Title: Quantization of the Ngô morphism

Abstract: We will discuss work, joint with Victor Ginzburg, which proves a conjecture of Nadler on the existence of a quantization (non-commutative deformation) of the Ngô morphism, a morphism of group schemes constructed by Ngô in his proof of the fundamental lemma in the Langlands program. We will first explain the representation theoretic background used in the construction of the Ngô morphism and discuss an extended example of this map for the group of invertible n x n complex matrices. Afterwards, we will review non-commutative deformations in general and discuss Hopf algebroids, which generalize Hopf algebras and play a key role in the construction of the quantization of the Ngô morphism, and use this to give a precise statement of the main theorem.

Time permitting, we will also discuss how the tools used to construct this morphism can be used to prove conjectures of Ben-Zvi—Gunningham on the behavior of a representation theoretic procedure known as parabolic induction.