This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified p-adic reductive group G in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex L-group of the unramified p-adic group G. Our partition functions specialize to Kostant's q-partition function for complex connected groups and also specialize to the Langlands L-function of a spherical representation. These partition functions are used to extend numerous results that were previously known only when the L-group is connected (that is, when the p-adic group is split). We give explicit formulas for branching rules, the inverse of the weight multiplicity matrix, the Kato-Lusztig formula for the inverse Satake transform, the Plancherel measure, and Macdonald's formula for the spherical Hecke algebra on a non-connected complex group (that is, non-split unramified p-adic group).Let N : X * (T ) → X * (A) be the norm map: Nα = α ′ ∈[α] α ′ . Lemma 1.1.1. If [α] ∈ Φ red is an indivisible restricted root, then the corresponding coroot is (1.1.2) [α] ∨ = k Nα ∨ when the diagram of α has type A k , for k = 1, 2. Proof. (See [24, 1.3.9].) Let W be the Weyl group attached to the root datum of G over a splitting field E. The restricted Weyl group W θ is the subgroup of W commuting with θ. The group W θ is a Coxeter group. The simple reflection in W θ associated with an orbit [α] of simple roots in Φ is the longest element in the Weyl group of the Levi component M [α] . We write ℓ(w) for the length of w ∈ W θ , computed relative to the set of simple reflections of the Coxeter group W θ . Lemma 1.3.5. The partition function evaluates to the local L-function. More precisely,Proof. Both sides are defined as the reciprocal of a determinant. On both sides it is the determinant of the same element acting on the same vector space.1.3.2. the partition function for the adjoint representation. A case of particular importance for us is the following. Let g be the adjoint representation ofĜ. It is an irreducible highest weight representation whose highest weight is θ-fixed. Hence g extends to an irreducible representation ofĜ ⋊ θ . Let n be the Lie algebra ofN. Then n = g R , where R is the set of positive roots. The set R is θ-stable. We have a partition function P(Ĝ, n, θ, E, q).Much of this article handles this particular case. WhenĜ, n, and θ are fixed, we abbreviate P(E, q) = P(Ĝ, n, θ, E, q). 6Casselman, Cely, HalesHecke algebras, partition functions, and motivic integrationRecall that each α ∈ Ψ red,+ has the form [β ∨ ] ∨ , with [β ∨ ] ∈ Φ red and some β in Ψ + . The root β has a diagram A 1 or A 2 , associated constant b = b(β), and kNβ = α for diagram type A k (Lemma 1.1.1). For each α ∈ Ψ red , we defineWe have the following factorization refining Lemma 1.3.4. Lemma 1.3.7. The determinant factors as det(1 − θEq; n) = α∈Ψ red,+ d α (q).Proof. Similar factori...
