Kazhdan-Lusztig conjecture via zastava spaces

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.

Automated summary

This article deduces the Kazhdan-Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category O using the equivariant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. Similar speculations are made for affine Lie algebras and W-algebras.