Tilting sheaves for real groups and Koszul duality

Abstract

For a certain class of real analytic varieties with the real Lie group action we define a tstructure on the category of equivariant-monodromic sheaves and develop the theory of tilting sheaves. In case of a quasi-split real form of an algebraic group acting on the flag variety we construct an analog of a Soergel functor, which fully-faithfully embeds the subcategory of tilting objects to the category of coherent sheaves on a block variety. We apply the results to give a new, purely geometric, proof of the Soergel’s conjecture for quasi-split groups. The thesis is based on a joint work with Zhiwei Yun.