)*

f rom G to G* is everywhere submersive.

When char f2 = 0, the proof is very easy. Moreover Borel assures me that this principle is actually true over an arbitrary field.

Let as verify it when char ~2 = 0. Put

q~: x I-+ (x~x-~) *.

Then q5, r (xy) = dpcy,-, (x) (x, y e G).

Hence it is enough to prove that ~b, t is submersive at x = 1. Fix a split component A of P and let P = M N be the corresponding Levi

decomposition of P. Let (/~, A) denote the p-pair opposite to (P, A). Then

= MN. We denote the Lie algebras of G, P, M, N, N by g, p, m, n, ff respec- tively. Then

p = m + n , ~ = n + m + n ,

and we have to verify that

(Ad (y-l) -- 1) g q- p = g.

Fix a symmetric, nondegenerate, G-invariant bilinear form B on g with values in g2. Since G is reductive, tlfis is possible. Let F be the Cartan subgroup of G determined by y and t the Lie algebra of F. Then t = ker (Ad (y) -- 1). For

A submerslon principle and its applicatlons 97

any linear subspace q of g, let qs denote tb.e space of all Y e g suck tha~ B (X, I0 = 0 for all X E q. Then

p• = l t , ((Ad ()'-0 - - 1) g)J- = ker (Ad 0') -- 1) = t.

Therefore ((Ad (7 -~) -- 1) g + p)• C t Cl n ---- {0},

and this implies the desired result.

3. The function fa,

Fix y and P as in theorem 1. Then the mapping*

(x,p) i-~ ~y . p ( x ~ G , p ~ P )

of G x P into G is everywhere submersive. Hence ([2], p. 49) there exists a unique linear mapping

a f . , , (a c y (G x P))

from C y ( G x P ) to C ~ ( 6 3 such that

j" ~(x :p)F(~7 . p) dxdzp = f f =, ,r (x) F (x) dx, G X P G

for all F~ C~ (63. Here dzp is the left Haar measure on P.

Lemma 1. Fix a e C~ (G x P). Then the mapping

y l ->f=, , (y~G')

from G' to C~ (63 is locally constant.

The mapping

(y, x, p) J-~ (y, "y. p)

from G' x G x P to G' x G is submersive. Hence there exists a unique Ill, ear mapping

ill-* ~ (,8 ~ C ~ (G' x G X P))

from C ~ (G' X G X P) to C ~ (G' x 63 such that

I / t ( y : x : p) ~ (y:~y . p) dydxd~p = I 4,t~ (y : x ) ~ ( y : x) dydx

for all ~ e C ~ ( G ' x G). Now let

5 0 ' :x) : 2(y) F(x),

where ~ C ~ ( G ' ) and F ~ C ~ ( 6 3 . Then

I2(y) dy I fl(Y :x :p) F ( ' y . p ) d x d ~ p = I2 (y )dy J'~bt~(Y : x )F(x )dx . G G X P G G

*We write " y = x T x - z for x, y ~ G.

P .(A)--7

98 Harish-Chandra

This being true for all 2, we conclude that

I fl(Y : x : p ) F ( ' y "p) dxd, p = Idp[~(y : x ) F ( x ) d x G •

for all y ~ G' and F e C ~ (G).

Now fix y0eG ' and put

f l ( y : x : p ) = / t ( y ) a ( x :p)

where a e C oO (G • P), /z e C~ (G') and H (Yo) = 1. Let Go be a neighbourhood

of yo in G' such that g = 1 on Go. Then f l ( y : x : p ) - - a ( x : p ) f o r y e G o a n d therefore

S fl(Y : x :p) F ('y . p ) d x & p = ,[ a ( x : p ) F ( ' y . p ) d x d z p G X P

= I fa, ~ (x) F (x) dx.

Hence ~ fa, , (x) F (x) dx = .f 49~ (Y : x) F (x) dx

for y e Go and F e C ~ (G). This shows that

A,,(x)-4~t3fy :x) (yeGo, X~G).

Since ~br e C~ (G' • G), our assertion is now obvious.

4. The operator T,

Fix a minimal p-pair (P, A) in G (P = M N ) and let K be an open compact sub- group of G of Bmhat-Tits ([2], p. 16) corresponding to A. Let n;be an admissible representation of G on V. We recall that End o V is defined to be the subspace of all T a End V such that the mappings

x I--, ~ (x) T, x D--, T~ (x),

of G into End V are both smooth.

Let dk denote the normalised Haar measure on K.

Theorem 2. Let rc be an admissible representation of G on V such that V is a finite G-module under ~. For x e G', define

I', = I ~ (kxk-~) dk. r

Then 1", e End o V and x i~ 7", is a smooth mapping from G' to End o V.

Let K0 be an open and normal subgroup of K and Vo the subspace of all vectors in V, which are left fixed by K0. Then dim Vo < oo. By choosing K0 sufficiently small, we may assume that V is spanned by elements of the form

(x) v (x ~ G, v e Vo).

A submersion principle and its applications 99

Let M + denote the set o f all m e M such that (a, Hkr (m))/> 0 for every root a o f (P, A) (see [3], w 7) for the definition o f H~r). Then G = K M +. K. Put A + = A O M +. Since M/A is compact, we conclude that M + C CA + where C is a com- pact subset of M. Hence it is clear that we can choose an open compact sub- group Pc of P such that m-lpo m C Ko for all m ~ M +.

Let a denote the characteristic function of K x P0 and put f~ = fa, u (Y e G') in the notation of lemma 1. Then y J~ f~ is a smooth mapping from G' to C ~ (G) and

I F e y . p) dkdp = I fu (x) F (x) dx, KXPo

for all locally summable functions F on G. (Here dp is the normalised Haar measure on P0.) F rom this we deduce immediately that

I p) & @ = I L (x) (x) dx = 0,). KxPo a

Let V, be the smallest K-invariant subspace of V containing rc (f~)V. Since f u e C, 0~ (G), it is clear that dim V u < co.

Since K0 is normal in K, 110 is stable under zr (K). Therefore since G = KM+K V is spanned by zr (KM +) 1Io. Fix k e K, m e M+ and v e Vo. Then

T w ~ (kin) v = ~ (k) T~ ~ (m) v (y e G').

But rr (fu) ir (m) v = T w I n (pm) vdp = T~ = (m) S n (m -x pm) vdp = T~ n (m) v Po Po

since m -1 Pom C Ko. Therefore

T u rc (km) v = rc (k) r~ (,fy) z~ (m) v e V u.

This shows that Ty V C Vy. Since T U commutes with zr (k) (k e k), dim V U < co and ~ is admissible, it is now clear that T, e End o g.

Fix k , m , v as above. Since the mapping Y~-*fu is smooth, it follows from the result obtained above that the mapping

y ,~ T w ~r (km) v = zr(k) zr (fu) rc (In) v (3, ~ G'),

f rom G' to V is also smooth. The second statement of the theorem is now obvious i f we recall that V is spanned by zt (KM +) Vo.

Corollary. Let O denote the character of zr. Then | coincides on G' with the locally constant function

x '-+ tr T, (x e G').

Pa t f 0 (x) = I f ( kxk -1 ) dk (x e G), E

for f ~ C ~ (G). Then

o f f ) = o o = tx 0"9.

lo0

Bat

for

n (fo) = j" f ( x ) T m dx G

f ~ C ~ (G'). Hence

O ( f ) = I f (x) tr T, dx G

Harish-Chandra

( f e c ~ (a')).

5. Some applications

Let F be a Cartan subgroup of G. For 7 e F ' = F I"1 G' and f e C~ (G), define

F t (?) = ] D (7) ]~ I f (x~ 'x-x) dx* GIA r

wkere D = DG, Ar is the split component of F and dx* is an invariant measure on G/Ar.

Theorem 3. Let Ko be an open compact subgroup of G. Given 7o e F', we can choose a neighbourhood co of Yo in F' such that F t is constant on o~ for every f e Co (a/Ko).

This resttlt had been proved by Howe some years ago in the case char /2 =- 0. Without loss of generality, we may assume that K0 is a normal subgroup of K.

Let us now use the notation of w 4. Then

S F (~? . p ) d k d p = l f ~ , ( x ) F ( x ) d x , KXPo G

for ? e F ' and F e C ~ (G). Fix f e C , ( G / K o ) and put

g (x) = I f ( kxk -1) dk (x ~ G). If

Since K0 is normal in K, g ~ C~ (G//Ko). Fix m e M + and let P (x )=-g (m -z xm) (x ~ G). Then

.f g (m -z . *y �9 pro) dkdp = Ifv (x) g (m -~ xm) dx. KXPa

But p m = m �9 m-l pm ~ raKe. Hence

I g ( m - ~ . * 7 . m ) d k = I f, y ( x ) g ( m - Z x m ) d x (y~F' ) .

Fix a neigkbottrkood Fo of Yo in F ' such that f i r=fir , for ?~Fo. This is possible by lemma 1. Then

I g ( m-1. *r �9 m) a~ = I g (m -~- % . m) dk (r ~ r0). K K

By standard arguments, the proof of theorem 3 is reduced to the case when F is elliptic. Then Ar = Z where Z is the maximal split toms lying in the centre of G. We know from the work of Brahat and Tits that

K ~ G/KZ ~_ M+] oMZ.

A submersion principle and its applications 101

Therefore Ft(7) = [ D (7) 1~ I f ( x T x -1) dx* GIg

= l D (7) [~ 2~ It (m) I g (m -1. ~7. m) dk, m~M+|oMZ E

where p (m) is the Haar measure of KmK. Let co be a neighbourhood of 70 in / '0 suck that [ D (7) [p is constant for 7 e o~. Then

F,(7) = F (70) (7 co).

Since co is independent o f f ~ C, (G/Ko), the theorem is proved. Theorem 3 makes it possible to prove Lemma 13 of([3], w 16) without any restric-

tion on char g2. I believe it is important, from the point of view of harmonic analysis, to obtain

an analogue of theorem 5 of ([1], p. 32). We give below a result which may be regarded as a step in this direction.

Take K0 ---- K and define fir (7 ~ F') as in w 4. Then f~ ~ 0. Put

/~ (7) ----- sup fv (x) (7 ~ F ' )

and define E as in ([3], w 14).

Theorem 4. Let co be a compact subset of F. Then we can number c such that

I ~ ( m-1 �9 ~ 7 �9 m) dk ~ cfl (7) E (m) 2, K

for all m ~ M + and 7 ~ og' = og N F'.

It is obvious that

Supp f " / C tO (*7. Po). (7 ~ F') .

Therefore Supp fv C Kco. Po ---- C (gay)

for 7 e co'. Since C is compact, we can choose a finite number y, (1 < i ~ r) in G such that

Suppf. r C U y~K l ~ - - - ~ r

for all 7 ~ co'.

Now fix 7 ~ 09', m e M+ and put

F (x) = E (m -1 xm) (x e G)

in the relation

I F ~ 7 . p ) d k d p = f f ~ ( x ) F ( x ) d x . K ~ o G

Observe that

F ~ 7 . P) = E (m -1 �9 ~7. pm) = E (rn -1 �9 ~7. m) @ eo)

choose a positive

of elements

102 Harish-Chandra

since m -1 p m e K. There fo re

I E ( m - t �9 *~'- m ) d k ~ f l ( 7 ) . Z I ~ ( m - l y l k m ) dk K r K

=/~ (r) 27 ~ (m-X y,) ~ (m)

f r o m the identi ty

I ~ ( x k y ) dk = ~ (x) ~ 0 ' ) (x, y ~ G). K

W e can choose a n u m b e r ct > 1 such tha t

E (xy,) < cl 2 (x) (1 < i < r)

fo r all x e G. Pu t c = rct and observe tha t E (x -1) = E (x).

I Z (m -1 �9 ~ . m) <_ ef t (7) E (m) 9 K

a n d this p roves our asser t ion.

One wotLld like to ver ify tha t

~ap I D (~) 1 ~, fl (y) < ~ o .

I believe this to be t rue bu t do not have a p roof .

Then we get

References

[1] Harish-Chandra 1966 Discrete series for semisimple Lie groups, II. Acta Math. 116 1~111, [2] Harish-Chandra 1970 Harmonic analysis on reductive p-adie groups, Lea ure Notes in A:ath.

(Berlin and New York: Springer-Verlag) Vol. 162. [3] Harish-Chandra 1973 Harmonic analysis on reductive p-adie groups, in Harmonic analysis

on homogeneous spaces (Providence: Am. Math. Soc.), pp. 167-192. [4] Harish-Chandra 1977 The characters of reductive p-adic groups, in Contributions to algebra

(New York: Academic Press), pp. 175-182. [5] Harish-Chandra 1978 Admissible invariant distributions on reducfive p-adic groups,

Lie theories and their applications, Queen's papers in pure and applied mathematics, No. 48 (t978), Queen's University, Kingston, Ontario, pp. 281-347.

Institute for Advanced Study, priaOeton, New Jersey 08540, USA