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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. [ABSTRACT FROM AUTHOR]

Let G be a complex reductive algebraic group with Lie algebra \mathfrak {g} and let G_{\mathbb {R}} be a real form of G with maximal compact subgroup K_{\mathbb {R}}. Associated to G_{\mathbb {R}} is a K \times \mathbb {C}^{\times }-invariant subvariety \mathcal {N}_{\theta } of the (usual) nilpotent cone \mathcal {N} \subset \mathfrak {g}^*. In this article, we will derive a formula for the ring of regular functions \mathbb {C}[\mathcal {N}_{\theta }] as a representation of K \times \mathbb {C}^{\times }. Some motivation comes from Hodge theory. In [ Hodge theory and unitary representations of reductive Lie groups, Frontiers of Mathematical Sciences , Int. Press, Somerville, MA, 2011, pp. 397–420], Schmid and Vilonen use ideas from Saito's theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If G_{\mathbb {R}} is split, and X is the spherical principal series representation of infinitesimal character 0, then conjecturally gr(X) \simeq \mathbb {C}[\mathcal {N}_{\theta }] as representations of K \times \mathbb {C}^{\times }. So a formula for \mathbb {C}[\mathcal {N}_{\theta }] is an essential ingredient for computing Hodge filtrations. [ABSTRACT FROM AUTHOR]