The Character Theory of a Complex Group

TL;DR

This paper explores the deep categorical and geometric structures of the Hecke category for complex reductive groups, revealing dualities, Calabi-Yau properties, and connections to the geometric Langlands program and supersymmetric gauge theories.

Contribution

It establishes the Calabi-Yau and duality properties of the Hecke category, linking it to character sheaves and Langlands duality within a geometric and topological framework.

Findings

Hecke category is a two-dualizable Calabi-Yau monoidal category.

Its center and trace coincide and relate to Lusztig's character sheaves.

Provides a geometric Langlands duality via Koszul duality.

Abstract

We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. Our focus is the Hecke category of Borel-equivariant D-modules on the flag variety of a complex reductive group G (equivalently, the category of Harish Chandra bimodules of trivial central character) and its monodromic variant. The Hecke category is a categorified analogue of the finite Hecke algebra, which is a finite-dimensional semi-simple symmetric Frobenius algebra. We establish parallel properties of the Hecke category, showing it is a two-dualizable Calabi-Yau monoidal category, so that in particular, its monoidal (Drinfeld) center and trace coincide. We calculate that they are identified through the Springer correspondence with Lusztig's unipotent character sheaves. It follows that Hecke module categories, such as categories of Lie algebra representations and Harish Chandra modules for G and its real forms, have characters which are themselves character sheaves. Furthermore, the Koszul duality for Hecke categories provides a Langlands duality for unipotent character sheaves. This can be viewed as part of a dimensionally reduced version of the geometric Langlands correspondence, or as S-duality for a maximally supersymmetric gauge theory in three dimensions.