Tilting sheaves for real groups and Koszul duality

TL;DR

This paper develops a theory of tilting sheaves for real analytic varieties with Lie group actions, establishing connections to Soergel bimodules and providing a geometric proof of a key conjecture for quasi-split groups.

Contribution

It introduces a new framework for tilting sheaves in the real setting and links it to Soergel bimodules, extending foundational theorems to real groups.

Findings

Constructed a fully faithful embedding into real Soergel bimodules

Established real analogs of Soergel's Structure and Endomorphism Theorems

Provided a geometric proof of Bezrukavnikov and Vilonen's theorem for quasi-split groups

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 Endomorphism Theorem. We apply these results to give a purely geometric proof of the theorem of Bezrukavnikov and Vilonen which proves Soergel's conjecture for quasi-split groups.