Generic unipotent standard modules

TL;DR

This paper uses Lusztig's geometric classification to identify reducibility points of standard modules for affine Hecke algebras with generic data, extending known results to broader contexts.

Contribution

It provides a geometric approach to determine reducibility points for affine Hecke algebra modules and extends results to non-generic cases.

Findings

Identifies reducibility points for generic inducing data

Recovers Muic-Shahidi's results for p-adic groups

Provides a necessary condition for reducibility in non-generic cases

Abstract

Using Lusztig's geometric classification, we find the reducibility points of a standard module for the affine Hecke algebra, in the case when the inducing data is generic. This recovers the known result of Muic-Shahidi for representations of split p-adic groups with Iwahori-spherical Whittaker vectors. We also give a necessary (insufficient) condition for reducibility in the non-generic case.