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.