Emmy-Noether-Seminar

Kazhdan-Lusztig equivalence via Soergel bimodules

Jonathan Gruber

Abstract: The category of finite-dimensional representations of a complex simple Lie algebra admits two canonical non-semisimple deformations: One is the parabolic BGG category O for an affine Lie algebra, the other is the category of representations of a quantum group at a root of unity. In a landmark result from the 1990s, Kazhdan and Lusztig have established an equivalence between these non-semisimple abelian categories. More recently, Gaitsgory has proposed a conjectural extension of the Kazhdan-Lusztig equivalence to a derived equivalence between the (non-parabolic) BGG category O of the affine Lie algebra and the category of representations of a mixed quantum group. In this talk, I will establish a variant of the derived equivalence conjectured by Gaitsgory, involving the principal blocks of the aforementioned categories, by relating the derived categories of these principal blocks to categories of Soergel bimodules. I will also explain how this derived equivalence can be used to recover a (non-derived) variant of the Kazhdan-Lusztig equivalence, again involving the principal blocks.

This is joint work with Peter Fiebig, partly based on recent results of Ivan Losev and Quan Situ.