## Columbia Mathematics Department Colloquium

Finite dimensional irreducible representations over quantizations of symplectic resolutions

by

# Ivan Loseu

# (Northeastern University)

Abstract:

A basic problem in Representation
theory is, given an algebraic object such as a group, an associative
algebra or a Lie algebra, to study its finite dimensional irreducible
representations. The first question, perhaps, is how many there
are.

In my talk I will address this question for associative algebras that are quantizations of algebraic varieties admitting symplectic resolutions. Algebras arising this way include universal enveloping algebras of semisimple Lie algebras, as well as W-algebras and symplectic reflection algebras.

The counting problem is a part of a more general program due to Bezrukavnikov and Okounkov relating the representation theory of quantizations to Quantum cohomology of the underlying symplectic varieties. It is also supposed to have other connections to Geometry. I will consider the case of quotient singularities and will make the exposition non-technical.

In my talk I will address this question for associative algebras that are quantizations of algebraic varieties admitting symplectic resolutions. Algebras arising this way include universal enveloping algebras of semisimple Lie algebras, as well as W-algebras and symplectic reflection algebras.

The counting problem is a part of a more general program due to Bezrukavnikov and Okounkov relating the representation theory of quantizations to Quantum cohomology of the underlying symplectic varieties. It is also supposed to have other connections to Geometry. I will consider the case of quotient singularities and will make the exposition non-technical.