Spherical Hecke algebras for Kac-Moody groups over local fields

TL;DR

This paper defines and analyzes the spherical Hecke algebra for Kac-Moody groups over local fields, establishing its structure, polynomial structure constants, and Satake isomorphism, extending concepts from reductive groups to more general Kac-Moody settings.

Contribution

It introduces the spherical Hecke algebra for Kac-Moody groups over local fields and proves its polynomial structure constants and Satake isomorphism, generalizing reductive group theory.

Findings

Structure constants are polynomials in the residue field size.

The algebra is commutative due to the Satake isomorphism.

Applicable to abstract hovels with unequal parameters.

Abstract

We define the spherical Hecke algebra H for an almost split Kac-Moody group G over a local non-archimedean field. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The stabilizer K of a special point on the standard apartment plays the role of a maximal open compact subgroup. We can define H as the algebra of K-bi-invariant functions on G with almost finite support. As two points in the hovel are not always in a same apartment, this support has to be in some large subsemigroup G+ of G. We prove that the structure constants of H are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We also prove the Satake isomorphism between H and the algebra of Weyl invariant elements in some completion of a Laurent polynomial algebra. In particular, H is always commutative. Actually, our results apply to abstract "locally finite" hovels, so that we can define the spherical algebra with unequal parameters.