The spherical Hecke algebra, partition functions, and motivic integration

Casselman, William; Cely, Jorge; Hales, Thomas

doi:10.1090/tran/7465

The spherical Hecke algebra, partition functions, and motivic integration
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by William Casselman, Jorge E. Cely and Thomas Hales PDF

Trans. Amer. Math. Soc. 371 (2019), 6169-6212

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 nonconnected complex group (that is, nonsplit unramified $p$-adic group).

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