Unipotent Representations and Microlocalization

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.