The Character Theory of a Complex Group

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.