Mathematics – Representation Theory
Scientific paper
2009-04-08
Mathematics
Representation Theory
substantial revision
Scientific paper
We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. We first establish good functoriality properties for ordinary and equivariant D-modules on schemes, in particular showing that the integral transforms studied in algebraic analysis give all continuous functors on D-modules. We then focus on the categorified Hecke algebra H_G of Borel bi-equivariant D-modules on a complex reductive group G. We show that its monoidal center and abelianization (Hochschild cohomology and homology categories) coincide and are identified through the Springer correspondence with the derived version of Lusztig's character sheaves. We further show that H_G is a categorified Calabi-Yau algebra, and thus satisfies the strong dualizability conditions of Lurie's proof of the cobordism hypothesis. This implies that H_G defines (the (0,1,2)-dimensional part of) a three-dimensional topological field theory which we call the character theory of G. It organizes much of the representation theory associated to G. For example, categories of Lie algebra representations and Harish Chandra modules for G and its real forms give natural boundary conditions in the theory. In particular, they have characters (or charges) as Hecke modules which are character sheaves. The Koszul duality for Hecke categories provides an equivalence between character theories for Langlands dual groups, and in particular a duality of character sheaves. It can be viewed as a dimensionally reduced version of the geometric Langlands correspondence, or as S-duality for a generically twisted maximally supersymmetric gauge theory in three dimensions.
Ben-Zvi David
Nadler David
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