Standard modules and intertwining operators for reductive p-adic groups

TL;DR

This paper proves that the Galois group action stabilizes standard modules in the representation theory of reductive p-adic groups, providing explicit formulas for Harish-Chandra e-functions and extending known results.

Contribution

It generalizes the stability of standard modules under Galois action and derives explicit formulas for Harish-Chandra e-functions for all Levi subgroups.

Findings

Galois group action stabilizes standard modules

Explicit formulas for Harish-Chandra e-functions

Extension of stability results to all irreducible representations

Abstract

Consider a reductive group G over a non-archimedean local field. The Galois group Gal(C/Q) acts naturally on the category of smooth complex G-representations. We prove that this action stabilizes the class of standard modules. This generalizes and relies on an analogous result about essentially square-integrable representations. Other important objects in the proof of our main result are intertwining operators between parabolically induced G-representations, and the associated Harish-Chandra \mu-functions. We determine an explicit formula for the \mu-function of any irreducible representation of any Levi subgroup of G.