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The fundamental lemma

by Bill Casselman


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Contents

1. The naked lemma .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3. The program for this seminar .............. 14

4. Unramified groups ........................... 185. Satake transform and Hecke algebra . . . . . 22

6. The L- g r o u p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 87. Endoscopic groups . . . . . . . . . . . . . .. . . . . . .. . . . . 31

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1. The naked lemma

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Suppose G to be an unramified reductive group defined over

a p-adic field k, H an unramified endoscopic group of G. Thetwo have the same rank.

Given a strongly regular semi-simple element t of H, let Tt

be its centalizer in H, a maximal torus. There exists a specialembedding of Tt in G. The element t gives rise to an elementtG of G, which we assume to be strongly regular in G.

Define orbital integrals of a locally constant function of com-

pact support on any reductive groupfor any strongly regular

t let

(f, t) = G

(t) 11/2vol T(o)vol G(o) CG(t)\G f(g1tg)dg

dt .

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(f, t) =

G (t) 1

1/2

vol T(o)

vol G(o)

CG(t)\G

f(g1tg)dg

dt.

The ratio of volumes is to eliminate the effect of choice of

measures on G and T.

Orbital integrals have something to with fixed-points, since

CG(t) is the set of points on G fixed under conjugation by t.We shall see that this connection extends to something very

deep. The product is over roots

with respect toT

. Such

factors are familiar in fixed point formulas.

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The data determining H have something to do a set of ele-ments s in the torus

T in the L-group LG. They all give rise

to the same map from the stable conjugacy class of tG,constant on the ordinary conjugacy classes. The -orbitalintegrals are linear combinations of the ordnary orbital inte-

grals:

G,H(f, t) = tt

(t

)G(f, t

) .

If 1 this is the stable sum stG(f, t).

Stable conjugacy means conjugacy in the algebrac closure.

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Any element f of the Hecke algebra HG//G(o) gives riseto an fH in H(H//H(o)

.

Fundamental Lemma:

G/H(f, tG) = stH(f

H, t)

for all strongly G-regular semi-simple elements t of H.

Very roughly, this says that analysis on G invariant underconjugation can be reduced to stable analysis on G and allits endoscopic groups.

Implicit in this are assertions about character sums in G match-ing stable character sums in H.

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2. History

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Bill Casselman, Notes on p-adic spherical functions, written

as a preface to a new printing of Ian Macdonalds book on

spherical functions.

Mark Goresky, Robert Kottwitz, and Robert MacPherson, Ho-

mology of affine Springer fibres in the unramified case, avail-

able on the http://www.arxiv.org.

Tom Hales, The Fundamental Lemma for Sp(4), SLProceedingsof the American Mathematical Society 125 (1997), 301-308

http://www.arxiv.org/math/0312227v2/

Tom Hales, A statement of the Fundamental Lemma, avail-

able on the http://www.arxiv.org.

http://www.arxiv.org/math/0312227v2/

David Kazhdan and George Lusztig, Fixed point varieties on

affine flag manifolds, Israel Journal of Mathematics 62 (1988),129168; appendix by Joseph Bernstein

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Robert Edward Kottwitz, Calculation of some orbital integral-

s, pages 349362 in The zeta functions of Picard modular

surfaces, CRM.

Jean-Pierre Labesse and Robert Langlands, L-indistinguishabilityfor SL(2), Canadian Journal of Mathematics 31 (1979), pages

726785Robert P. Langlands, Les debuts dune formule des traces

stables, Publications mathematiques de lUniversit e Paris VII,1983

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Gerard Laumon, On the Fundamental Lemma for unitary group-

s, audio file at

http://www.fields.utoronto.ca/audio/02-03/shimura/laumon/

Ngo Bao Chau, Hitchin fibration and the Fundamental Lem-ma, talk at the IAS (October 9, 2006):

http://www.math.ias.edu/lg40/ngo.pdf

Jonathan Rogawski, Automorphic representations of unitarygroups in three variables, Princeton University Press, 1990.

Jonathan Rogawski and Don Blasius, Fundamental Lemmas

for U(3) and related groups, Pages 363394 in The zeta func-tions of Picard modular surfaces, CRM.

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The first result of this type was in the paper by Labesse and

Langlands on SL(2). It was first formulated in complete gen-erality in Langlands Paris VII booklet. Both of these were

motivated by the problems of stable conjugacy arising natu-rally in the trace formula.

It was proven for SL(3) by Kottwitz. He calculated orbital in-tegrals by means of geometry on the Bruhat-Tits building of

G, following Langlands work on base change.

The group SL(n) was dealt with by Waldspurger. Rogawskiand others proved it for U(3), Hales for Sp(4). I do not know

what techniques they used.

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A paper by Kazhdan & Lusztig, with an appendix by Bern-

stein, changed the situation drastically. In this it was pointedout that orbital integrals count points on algebraic varieties

over finite fields. Goresky, Kottwitz, and MacPherson used

this idea and formulated the Fundamental Lemma in terms of

equivariant cohomology. They proved it for unramified conju-

gacy classes.

Laumon, working at first by himself and then with Ngo, proved

it for unitary groups U(n). It appears that recently Ngo, usingsomewhat related techniques, has proved it in all cases.

All along, various important technical results and reductions

have been added by various people, including Labesse, Lang-lands, Shelstad, Kottwitz, Waldspurger, Hales, Cunningham,

Kazhdan, Lusztig, and Bernstein.

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3. The program for this seminar

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There are many relatively elementary items to be explained.

These include

the structure of unramified groups

the Hecke algebras

the L-group (which contains the torus T) the map from the Hecke algebra of G to that of H

characterization and properties of (unramified) endoscopic

groups stable conjugacy and Galois cohomology

the characters

orbital integrals and the geometry of buildings the work of Kazhdan, Lusztig & Bernstein relating orbitalintegrals to the geometry of the affine Grassmannian

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There are many nastier items to be explained. These include

why LH does not always embed into LG

specification of the embedding of TH into G (transfer fac-tors)

reduction of the general case to that in which f and fH areHecke units

relationship very generally between orbital integrals and

varieties over finite fields role of equivariant cohomology

Ngos use of what he calls the Hitchin fibration

the transfer from the geometric fields to p-adic groups the geometric version of the Satake transform in terms ofsheaves on the affine Grassmannian

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It would be nice to exhibit some applications, but thats too

much to expect.

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4. Unramified groups

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Split reductive groups over k are classified up to isomorphis-

m by root dataquadruples

(L,, L,)

where L is a lattice, a root system in L, to coroots inthe dual lattice L. Here L is to be identified with the groupof characters of a maximal split torus T in G. Given a Borelsubgroup B we get a based root datum

(L,, L,) .

The lattice L is the character group X

(T) = Hom(T,Gm),L the cocharacter group X(T) = Hom(Gm, T). We may

identify T(k) with Hom(L, k) or L C.

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An unramified group has two characterizations. (1) It is ob-

tained by base extension from a smooth reductive group

scheme over o. (2) It is one that splits over an unramfied ex-

tension of k. Essentially because reductive groups over finitefields possess Borel subgroups, so do G(o) and G. Let it beB, T a torus T contained in B, and A a maximal split torusin T. Such groups are classified through Galois descent the-

ory by quintuples(L,, L,, )

where is an automorphism of the based root datum, giving

rise to the Galois action on roots.

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The lattice L and determine the unramified torus T. The

embedding of A in T induces an isomorphism of A = A/A(o)with T = T/T(o). These may both be identified with X(A) =(L) via images of the generator of p.

The relative Weyl group is the subgroup of the split Weyl

group that takes X(A) to itself. Elements of W may alsobe characterized as elements of the split Weyl group fixed by

the Frobenius.

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5. Satake transform and Hecke algebra

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The Hecke algebra is the convolution ring HZG//G(o). offunctions on G of compact support, bi-invariant under G(o).If (, V) is an admissible representation of G, it is called un-ramfied if it subspace of vectors fixed by G(o) is not 0. A

function in the Hecke algebra acts by convolution on thissubspace.

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There is a well known way to obtain all such representations.

Let be a character of T/T(o) = A/A(o) with values in C,and let

I() = {f:G R | f(bg) = 1/2

(b)(b)f(g)}

where

1/2 : b | det Ad(b)|

is the modulus character of B. It is there because B is nota unimodular group. Because of the Iwasawa decomposition

G = B G(o), the subspace of vectors fixed by G(o) has

dimension one. The Hecke algebra acts on it by scalar multi-plication, so we get a ring homomorphism from H to C.

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Choose a basis (ti) for the lattice T/T(o). Characters of T

are parametrized by an array (si), where (ti) = si. Neces-sarily, each si is invertible. For a given f in H, the functiontaking f to as a function of is a polynomial in the vari-ables si . Let R be the ring of all such polynomials. The rep-

resentations I() and I(w) are generically isomorophic forw in the Weyl group. We therefore have a map from H to thering of invariants RW. It is an isomorphism.

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The points (si) make up a copy of (C)n, and can be seen

as the complex points on a torus T. It may be canonicallyidentified with Hom(L,C). The character group of T is thecocharacter group of T.

This torus sits in turn in a complex reductive group called theL-group.

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Let me summarize. If F is a field, T a torus defined over F,and t in T(F), then the map taking an algebraic character

to (t) identifies T(F) with Hom(X

(T), F|times

).An unramified character of T is a homomorphism from thelattice L = T/T(o) to C. It may be identified with the

complex points on a torus T whose character lattice is L,whereas the character lattice of T is L. Principal series areparametrized by W-orbits on T, and also by homomorphismsfrom H

G//G(o)

to C.

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6. The L-group

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The complex group dual to G is the reductive group G de-fined over C associated to the dual root datum(L,, L,) .

The automorphism corresponds to an automorphism of thisgroup preserving a Borel subgroup B containing T. Thence asemi-direct product LG =

G .

The lattice L is now the character group of a complex torusT. Unramified characters of T correspond to elements of T.If G is split and is trivial, a Weyl group orbit of unramifiedcharacters corresponds to a semi-simple conjugacy class in

the coset G. In the general case, it turns out that they corre-spond to conjugacy classes in the coset G .

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Sem

i-simpleconjugacyclassesinL

G

correspo

ndtoho

phis

msfromH

toC.

A

co

njugacyclassin

G

correspondstoa

homom

from

HtoC.S

odoesthecharacterofarepre

sentatio

LG.

The

integralH

eckealgeb

ramaybe

identified

withari

certainreprese

ntationsofLG

.

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7. Endoscopic groups

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An unramified endoscopic group of G is an unramified reduc-

tive group H with datum

(L,H, L,H, H)

subject to the condition that there exist s in T = Hom(L,C)with

(a) H = Ker(s) G

(b) H = w G for some w in W

(c) H(s) = s

The element s is by no means unique it is not actually partof the data describing an endoscopic group. (Maybe it should

be?)

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The group LH often embeds into LG. When it does, eachemi-simpe class in H givs rise to one of G , and themap backwards is the restriction of a representation of LG toone of LH, or in othr words a map from the Hecke algebra of

G to that of H.But sometimes LH does not embed into LG . . .

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Example. G = SL(2). The group is split, so G = I.

The lattices L and L may be identified with Z, with 1, with 2. Then T = Hom(L,C) is C.The only s annihilating any roots is s = 1. So H = if s = 1and all of

Gotherwise.

If w = 1, so is H. (c) makes is no condition on s. So thereare two cases: (i) s = 1 when H = G, and (ii) s = 1 when His the split one-dimensional torus.

If w = 1, then H is non-trivial, and the torus T is the norm-one subgroup of the multiplicative group of the unramified

quadratic extension. It must stabilize positive roots if there

are any, amd this does not happen, and s must be 1. Condi-tion (c) is vacuous. Therefore H itself is (iii the unramifiednon-split torus.

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Example. G = PGL(2). The group is again split and G = I.Here L = Z but = 2. So both s = 1 fix all roots.

When w = 1 there are again two possibilities, (i) the splittorus and (ii) PGL(2) itself. In the second case, both s = 1will be compatible.

If w = 1, then (c) tells us that w(s) = s, so s = 1 andH = G. But in this case w does not fix the positive roots,so this case is disallowed.

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Example. G = U(3)?

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It would be nice to have a simple algorithm for listing all pos-

sible endoscopic groups, along with information about em-beddings of LH into LG.

It is not s, but at most the W-orbit of s that matters. This or-bit corresponds to an unramified representation, and presum-

ably endoscopy says something about its character. What is

conjectured and what proven?

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