Kazhdan-Lusztig conjecture via Zastava spaces

TL;DR

This paper proves the Kazhdan-Lusztig conjecture for simple complex Lie algebras using geometric Satake correspondence and zastava spaces, and explores extensions to affine Lie algebras and W-algebras.

Contribution

It provides a geometric proof of the Kazhdan-Lusztig conjecture via zastava spaces and extends the approach to affine Lie algebras and W-algebras.

Findings

Proof of the Kazhdan-Lusztig conjecture using geometric methods.

Analysis of torus fixed points in zastava spaces.

Speculations on affine Lie algebras and W-algebras.

Abstract

We deduce the Kazhdan-Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category O from the equivariant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. We make similar speculations for the affine Lie algebras and W-algebras.