Speaker: Tom Gannon (University of California, Los Angeles)

Title: Quantization of the universal centralizer and central $D$-modules

Abstract: We will discuss work, joint with Victor Ginzburg, on certain quantizations (or non-commutative deformations) of objects and morphisms of interest in the geometric Langlands program. First, we will discuss the quantization of a morphism of group schemes used by Ngô in his proof of the fundamental lemma, which confirms a conjecture of Nadler. Afterwards, we will discuss how this morphism can be used to construct a $D$-module analogue of a recent equivalence proved by Bezrukavnikov--Deshpande in the ℓ-adic setting, identifying a certain braided monoidal subcategory of the category of $G$-equivariant $D$-modules on $G$ known as vanishing $D$-modules with the category of a modules for a ring known as the spherical nil-DAHA. We will also explain the construction of a certain bimodule isomorphism used to construct this braided monoidal equivalence, whose existence was originally conjectured by Ben-Zvi--Gunningham. 

Time permitting, we will also discuss another application of our methods: a proof of a conjecture of Braverman and Kazhdan, known as the exactness conjecture, in the $D$-module setting.