The Spherical Hecke algebra, partition functions, and motivic integration

TL;DR

This paper proves the fundamental lemma for spherical Hecke algebras in large positive characteristic p-adic groups using motivic integration, extending key results to non-split groups via new partition functions.

Contribution

It introduces a family of partition functions linking L-groups to motivic integration, enabling extension of classical results to non-connected p-adic groups.

Findings

Proof of the fundamental lemma for unramified p-adic groups in large characteristic

Explicit formulas for branching rules and inverse Satake transform

Extension of classical formulas to non-split unramified p-adic groups

Abstract

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).