On Satake parameters for representations with parahoric fixed vectors

TL;DR

This paper constructs Satake parameters for irreducible smooth representations with parahoric fixed vectors in p-adic groups, especially for non-quasi-split groups, aiding in geometric Satake isomorphism development.

Contribution

It introduces a new construction of Satake parameters for all parahoric fixed vector representations, including non-quasi-split groups, extending previous theories.

Findings

Constructed Satake parameters for all parahoric fixed vector representations.

Parametrization is complete for special maximal parahoric subgroups.

Facilitates potential geometric Satake isomorphism for non-quasi-split groups.

Abstract

This article constructs the Satake parameter for any irreducible smooth $J$-spherical representation of a $p$-adic group, where $J$ is any parahoric subgroup. This parametrizes such representations when $J$ is a special maximal parahoric subgroup. The main novelty is for groups which are not quasi-split, and the construction should play a role in formulating a geometric Satake isomorphism for such groups over local function fields.