The Satake isomorphism for special maximal parahoric Hecke algebras

TL;DR

This paper establishes a Satake isomorphism for the Hecke algebra associated with a special maximal parahoric subgroup of a reductive group over a nonarchimedean local field, extending previous frameworks.

Contribution

It introduces a Satake isomorphism for Hecke algebras of special maximal parahoric subgroups, utilizing a Cartan decomposition for the double coset space.

Findings

Derived a Cartan decomposition for Kackslash G(F)/K

Established the Satake isomorphism for the Hecke algebra

Connected results to Cartier's treatment with different subgroups

Abstract

Let G denote a connected reductive group over a nonarchimedean local field F. Let K denote a special maximal parahoric subgroup of G(F). We establish a Satake isomorphism for the Hecke algebra H of K-bi-invariant compactly supported functions on G(F). The key ingredient is a Cartan decomposition describing the double coset space K\G(F)/K. We also describe how our results relate to the treatment of Cartier, where K is replaced by a special maximal compact open subgroup K' of G(F) and where a Satake isomorphism is established for the Hecke algebra of K'-bi-invariant compactly supported functions on G(F).