Tilting sheaves for real groups and Koszul duality

Abstract For a certain class of real analytic varieties with Lie group actions, we develop a theory of (free-monodromic) tilting sheaves, and apply it to flag varieties stratified by real group orbits. For quasi-split real groups, we construct a fully faithful embedding of the category of tilting sheaves to a real analog of the category of Soergel bimodules, establishing real group analogs of Soergel’s structure theorem and the endomorphism theorem. We apply these results to give a purely geometric proof of the main result of Bezrukavnikov and Vilonen Koszul duality for quasi-split real groups , Invent. Math. 226 (2021), 139–193, which proves Soergel’s conjecture Langlands’ philosophy and Koszul duality , in Algebra – representation theory (Constanta, 2000) , NATO Science Series II: Mathematics, Physics and Chemistry, vol. 28 (Kluwer Academic Publishers, Dordrecht, 2001), 379–414 for quasi-split groups.