Characteristic cycles and Harish-Chandra modules

Abstract

Recall that Theorem 8.5 related the category of equivariant perverse sheaves (or, equivalently, equivariant D-modules) on the geometric parameter space to certain categories of Harish-Chandra modules. Our goal in this chapter is to see what this relationship has to say about the characteristic cycles. Theorem 8.5 implies that the characteristic cycles must somehow be encoded in the Harish-Chandra modules. We are not able to break that code, but we get some useful information about it (Theorem 20.18). This will later be the key to relating our definition of Arthur’s unipotent representations to the original one of Barbasch and Vogan. Roughly speaking, Arthur’s representations will be characterized on the E-group side by the occurrence of certain “regular” components in a characteristic cycle (Definition 20.7). We therefore seek to understand that occurrence on the level of Harish-Chandra modules.