On the Satake isomorphism

TL;DR

This paper revisits the Satake isomorphism for affine Hecke algebras, providing new proofs and extending character formulas to disconnected groups, enhancing understanding of representation theory in positive characteristic.

Contribution

It offers alternative proofs of existing results and extends character formulas to disconnected groups within the context of the Satake isomorphism.

Findings

New proofs based on J-ring theory and character formulas

Extension of character formulas to disconnected groups

Deeper understanding of affine Hecke algebra representations

Abstract

In a 1983 paper the author has established a (decategorified) Satake equivalence for affine Hecke algebras. In this paper we give new proofs for some results of that paper, one based on the theory of J-rings and one based on the known character formula for rational representations of a reductive group in positive, large, characteristic. We also give an extension of that character formula to disconnected groups.