Characteristic Cycles and Wave Front Cycles of Representations of Reductive Lie Groups

Abstract

In the papers [V1] and [BV], Vogan and Barbasch-Vogan attach two similar invariants to representations of a reductive Lie group, one by an algebraic process, the other analytic. They conjectured that the two invariants determine each other in a deflnite manner. Here we prove the conjecture. Our arguments involve two flner invariants { the characteristic cycles of representations { which are interesting in their own right. To describe the invariants, we consider a linear, reductive Lie group G and flx a maximal compact subgroup K ‰ G . We denote their Lie algebras by and , and the complexifled Lie algebras by , . An element ‡ of the dual space ⁄ is said to be nilpotent if it corresponds to a nilpotent element of [ ; ] via the isomorphism [ ; ] » [ ; ] ⁄ ‰ ⁄ induced by the Killing form. Via the adjoint action, the complexiflcation G of G acts with flnitely many orbits onN ⁄ , the cone of all nilpotents in ⁄ . Like all coadjoint orbits, each G-orbit O‰N ⁄ carries a distinguished (complex algebraic) symplectic structure; the intersection ofO with i ⁄ =f‡ 2 ⁄ jh‡; i‰i g consists of flnitely many G -orbits, each of which is Lagrangian inO. The choice of a maximal compact subgroup determines a Cartan decomposition = ' , and dually ⁄ = ⁄ ' ⁄ . The complexiflcationK of the groupK operates onN ⁄ ⁄ with flnitely many orbits, and each of these orbits is Lagrangian in the G-orbit which contains it [KR]. Now let … be an irreducible, admissible representation of G { for example an irreducible unitary representation. To such a representation, one can associate its Harish-Chandra module V , which is simultaneously and compatibly a module for the Lie algebra and a locally flnite module for the algebraic group K. Then V admits K-invariant \good flltrations, as module over the universal enveloping algebra U( ), relative to its canonical flltered structure. Vogan [V1] shows that the annihilator of the graded module deflnes an equidi