On Satake parameters for representations with parahoric fixed vectors

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.

Theorem 1.3 and Lemma 4.9 added. Minor correction made in proof of Proposition 4.1. Lemma 10.2 added. Accepted version. 26 pages