Geometric Representation Theory
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.
We will review a set of conjectures related to the structure of cohomological Hall algebras (COHA) of categories of Higgs sheaves on curves. We then focus on the case of P^1, and relate its COHA to the affine Yangian of sl_2.
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