Unipotent Representations and Microlocalization

TL;DR

This paper develops a microlocalization theory for Harish-Chandra modules, applying it to unipotent representations of real reductive groups and proving a conjecture on K-multiplicities for complex groups.

Contribution

It introduces a microlocalization framework for Harish-Chandra modules and verifies Vogan's conjecture on K-multiplicities in many cases.

Findings

Derived a formula for K-multiplicities of unipotent representations

Proved Vogan's conjecture for a large class of cases

Extended microlocalization techniques to Harish-Chandra modules

Abstract

We develop a theory of microlocalization for Harish-Chandra modules, adapting a construction of Losev (\cite{Losev2011}). We explore the applications of this theory to unipotent representations of real reductive groups. For complex groups, we deduce a formula for the $K$-multiplicities of unipotent representations attached to a nilpotent orbit $\OO$, proving an old conjecture of Vogan (\cite{Vogan1991}) in a large family of cases.