Perverse sheaves on affine flag varieties, geometry of the dual group, and representations

Abstract: The description of regular blocks of the category O of a complex semisimple Lie algebra in terms of perverse sheaves on a flag variety has been a crucial tool for its study, and in particular for the proof of the Kazhdan-Lusztig character formula. This description has a conjectural analogue for representations of reductive groups over fields of positive characteristic, as predicted by Finkelberg-Mirkovic, which involves a category of perverse sheaves on the affine Grassmannian of the Langlands dual group, with coefficients in a field of positive characteristic. In this talk I will present a work in progress with Bezrukavnikov which we expect will lead to a proof of this conjecture. As a step towards this goal, we obtain a description of tilting perverse sheaves on the affine flag variety reminiscent of the corresponding result for usual flag varieties due to Soergel.

algebraic geometryalgebraic topologycomplex variablesdifferential geometrygeometric topologymetric geometryquantum algebrarepresentation theory

Audience: researchers in the topic

Sapienza A&G Seminar

Series comments: Weekly research seminar in algebra and geometry.

"Sapienza" Università di Roma, Department of Mathematics "Guido Castelnuovo".