Annals of Mathematics
A Theorem on Semi-Simple P-adic Groups
Author(s): J. A. Shalika
Reviewed work(s):
Source: The Annals of Mathematics, Second Series, Vol. 95, No. 2 (Mar., 1972), pp. 226-242
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1970797 .
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?T-adicgroups
A theoremon semi-simple
By J. A. SHALIKA
TABLE OF CONTENTS
Introduction
Invariantdistributions
?1
on G
1.1 Invariantdistributions
1.2 Measureson conjugacyclasses
2
Asymptoticexpansionof Ff
2.1 Germsof functions
2.2 Determinationof re for SL2
References
Introduction
local fieldand G a connected,semi-simplealgeLet kbe a non-archimedean
braicgroupdefinedoverk. If G = G(k) denotesthe groupof k-rationalpoints
of G, thenG, withits naturaltopology,is locallycompact. Let G' and V denote
respectivelythe sets of regular and unipotentelementsof G, and let Cc(G)
denotethe space of locallyconstant,complexvalued functionson G having
compactsupport. For x E G' U V, let G(x) denotethe conjugacyclass ofG containingx. G(x) carriesan essentiallyunique G-invariantmeasure Pa (? 1.2).
Fix a normalizationof pa. For f E Cc(G) and x as above, let
If(X) =
5fd2.
G(x)
It is known [5(c)] that the integral convergesfor x regular. It has been
conjecturedby Harish-Chandrathat the integralalso convergesfor x unipotent. We verifythisby directcomputationforG = SL2 (? 1.2). The general
resultsdescribedbelowdependonthisconjectureand we assumeitthroughout.'
The main purposeof this paper is the studyof the functionIf. We considerthe followingproblem.
For x E G', determinethe behaviorof If(x) as x approaches the singular
set (that is, the complementof G' in G). We show in the presentpaper that
If has, in the precisesense definedbelow,an asymptoticexpansionin terms
of the integrals
AO(f)
5f du
1 Since the preparationof this manuscript,the author has been informedthat this conby Deligne and Ranga Rao.
jecture has been provedin characteristiczero, independently
SEMI-SIMPLE
P-ADICGROUPS
227
of f over the unipotentconjugacyclasses O. (For e = {1}, the trivial class,
we may assume that A0(f) = f (1).) The coefficients
rF(x) (x E G') in this expansionare independent
ofL The functionsre are determined
explicitlyonly
in the case G = SL2. We believe that the explicit determinationof these
functionsin the generalcase is desirableforthe followingreasons.
(a) Firstly,one expects a formulaforf(1) obtained fromthe values of
If on G'. We presentsome heuristicevidence(?2.2), involvingthe Steinberg
{1} and x elliptic, re(x) may
representation
of G, to the effectthat, for e
be expressed in a simple fashionin terms of invariantsof the affineWeyl
group, W,, of G. The importanceof such a formulaas a step in provingthe
Plancherelformulaforreal groupsis well known. It was suggestedby HarishChandrathat a similarformulacouldplay an importantrolein the p-adiccase
as well.
of a representation
(b) Secondly,by taking f to be a matrixcoefficient
fI of G of compactsupport,one expects to obtainan explicitformulaforthe
characterof II near the identityon each compacttorus. The leading termin
this formulashould be expressed in a simplefashionin termsof the formal
degreeof II and invariantsof W,. This, of course, agrees with the explicit
formof such characterswhen G = SL2 [12(a)].
We outlinehowexplicitexpressionsforthe functionsre lead to a proofof
the Plancherelformulafor G = SL2, (assumingk of odd characteristic). Let
G denotethe set of equivalenceclasses ofirreducible,unitaryrepresentations
of G. For II E G acting on a Hilbertspace 9, let 11(f) denotethe operatoron
9 associatedto fe CC(G). It is known[5(c)], [6], [12(a)], that H(f) is of trace
class. Let f(11) = trace 11(f). f is called the Fourier transformof f. For
i EG, let h = h(11)denote the conductorof 11 [12(a)]. Let dil denote the
Plancherelmeasureon G. Put
Ah(f)
fdih,
Gh
whereGh denotesthe set of II E G having conductorless than or equal to h.
A simpleformulaforAh, whichis a locally integrablefunctionon G, is given
in [12(b)]. By using Weyl's integrationformula(? 1.1), it is easily seen that
the formula
limha A
Ah(f)
=f
(1)
is equivalentto the formula
(T)
|(T
I
D(t)I re(t)Ah(t)dt= 0
fore unipotentof positivedimensionand h sufficiently
large, togetherwith
J. A. SHALIKA
228
the knowledgeof re for 0 = {1}. In this last formula,the sum is extended
over the equivalenceclasses of maximaltoriin G, and D(t) is the well-known
discriminant.For the case of general G, one may have to guess the formula
for Ah The above identity (*) then gives a necessaryconditionthat such
functionsconvergeto the delta distribution.
The resultsof this paper will be applied elsewhereto a detailedstudyof
local field[13].
the Fouriertransformon SL2 over a non-archimedean
The author would like to take this opportunityto express his sincere
appreciationto Harish-Chandraforhis encouragementand suggestionsover
the last several years. Many of the resultsdescribedhere.grew out of conversationsconcerninghis workon invariantdistributionsin boththe real and
p-adic cases. I referespeciallyto his lectureson p-adic groups[5(c)].
Let G be a connectedsemi-simplealgebraic group definedover a nonarchimedeanlocal fieldk. Let V denote the set of k-rationalunipotentelementsof G.
Throughoutthe paper we will make various assumptionsconcerningthe
set V. They have been verified,by several authors,forchar(k)= 0. The assumptionswill be stated in this paragraphand referredto whenappropriate.
We will also discuss underwhat conditionstheyare knownto be true. The
assumptionsare as follows:
(Ul) For all x0E V, themorphism
(p(xO):G
defined by (p(x,)g
=
gxg-'
G
(g E G) is separable.
This is obviousforchar(k) = 0. It is false forSL2 in characteristictwo.
(Ul) will be assumed throughoutthe paper.
Let G = G(k) denotethe groupofk-rationalpointsof G. G has the natural
structure of a locally compact group. For x0 E V, let ZG(xO) denote the centralizer of x0in G. ZG(xO)is a closed subgroupof G.
(U2) ZG(xO) is unimodular.
For char(k)= 0, this has been observed by Harish-Chandra. For a proof
and a detaileddiscussionof when the secondassumptionhas been verifiedin
in [3]. For furtherdisgeneral,we referto the articleof Springer-Steinberg
cussionsee also [11], [16].
carriesan invariant
For Z,(x0)unimodular,thehomogeneousspace G/Z,(x0)
measure. Denote this measureby dx*. Let x* denotethe canonicalimage of
an elementx of G in G/ZG(xo)
(U3) The integral
SEMI-SIMPLE 9P-ADIC GROUPS
|
GI ZG (xo)
229
(X0 G V)
f(X*xo(x-1)*)dx*
for all f E Cc(G).
converges
As stated in the Introduction,the thirdassertionhas been provedin characby Deligne and Ranga Rao [101. (U3) is readily
teristiczero, independently
verifiedforG = SL2 in all characteristics(see Prop. 1.2.2 below).
Finally, we assume
(U4) The numberof unipotentconjugacyclasses in G is finite.
It followsdirectlyfromKostant [7] and Borel-Serre[2], thatthisassumption is true in characteristiczero. It is false forSL2 in characteristictwo.
1. Invariant distributions
1.1.
INVARIANT DISTRIBUTIONS ON G.
In this section,G = G(k), the group of k-rationalpoints of a connected
local fieldk.
semi-simplealgebraic group G definedover a non-archimedean
For 0 an opensubsetof G, denoteby Cc(O) the vectorspace oflocallyconstant
functionsdefinedon 0 whichhave compactsupport. Cc(O) may be regarded
in a natural way as a subspace of Cc(G). If A is a distributionon G, we say
ofA ifthereexists a neighborhood
that an elementx ofG is in the non-support
0 of x such that the restrictionof A to Cc(O) is zero. The supportof A, denotedby Supp A, is then the complementin G of the non-support.
Let V denotethe set of unipotentelementsin G and G' the set of regular
elementsofG. Let A be an invariantdistributionon G, that is, a linearfunctionalon Cc (G) invariantby the actionof G inducedby innerautomorphisms.
In the followingwe assume that Supp A is containedin the closed set V or in
the open set G' and "determine"all such A. For this, we need some general
on real maniresultsofHarish-Chandra[5(a)] fromthe theoryofdistributions
foldswhose analogues are true forall local fields[5(c)].
local field,let Cc(M) denote
For any manifoldM over a non-archimedean
functions
on
M whichhave compactsupthe vectorspace of locallyconstant
concerningmanifoldsoverlocal fieldssee
port. (For the standardterminology
[14].) Suppose that M1and M2are analyticmanifoldsover k of dimensionm,
formson M, and
and m2respectively. Let co and c2 be analyticdifferential
M2of degreesm1and m2, and co
I I and Ico,I the correspondingpositiveBorel
measures[17]. Let 7wbe a surjective,analyticsubmersionfromM1,to M2.
THEOREM.(Harish-Chandra)Suppose thatco2is nowherezero. Then,for
everya E:Cc(M1),thereexistsa unique functionfa E:Cc(M2) such that
(1.1.1)
5M(Fo
M-
) aI
=
|
M
Ffa.1)21
J. A. SHALIKA
230
for all F E Cc(M2). Moreover
(1) Supp fa ci 7(Supp a),
(2) if co,is nowherezero, a v-.fa is surjective,
(3) if F is a Borel measurablefunctionon M2,thenF is locallyintegrable
(withrespectto Ic,21) if and onlyif Fo 7 is locallyintegrable(withrespectto
0), I) and (1.1.1) continuesto holdin thiscase.
We applythe above resultin two cases.
First application. Let G be a connectedsemi-simplealgebraicgroupdefinedover k and let 2 = Lie (G). Let x0be a k-rationalunipotentelementof
G. Let the morphism (x0): G
-
G be definedby p(x0)(g)
=
gxg-1
(g G G).
The kernel,Z(x0),of the tangentmap dp(x0)at any pointof G is seen to
be the set ofD E 2 satisfyingAd(x0)D = D. Let ZG(xO) denotethe centralizer
of xOin G. Then (assuming(Ul)) one has [1]
Lie Z,(xO) = Z(xO)
(1.1.2)
Let p(xo) also denote the induced map of G G(k) into itself,and let
ZG(XO)= ZG(XO)(k) denote the centralizerof x0 in G. Then ZG(xO)is a closed
subgroup (submanifold)of G, and the quotientG/ZG(xO)is an analyticmanifold. It followsfrom(1.1.2) that the inducedmap of G/ZG(xO)into G defined
by 'p(xo)is immersive. Let G(xO), the conjugacyclass of x0in G, denotethe
image of this map. Then G(xO) is a submanifoldof G and is, in particular,
locallyclosed. Let ( = Lie (G). The tangentspace at any pointof G(xO)may
with (b/LieZG(xO).
be identified
of x0in G, and let x1,* X, be the
Let J be a coordinateneighborhood
coordinatefunctions.Supposethat G(xO)is definedin J by the equationsx1=
of G containingthe
x2 = *.* = XI = O. Choose U a local analyticsubmanifold
0..= Xh = O. Considerthe map 7 of
identityso that x0U is definedby x, =
G x U into G definedby 7r(x,u) = x(xou)x-1(x E G, u E U). We may identify
the tangent space at any point of U with a subspace U of (X. With these
we findthat d, the jacobianof 7r,evaluated at (x, u) in G x U
identifications,
is given by
d(D, 0))
Ad(x){(Ad(xou)-
-
1)D + c)},
0) U).
(D E (, cE
As in [5(a)], we have
PROPOSITION1.1.3. Thereexistsa local submanifoldU' of U so thatd is
surjectivefor x E G, u E U'.
Let Ml = G x U' and let 7walso denotethe restrictionof 7wto M1. Let co
be an (analytic)formon G so that I oI is a Haar measure dx on G. We set
SEMI-SIMPLE
231
P-ADICGROUPS
du = dx,A A. A dx, and co),= c x du. If M2= Q denotesthe image of 7r,
then,by Proposition1.1.3, M, is an open submanifoldof G. We let co2denote
the restrictionof coto M2
Secondapplication. Let T be a maximalk-torusin G, and let T= T(k).
of1 - Ad(t) acting
Put T' = Tn G'. For t e T, let D(t) denotethe determinant
on G/Lie(T). T' is definedby the equation D(t) # 0, and hence is an open
submanifoldof T. Let N(T) denote the normalizerof T in G, and let WT
denotethe finitegroupN(T)/T. Let M1= G/T x T'. For x E G, let x* denote
the image of x underthe canonicalmap fromG to GIT.
Now define7wfromM, to G by setting 11(x*,t) = xtx-'. 7wis well-defined.
Choose Co* and oT, differentialformson GIT and T' respectively,so that
dx* is an invariantmeasureon G/T,and IcTI is the restrictionto T'
CO*
of a Haar measuredt on T. We may assume that dx = dx*dt. In this case,
it is knownand easily verifiedthat the jacobianof 7wat (x*, t) is j D(t) l. Thus
7wis submersive. The image, M2, of M, under7r,whichwe also denoteby (TP)G,
is thusan open submanifoldofG. Let o,),= o0 X (0)T and let w2be the restrictionof c to M2. Withthesedefinitions,
as in [5(a)], [5(c)], we have the following corollaryto the above theorem.
COROLLARY
1.1.4. (1) For all a e Cc (G x U'), thereexistsa uniquefunctionfa e Cc (Q) such that
(1.1.5)
|U
u)dxdu=
F(x(xou)x-')a(x,
F(x)fa,(x)dx.
Themap a v-.fa of Cc (G x U') intoCc (Q) is surjectiveand SuppfaG w(Suppa).
Moreover,if F is locallyintegrableon Q, thenthemap (x, u) v-.F(x(xou)x-'),
(x E G, u E U'), definesa locallyintegrablefunctionon G x U' and (1.1.5) holds
in thiscase.
(2) For all a E Cc(G/T x T'), thereexistsa unique functionfa EGCc((T')G)
such that
(1.1.6)
(x)dx .
t)dx*dt |GF(x)f,,
(T"
(T')G
F(x*t(x-')*)ar(x*,
JG/TXT'
|IXT
The map a v--fi of CcGG/T x T') into Cc ((T')G) is surjective and Supp fi C
7r(Suppa). Moreover,if Fis locallyintegrableon (T')G, thenthemap (x*,t) H
F(xtx-1),(x E G, t E T'), definesa locallyintegrablefunctionon G/T x T' and
(1.1.6) holdsin thiscase.
As in [5(a)], one deduces the following.
1.1.7. (1) Let A be an invariantdistributionon Q. Thenthere
THEOREM
existsa unique distributionUA on U' such that
232
J. A. SHALIKA
A(fa)=
V-)
where ,0 e CP( U') is definedby /Ja(u) = a(x, u)dx, (u e U').
G
(2) Let A bean invariantdistributionon ( T) G. Thenthereexistsa unique
distribution sA on T' such that
A(fa) =UA(j),
where80t e Cc(T') is definedby/5'a(t) = |a(x*,
t)dx*, (t e T')
GIT
In both(1) and (2) above,AA = 0 implies thatA = 0.
The expressionforS, maybe made moreexplicitby using
Weyl's Lemma. Suppose that dx, dx*, and dt are normalized so that
dx = dx*dt. Then,for any integrablefunctionf on (T')G,
f(x)dx
=
[WT: Ii-j
ID(t) l|
f(x*t(x )*)dx*dt
The prooffollowsfromthe explicitformof the jacobian of 7win this case.
UsingWeyl's Lemmaand takingF to be a locallyintegrableclass function
in part (2) of Corollary1.1.4, we see immediatelythat, fora e Cc(G/T x T')
f3a(t)
[ WT:11- ID(t) 1fG(xIt(x
) ,)dx,
(t e T')
For anyfe Cc-(G)and any maximaltorus T in G, we set
(1.1.8)
FfT(t)= ID(t)
11/2
f(x*t(x-1)*)dx*,(t e T')
wheredx* is the measureappearingin Weyl's Lemma.
Throughoutthe remainderof the paper,we assume that the measuresdx,
dx*, and dt are normalizedas in Weyl's Lemma.
1.2.
MEASURES ON CONJUGACYCLASSES
In this paragraph,we associate to each regular or unipotentconjugacy
class in G (see assumptionsbelow) an essentiallyunique distributionon G.
We also determineall invariantdistributionson G with supportin the set V
of unipotentelements.
Let x, be a regularor unipotentelementin G. As in ? 1.1, let G(x0)denote
the conjugacyclass in G containingx0,and let ZG(x0)denotethe centralizerof
X0in G.
G(xO)is locallyclosed (? 1.1) and is thereforelocallycompact. Hence, by
Arens' Theorem[9], G/ZG(xO)
and G(xO)are homeomorphic.Let P be the Borel
measure on G(xO)obtained by transportingan invariant measure dx* on
We quote a resultof Harish-Chandra[7(c)] which states that, for
G/ZG(xO).
233
P-ADICGROUPS
SEMI-SIMPLE
x0e G', the integral
(1.2.1)
G (xo)
= |
~aizc)fd
GI ZG (X0)
f(x*xo(x-1)*)dx*
convergesforall f in Cc(G).
For later purposes,we provethe following.
PROPOSITION1.2.2. Let G = SL2. For xoe V, the integral 5fd1Le conG (xo)
verges.
Proof. For x0= 1, the propositionis trivial. We may thereforeassume
that x0is of theform(1 whereCe Vc. Let r denotethe
by conjugation
ringof integersin k and put K = SL2(r). Let A denotesthe groupof diagonal
ofelements
inG andN thesubgroup
oftheform(j $) for~eEk. Let
matrices
dn
denoteHaar measureson K, A, and N respectively.It is well
dK,da, and
knownthat G -KAN, and, forf integrableon G,
j)
G
f (x)dx
5
IX I2di da dn,
|Af(Kan)
KXAXN
provideda is of the form(0 ?-), X e kx,and dx is someHaar measureon G.
It followsreadilythat,if a is a functionon G satisfyinga(xn) = a(x), x E G,
n e N, and a is integrableon GIN, then
(1.2.3)
|cx(x*)dx*=
GIN
|cx(ra)
KXA
X12 di
da.
Now, forf e Cc(G), definea by a(x) f(xx0x-1),x e G. Using (1.2.3) and
IX IdX,where
|5 ((o
makinga changeofvariables,we obtain 5 fd
and dX is a suitable Haar measureon k. The proposition
7(x) =
f(Kcxrc-')dKc,
K
is now clear.
For any unipotentor regularconjugacy class (9, fix xoe () and ,cean invariant measure on e as above. Assuming (U2) and (U3) we define,for
fE Cc-(G),
As(f)
|
GI ZG (XO)
f(x*xo(x-')*)dx*
A0 is an invariantdistributionon G with supportin the closure(' of (9. Assume (Ul) and (U4). We then have the followingresult.
PROPOSITION1.2.4. Let C0be a unipotentconjugacyclass. If A is an in-
variantdistributionon G withsupportin theclosureO' of(D0,then,forsuitable
constantsCC,,
A = Le COAO
J. A. SHALIKA
234
wherethesummationis extendedoverall conjugacyclasses C containedin C0.
Let 00 be a regular class. If A is an invariant distributionon G with
supportin C0,then
A = cAo,,
for a suitableconstantc.
The proofof the propositionis containedin the followinglemmas.
LEMMA1.2.5. (Localization) Let M be a manifold over k, and let
f e Cc(M). Suppose that { Vi}iGe is an open coveringof Suppf. Then there
existfunctionsfi e Cc(M) with Suppf, c V, such that
(1) f = 0 for all butfinitelymany i e I,
(2) f = Ye GI fi
Proof. We may assume withoutloss of generalitythat f is the characteristicfunctionof a compactopen set S, that I is finite,and that each Vi is
containedin S. S is a compactmanifoldand, hence, is the disjoint union of
a finitenumberof balls. We may thereforeassume that S is a ball in Euclideanspace.
Supposethat S = V1U V2whereV, and V2are open. Take x e S. If x e V1,
let B(x) be a ball containingx and containedin V1. If x ? V1,let B(x) be a ball
containingx and containedin V2. We have UxeS B(x) = S. Select a finite
subcover B1, ***, B,. We may assume that B. f B3 = 0 for i = j. Thus S =
US= Bi (disjointunion). Suppose that B1, * , B, are containedin V1 and
**...I B. are not. Let ao denote the characteristic functionof B., 1 < i _ s.
B+,
at. Then f1and f2are in Cc(S), f
and f2 = f
Let f, = f 1a,
f1+ f2, and fi has supportin Vi (i = 1, 2). This proves the lemma in this
case. The generalcase followsby a simpleinduction.
-
?
Assume (Ul) and (U4). Then we have
Let C0bea unipotentconjugacyclass. Let C' denotetheclothendimk 0 < dimk 00.
sureofC0. If C is a conjugacyclass in G contained \0\C0,
LEMMA 1.2.6.2
Proof. Fix xOe C0. Let G(xo)denotethe image of xOunder9(xo), that is,
the G conjugacyclass containingxO. Let CL G(xo)denotethe Zariski-closure
of G(xo)in G. Then G(xo) is a smoothvarietywhichis open in CL G(xo) [1].
Moreover,the algebraicdimensionof any G orbitin CL G(xo)\G(xo)is smaller
than that of G(xo).
DecomposeG(xo)(k)into G orbits:
G(xo)(k) = G(xo) U G(x1) U
2
...
U G(x7),
The author is indebtedto W. Casselmanfor the proofof this lemma. See also [5(c)].
235
SEMI-SIMPLE 9)-ADIC GROUPS
where 00= G(x0). By (U4), the (disjoint) union is finite. From the fact
that 9(x0) is submersive, it follows that the induced map G(k) v--G(x0)(k) is
submersive and thus the image G(x0) is open in G(x0)(k). Similarly, each
G(xi), 1 < i < r, is open and hence closed in G(x0)(k). It follows that
(DcO-(\D0czCLG(xJ)\G(xJ). Thus dimk C <dim G(x) =dimG(xJ). This proves
the lemma.
Now let Vs = UdimO<sC,where C is a unipotent class. By Lemma 1.2.6,
V8 is closed and each class of dimension s is open in Vs. Fix x0e V. Since G(x0)
is op.n in its closure G(x0)', we can choose Q0 open in G, invariant by inner
automorphisms, so that Q0 nG(x0)' = G(x0). Then we can choose U0 open in
U containing 1 so that r(G x UJ)= Q,. With this notation, we have (assuming (Ul) and (U4))
1.2.7. Suppose that A is an invariant distributionon Q0 with
supportin C = G(x0). Then
A= cAr
LEMMA
for a suitableconstantc.
Proof. Let a, be the distributionon U0 associated to A by Theorem 1.1.7.
Clearly, xx0x-1e x0U, x e G, implies that xx0x-1= x,. Proceeding as in Lemma
23 of [5(a)], we have Supp A ci {1}. Thus, UA c=uA, where Al is the delta function at 1 in U,. Since AO clearly satisfies the assumptions of the lemma, we
are through.
We now prove Proposition 1.2.4 forunipotent classes C0. Suppose dimC0= s.
CN be all the unipotent classes of dimension s. Then, by the
Let (9, 02@
above, each (9, 1 _ i < N, is open in V, As above, choose Q1,* , Q, open in
G, invariant by inner automorphisms, of the form r(G x UJ), and such that
...,
Q, 0nV. = Ci,
Lemma 1.2.7,
1 < j < N. Let Aj denote the restriction of A to Qj. Then, by
Aj = cjAoj ,
1_ j
_
N
for suitable constants cj. Fix fe Cc(G) so that the restriction of f to
Therefore,byLemma
C0j)iszero. Then Suppfcz U1=1QjU(G\V8).
V.\(U>
1.2.5, we may write
f =fE1 fi + g,
where Supp fj ci Qj, 1 < j < N, and SuppgczG
vanishes on the closure of Q,
A(f) =
Thus, A
-
V,. In this case, since f-fl
1A(f5) = #'Y1cjAO(fj) =
N c5jAOhas support in V8\(Ui'
ccjAo0(f)
C9j)
V,,1. The proposition
J. A. SHALIKA
236
followseasily by induction.
2. Asymptoticexpansion of Ff
In this sectionwe obtain a limitingrelation between distributionssupportedon G' and those supportedon V. In the case whenG = SL2 and k has
odd residualcharacteristicthis result will be appliedelsewhereto derivethe
Fouriertransformof a distributionwith supportin V fromthat ofa distributionwith supportin G'. In particular,one findsan expressionforf(I) from
the values of If on G' [13]. This is similar in spirit to the resultof HarishChandraforreal semi-simpleLie groups.
2.1.
GERMS OF FUNCTIONS
In orderto state the main result of this paper, we must firstdefinethe
germ of a functionnear 1 on a maximal torus in G. As above, let G be a
groupdefinedoverk. If p is a fixedk-rationalfaithfullinearrepsemi-simple
then
resentationof G, and t is an indeterminate,
Det(t - (p(x) - 1)) - t + PV_1(x)tN + ... + PO(x)
Each P3 is a regularfunctionon G definedover k, and PNl1(X)
= P0(x) 00 if and only if x is unipotent.
Let P denotethe map of G intokN definedby
= PN2(X)
xc G
>(PV_(x), ... , PO(x)),
Now supposethat d is a positiveintegerand that 9P is the prime ideal in k.
We set k1N= C1dX ... X ?fd C k x *.. x k, wherethereare N termsin each
product. Let Gd denote the completeinverse image of kv in G under the
mappingP. Then, since kI is open and closed in klN, Gd is open and closed in
and nd?o Gd = V.
G, invariantby innerautomorphisms,
Let T be a maximaltorusin G, T' =G'
T, and put Td = T0nGd. Two
complexvalued functions,r1,and J2, definedon Td, and Td2respectively,are
calledequivalent iftheyhave thesame restrictionto Td1n T'2. An equivalence
class definedby this last relationis called the germof a functionon T'. If f
is a complexvalued functionon T', we denotethe germoff by {f}.
In the remainderof this paragraphwe will assume (Ul) through(U4).
xi
2.1.1.3 Let T be a naximal torusin G. Thenthereexistunique
THEOREM
with the unipotentclasses 0,
correspondence
germs{rF} on T' in one-to-one
and depending only on T, such that, for all f e Cc(G),
{FfJ} =
{1}Al(f) .
3 The theorem is valid without reservation in characteristic zero.
SEMI-SIMPLE P-ADIC GROUPS
Let so > s1 >
...
>
SN
237
be the dimensions of the various unipotent classes.
Let s be one of these integers. In the following,we shall deal with a family
of distributionsAs, t e T', satisfyingthe followingcondition:
(2.1.2)
forall f e Cc(G) vanishingon Vs, there
exists an integerd, possiblydependingon f,
so that As(f) = 0 forall t E Td'.
Let 0 be a unipotentclass of dimensions. Choose Quaopen in G of the
formr(G x UO)so that Q2 nVS= O. Let oas, t e T', be the distributionon
UOassociatedwith As,.
LEMMA 2.1.3. Fix 8 e Cc (UO) such that fl(1) 0. Then thereexists an
0 for all t e Td0.
integerd0,possiblydependingon fi,so that ao(f)
Proof. Take ,8as in thehypothesis.Since,8is locallyconstant,1 X Suppfl.
Take ye Cc(G) so that 5dx = 1. Put a = y x 6' and set f = fa. Then
G
(2.1.4)
As(f)
=
as(8
7
Supp f c r(Suppa) .
(2.1.5)
As in the proofof Lemma 23 of [7(a)], we have Suppf ne 0 0. Sincef has
supportin Q,, it followsthat f vanishes on Vs. Therefore,by (2.1.2) and
(2.1.4), the proofis immediate.
For t e T', f e Cc(G), definea function(q,,fon T' by
9,f (t)
=
As(f ) t e T'
2.1.6. Fix a unipotentclass O. Then thereexists a unique
COROLLARY
dependingonlyon T, such that,for all f e Cc(QO),
T',
on
germ{fr}
{9(sf} = {PfO}AO(f).
Proof. Choosea e Cc(G x Uj) so thatf = fa. Let 8 = 8,8. Take L to be
a fixedopencompactset in UOcontaining1, and let C denotethecharacteristic
functionof L. Put Po = f8-l8(1)C. Then fl0(1)= 0. Therefore,by Lemma
= 0, for
2.1.3, there exists an integer d so that a-9(0) = t(fl) - (1)S(c)
te Td. By the proofof Lemma 1.2.7, e(1) = cA(f), for some constantc,
independentoff. We may take f'o(t)= cat(C). The uniquenessis obvious.
For each unipotentclass 0 of dimension
s, thereexistsa unique germI{0} on T', dependingonlyon T, such that,for
all fe Cc(G) which vanish on
,
V0
COROLLARY2.1.7. Let s
{(sf}
= Sk
= Edim0=s
{rO}AO(f)
Proof. For each class 0 chooseQ0 as above. If f vanisheson V8kl, then
238
J. A. SHALIKA
SUPPf C Udim.C=. QO U (G\ Vs). Hence, by Lemma 1.2.5, f = dimt9=s fO + g,
whereSuppft(c QO, Supp g c G\ Vs, and each fe belongsto Cc(G). Since g
vanisheson Vs,by (2.1.2) thereis an integerdoso that AS(g)= 0, forall t e Td0.
Therefore,by Corollary2.1.6,
{9(sf}
I
:dimm=s
9)s'fo
EdimmD=s
{If}AO(f0)
Sincef - ft is zero on 0, this last sum is equal to
Edim.O=s
{I
C}IAO(f)
PROPOSITION2.1.8. Thereexistunique germs{FI} on T' so that,for all f
vanishingon V8s,
{FJ'} =
Edimo>s
{}(f)
Proof. First supposethatf vanisheson V= V... Then, since nd Gd= V
andf has compactsupport,Suppf c G\Gd ford > do. In thiscase, Ffr(t)= 0
for t e Td.. Suppose the statementis true for s = Sk, and supposef0e Cc(G)
vanisheson Vsk~l For t e T', f e Cc(G), put
As(f) = Fr(t) -
IdimD>s
PC(t)Af
By induction,As satisfies(2.1.2). Hence, by Corollary2.1.7, there exists an
integer d, so that As(fo) = Fdim s JI%(t)AO(fo),
for t e T' . Combining these
two equalities provesthe assertionforSk+li
The proofof Theorem2.1.1 is now obvious.
2.2. DETERMINATIONOF re FOR SL2
In thisparagraph,we determineexplicitlythe germs{J7O},
definedin ?2.1,
in the case G = SL2. We assume that the characteristicof k is not two.4
Let N and N- denotethe subgroupsof G consistingof elementsof the
form(1 x) and (1 0), xe k, respectively. Then V = NG. For x0 in N, it is
easily seen that we may take the local submanifoldU' (? 1.1) to be N-. As
above, let Q = Q(xo)denotethe set ofelementsin G of the formxxonx-1
(x e G,
neN-). LetT =T' nQ.
PROPOSITION2.2.1. TQ depends only on the conjugacyclass ( of G con-
tainingx0 (xoe N).
The prooffollowsimmediatelyfromthe fact that two non-identity
elementsof N are conjugateif and onlyif theyare conjugateby an elementof
A togetherwith the fact that A normalizesN-. Thus we may write Ti0for
TQWe normalizeAOas follows. If C = {1}, we take AeKf)= f(1), (f e C-(G)).
4In
this case all of the assumptions(U1)-(U4) are valid.
239
SEMI-SIMPLE 9-ADIC GROUPS
If C containsan elementof the form(1
AO(f)
=
j), we take
1))
(kx)2d((O
(see ?1.2) .
With these normalizations,we have
2.2.2. Let T be a compacttorusin G. Let C.,,denotethecharacteristicfunctionof To. Then,for f e C-(G),
{Ff} = -AT{1D 11}f(1) + BT EdimO>O {CcJA0(f)
THEOREM
whereAT and BT are positiveconstants.5
The proofis containedin the followinglemmas.
For t e To, we maywritet= xx0nx-'(x e G, n e N-). It is easy to see that,
givent, n is unique. Let n = n(t). By directcomputation,one sees that for
t1, t2e Te, n(t1)= n(t2)if and onlyif thereexists w e WT such that t1 wt2w-1
(see the proofof the lemma below). Thus we obtain an analytic map from
T@/WT intoa subsetN7 of N . The jacobian at t e To is seen to be cr I D(t) 11/2,
whereCT is a positiveconstant[13]. Therefore,the above mappingis an analyticisomorphism.
LEMMA 2.2.3. Suppose that T is a fixedcompacttorusin G. Let ( be a
non-trivialunipotentconjugacyclass. Then
{rP} = cl{Co}
wherec3 is a constantpossiblydependingon O.
Proof. Choose Q0 as above. Fix ,8e Cc(N-) so that f8(1)# 0. Choose
re Cc(G) so that 5 dx = 1. Let a = y x S. Let f = f, be the corresponding
G
functionon Q0. By Corollary2.1.6, thereexists an integerd so that Ff (t) =
rF(t)Ao(f) fort e Td. Since 8 is locally constant,we may choose d so large
that ,(n) = 8(1) for all n e N
satisfying xn e
Gd.
Let F be any locallyintegrableclass functionwithsupportin (Td)G. Then,
by Corollary1.1.4,
|F(x)f(x)dx =
QCa
N-
F(xon),(n)dn=,8(l)| F(xon)dn.
NT
On the otherhand, by Weyl's Lemma, sincef has supportin Qu
ByQchanging
|F(x)f(x)dx
wc
variables,
=
r
WT
D(t) l112F(t)FfT(t)dt
Bychangingvariables,we can writethelast integralas
5
For k not of characteristictwo, explicitvalues for AT
(CA)-1
F(x(,n)Ff'xon)dn,
N-T
and BT may be foundin [13].
J. A. SHALIKA
240
wheretheintegrationmaybe takenoverthosen satisfyingx,,ne Gd. Sincere is
invariantby the action of W, on T', we may extendit to a functionon(T,)G;
similarly forFf}. The last integral then becomes (c)-'Ac(f)
By Lemma 1.2.7, ,(1) = cAo(f),
A8(1)
#
NT
rO(xof)F(xon)dn-
where c' is a constant. Therefore,since
0,
C@CT
F(xOf)dn
=
Nr(x)F(xTn)dn
Changingvariablesagain, we obtain
CfCT
Tel W 7
5
F(t) ID(t) 11"2dt
Tel IFT
.
.
F(t)rF(t) ID(t) "1/2dt
Since thislast equalityholdsforany boundedfunctionon T/WT with support
in T/ WT, we have ra(t) = Cb'CT forall t in To n Td. To finishthe proof,we
observethat, forfe Cc(Qu_),Ff has supportin To.
LEMMA
is independentof C.
2.2.4. For ( a non-trivialunipotentclass, coD
g = (1 1), aekx. For xe G, letx = gxg-'. Themapx-* xg
inducesa transitiveaction, e --*09, of kx on the non-trivialunipotentclasses.
Proof.
For f e
Let
define f9 e Cc (G) by fg(x) = f(xg), x e G. Assume f satisfies
f(x), x e G, K e K. Using the explicitformulaforAQ,we have
CC (G),
f(Kxi-1) =
(2.2.5)
A((fg)
=
Ia I-' Aeq(f)
We also have Qog = gQo9g-1.
Now fix 8 e Cc(N) so that 8(1) 0O. Choose -ye Cc(G) so that dx = 1.
functionon Cc(Q,,). Then
Put x = y x , and let f = fa be the corresponding
f9 e Cc(g-'Q~g). Definea' on G x N by
a'(x, n)
(2.2.6)
ja 1-1a(gxg-, gng-1)
Using the definingconditionsatisfiedbyfa, we see that the functionon g-'Qcg
to a' is f 9. Definec' as in the proofof Lemma2.2.3. It suffices
corresponding
to provethat ct9is independentof 0. By (2.2.5) and (2.2.6), we have
a 1-' , c(1)= 'g-1Ag-1(f9) = a j-'c' Aq(f)
-
Consequently,c; = csg-1,and the lemmais proved.
D 1/2} follows from the explicit form of Ff}
forspecialf e Cc(G) [12(b)],[13].
In the remainderof thisparagraph,we give heuristicevidenceto support
.a conjecturalformulafor]e0 when G is connected,semi-simpleand C = {1} is
-thetrivialclass. For simplicity,we assume that G is split and referto [8]
The fact that {Fl}
=-AT{I
SEMI-SIMPLE
241
SP-ADIC GROUPS
forthe moregeneralcase. As in [15], let fl = fl, be the Steinbergrepresentation of G associated with the trivial characterX. Then d(fl), the formal
degreeof H, is given by d(fl)-' = meas(B) CG, where B is an Iwahori subgroupand C0 dependsonlyon the affineWeyl groupof G. Let (p be a B-finite
of l((p(l) = 1). In generalq' does nothave compactsupport.
matrixcoefficient
It is reasonableto expect that H has a characterhe given by
on(X) = d(H)|
(p(yxy-')dy ,
forx elliptic. Moreover,by analogy with the formulaof Curtis [4] for the
Steinbergcharacterof a finiteB-N pair, we expect that, for x elliptic,
e11(x)- (-_ ), where r is the rank of G.6 Thus, formally(assuming the
asymptoticformulaforqA),we have
F,(x)
(-_)r meas(B)CG"(T) 1 ID(t)
x1/2 -
F()
+
FdimO>O
t(x))A0(q)
By using homogeneitypropertiesof the functions1F,0under"stretching"(assumingk has characteristiczero) as in [5c] we concludethat
F1}
1(X)
=
(-1)r
meas(B) CGe(T)-'
ID(t)
11/2 ,
forx ellipticand sufficiently
small. One maycheckthisdirectlyforG = SL2(k),
local field(chark # 2), by using the resultsof [13].
k any non-archimedean
THE JOHNS HOPKINS UNIVERSITY,
BALTIMORE, MD.
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(Received October4, 1970)
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