THE SPHERICAL HECKE ALGEBRA, PARTITION FUNCTIONS, AND MOTIVIC
INTEGRATION
THOMAS HALES0
1. I NTRODUCTION
Let G = GL4 (Qp ) and K = GL(4, Zp ) ⊂ G be a maximal compact subgroup. The Hecke
algebra K(G//K) is defined as the convolution algebra on the function space
{f ∈ C∞
c (G) | f(k1 gk2 ) = f(g), ki ∈ K}.
Satake constructed an isomorphism
∼ C[x± , . . . , x± ]S4 .
K(G//K) =
1
4
We can do this for other p-adic Lie groups. For example if we take H = GL2 (Qp ) ×
GL2 (Qp ) with compact KH = GL2 (Zp ) × GL2 (Zp ). Then Satake gives an isomorphism
∼ C[y± , y± ]S2 ⊗ C[z± , z± ]S2 =
∼ C[x± , . . . , x± ]S2 ×S2 .
K(H//KH ) =
1
2
1
2
1
4
In particular, there is a natural homomorphism b : K(G//K) → K(H//KH ) induced by the
± S4
± S2 ×S2
inclusion C[x±
→ C[x±
.
1 , . . . , x4 ]
1 , . . . , x4 ]
More generally, let F be a p-adic field, G an unramified reductive group over F, and H an
endoscopic group of G (we won’t define this here). Then Langlands proved there is a ring
homomorphism of Hecke algebras b : K(G//K) → K(H//KH ).
In the 1980s, Langlands formulated the fundamental lemma which at the time was a conjecture but now is a very hard theorem. Roughly speaking this predicted an equatlity of the
form
XZ
XZ
f=
b(f)
conjugacy classes of G
conjugacy classes of H
where f is a characteristic function of G(OF ) and the sum is over conjugacy classes with the
same characteristic polynomial.1
1.1. History. The fundamental lemma for the unit element 1 ∈ K implies fundamental lemma
for any function f ∈ K (due to Hales). This uses the exponential map on Lie algebras as well
as descent. Waldspurger showed that the fundamental lemma for characteristic functions in
characteristic p implies the fundamental lemma for characteristic functions in characteristic
0. Finally Ngô Bao Châu proved fundamental lemma in general using a study of the Hitchin
fibration.
Date: March 27, 2017.
0
Notes by Dori Bejleri who takes all responsibility for any mistakes or inconsistencies.
1
Recall the abstract characteristic polynomial is the natural map G → T/W where T is the maximal torus and
W is the Weyl group.
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2. M AIN QUESTION
Question 1. Does the fundamental lemma hold for any f ∈ K in characteristic p?
Theorem 2.1. (Hales - Cely - Casselman) Yes for large enough characteristic.
There are four approaches to this result:
• using trace formulas (this is a global argument and requires some care at Archamedean
places) – not done;
• lifting the ideas of Ngô Bao Châu on the Hitchin fibrations to Lie groups – done by
Bouthier in 2015;
• using close local fields to transfer from characteristic 0 to characteristic p – proposed
by Lemaire, Moeglin and Waldspurger in 2015 but hasn’t been carried out;
• our approach uses motivic integration.
Let’s consider again the case G = GL4 (Qp ). The Satake isomorphism gives an identification
∼ C[x± , . . . , x± ]S4 = C[Λ]S4 where Λ = Z4 is a lattice. We write eλ ∈ C[Λ] for λ ∈ Λ.
K=
4
1
We can think of Λ as the character lattice X∗ (Tb) for a complex torus Tb of dimension 4 and so
the isomorphism becomes
∼ C[G//Adj]
∼ C[Tb/S4 ] =
b
K(G//K) =
b and Adj is the adjoint action.
where Tb ⊂ GL4 (C) =: G
This is the form in which Langlands thought about the fundamental lemma. If G is a split
b is the complex group with
group, it is classified by the root datum (X∗ , Φ, X∗ , Φ∨ ). Then G
∨
∗
dual root datum (X∗ , Φ , X , Φ). Using the above interpretation of the Satake isomorphism
b then the inclusion H
b ⊂ G
b leads to the map b : K(G//K) →
in terms of the dual group G,
K(H//KH ).
The main problem for proving the fundamental lemma then is that this requires passing
information between p-adic integrals for this p-adic group G/F to integration on this complex
b
group G/C.
3. M OTIVIC INTEGRATION
Motivic integration is an abstract version of p-adic integration. Let k((t)) be the field of
Laurent series with its valuation v : k((t)) → Z and residue field k. Motivic integration
combines integration over the valued field, as well as integrating over k (e.g. point counting
over a finite field) and integration over Z (integer sums). So we are dealing with subsets
n
r
X ⊂ Am
k((t)) × Ak × Z .
Motivic integration comes with a transfer principle: identities of integrals transfer from one
field to another. In particular, motivic integration results can pass between positive and zero
characteristic.
C(Zr ) will be a ring of functions on Zr that we can integrate. Presburger defined a language
∼ n ), n = 2, 3, . . . where =
∼ n is congruence mod n. We consider subsets of Zr
(0, +, 6, =
that can be defined in this language using =, ∨, ∧, ¬, ∀, ∃. We call these Presburger sets. This
includes things like polyhedra and cones. Given Presburger subsets X ⊂ Zr and Y ⊂ Zm and
2
a function f : X → Y, we say f is Presburger if the graph Γf ⊂ Zr+m is Presburger. This makes
Presburger sets into a category.
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Let A = Z[q± , 1−q
a ] such that a ∈ Z, a 6= 0. The ring C(X) of Presburger constructible
1
functions is the A-algebra generated by qα , β, 1−q
a for α, β : X → Z Presburger functions.
Furthermore, given a Presburger function f : X → Y with nice integrability properties, Cluckers
and Loeser define a map C(X) → C(Y) by integrating along the fibers.
b or L G = G
b o hθi from the point
Problem 1. Can we investigate the representation theory of G
of view of Presburger arithmetic?
Answer 1. Yes.
Let A be a maximal split torus of G and P+ ⊂ X∗ (A) a positive Weyl chamber and let $ be
a uniformizer for F. The characteristic functions fλ ∈ K(G//K) of K$λ K for λ ∈ P+ form a
basis of K(G//K). The function
B : (λ, h) ∈ P+ × H(F) 7→ b(f)(h) ∈ C
then allows us to use Presburger arithmetic to study K(G//K).
Theorem 3.1. (Hales - Cely - Casselman) B is a motivic constructible function.
Using the transfer principle for motivic integration, it follows that the fundamental lemma
holds in positive characteristic for the full Hecke algebra. However, to use the transfer principle
we lose finitely many places.
3.1. Idea of proof.
I Define partition functions of the form det(1 − qθE, V)−1 where V is a representation and
E a certain operator.
II Partition functions are Presburger constructible.
III Examples of partition functions include
• denominator of Weyl character formula,
• Kostant q-partition functions,
• Harish-Chandra c-functions,
• Langlands local L-functions for unramified p-adic groups
IV Use restriction of characters along L H →L G, Macdonalds formula for the Satake transform fλ 7→ f̂λ , and various changes of basis for the Hecke algebra (Kato-Lusztig, geometric Satake, etc) which can all be expressed in terms of partition functions to express
f 7→ b(f) as a constructible functions.
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