Unipotent representations and microlocalization

Abstract

We develop a theory of microlocalization for Harish-Chandra modules, adapting a construction of Losev [Duke Math. J. 159 (2011), pp. 99–143]. We explore the applications of this theory to unipotent representations of real reductive groups. For a unipotent representation of a complex group, we deduce a formula for the restriction to a maximal compact subgroup, proving an old conjecture of Vogan [Associated varieities and unipotent representations, Birkhäuser Boston, Boston, MA, 1991] in a large family of cases.