With D. Nadler. Langlands Duality for Character Sheaves.
We establish a Langlands duality for the representation theory of complex reductive groups. By applying S^1 equivariant localization techniques developed in our paper arXiv:0706.0322 to the tamely ramified local geometric Langlands theorem of Bezrukavnikov, we obtain a canonical monoidal equivalence between the category of Harish-Chandra modules for a complex group G and the finite Hecke category for the Langlands dual group G^ (generalizing results of Beilinson, Ginzburg and Soergel). We also prove a result describing the derived centers (Drinfeld doubles or Hochschild categories) of Hecke categories in terms of Lusztig's theory of character sheaves. Combining these results we obtain an equivalence (as braided or ribbon categories) between the categories of character sheaves associated to the groups G and G^. This result is interpreted as a Langlands duality for three dimensional topological gauge theories, obtained by dimensional reduction from the four dimensional gauge theories behind geometric Langlands a la Kapustin-Witten.