Steinberg-Whittaker localization and affine Harish--Chandra bimodules

Abstract: A fundamental result in representation theory is Beilinson--Bernstein localization, which identifies the representations of a reductive Lie algebra with fixed central character with D-modules on (partial) flag varieties. We will discuss a localization theorem which identifies the same representations instead with (partial) Whittaker D-modules on the group. In this perspective, representations with a fixed central character are equivalent to the parabolic induction of a 'Steinberg' category of D-modules for a Levi.

Time permitting, we will explain how these methods can be used to identify a subcategory of Harish--Chandra bimodules for an affine Lie algebra and prove that it behaves analogously to Harish--Chandra bimodules with fixed central characters for a reductive Lie algebra. In particular, it contains candidate principal series representations for loop groups. This a report on work with Justin Campbell.

algebraic geometry

Audience: researchers in the topic

UC Davis algebraic geometry seminar