The Character Theory of a Complex Group
David Ben-Zvi, David Nadler
TL;DR
The paper develops a derived, higher-categorical framework for the Hecke category of Borel-biequivariant ${\mathcal D}$-modules on a complex reductive group and its monodromic variant. It proves that these as monoidal categories are semi-rigid Calabi–Yau, and that their monoidal center and trace coincide with Lusztig’s unipotent character sheaves, thereby giving character-theoretic interpretations for dualizable Hecke modules. Through Koszul duality, it establishes Langlands duality for unipotent character sheaves and frames these results in an extended 2d TFT perspective, offering a dimensionally reduced viewpoint on geometric Langlands and S-duality in gauge theory. The work also develops a functional-analytic toolkit for ${\mathcal D}$-modules on stacks, explains integral transforms via kernels, and extends these ideas to monodromic settings, linking boundary conditions to Harish-Chandra-type categories. Collectively, the results provide a cohesive bridge between higher category theory, ${\mathcal D}$-module representation theory, character sheaves, and topological field theory, with broader implications for geometric Langlands and quantum field theory interpretations.
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.