Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

TL;DR

This paper develops a new categorical framework for affine Harish-Chandra bimodules, establishing equivalences with affine Hecke categories and introducing a novel localization approach on Whittaker flag manifolds.

Contribution

It constructs affine Harish-Chandra bimodule categories, proves monoidal equivalences with affine Hecke categories, and introduces a new singular localization technique.

Findings

Established monoidal equivalences under integrality conditions.

Introduced a new singular localization on Whittaker flag manifolds.

Connected Lie algebra representations with parabolic induction of Steinberg modules.

Abstract

We construct categories of Harish-Chandra bimodules for affine Lie algebras analogous to Harish-Chandra bimodules with infinitesimal characters for simple Lie algebras, addressing an old problem raised by I. Frenkel and Malikov. Under an integrality hypothesis, we establish monoidal equivalences between blocks of our affine Harish-Chandra bimodules and versions of the affine Hecke category. To do so, we introduce a new singular localization onto Whittaker flag manifolds. The argument for the latter specializes to a somewhat alternative treatment of regular localization, and identifies the category of Lie algebra representations with a generalized infinitesimal character as the parabolic induction, in the sense of categorical representation theory, of a Steinberg module.