Abstracts for Geometric Representation Theory

We describe an algorithm which takes as input any pair of permutations and gives as output two permutations lying in the same Kazhdan-Lusztig right cell. There is an isomorphism between the Richardson varieties corresponding to the two pairs of permutations which preserves the singularity type. This fact has applications in the study of W-graphs for symmetric groups, as well as in finding examples of reducible associated varieties of sln-highest weight modules, and comparing various bases of irreducible representations of the symmetric group or its Hecke algebra.

This is joint work with Justin Hilburn. We will explain a theorem showing that D-modules on the Tate vector space of Laurent series are equivalent to ind-coherent sheaves on the space of rank 1 de Rham local systems on the punctured disc equipped with a flat section. Time permitting, we will also describe an application of this result in the global setting. Our results may be understood as a geometric refinement of Tate's ideas in the setting of harmonic analysis.

I will explain how to obtain the Gordon-Stafford construction and some related constructions of Z-algebras in the literature, using certain mathematical avatars of line defects in 3d N=4 theories. ​Time permitting, I will discuss the K-theoretic and elliptic cases as well.

In this talk, I will give a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic group in terms of perverse sheaves on the Langlands dual affine flag variety. This is joint work with R. Bezrukavnikov and S. Riche.