Title: Deformations of singular symplectic varieties and the Orbit method
Speaker: Ivan Losev (Northeastern University)
Abstract: This is a story of how modern developments in Algebraic geometry (incl. Minimal Model Program) help to solve classical problems in Representation theory.
One of the cornerstones of the infinite dimensional Lie representation theory is Kirillov's Orbit method (1961). It says that the irreducible unitary representations of a nilpotent Lie group are in a natural bijection with the orbits for the action of the group on the dual space of its Lie algebra. There is an analog of this result for nilpotent Lie algebras, due to Dixmier (1963): instead of unitary representations one considers so called primitive ideals (=annihilators of irreducible modules) in universal enveloping algebras.
An immediate question is how to generalize these results to semisimple Lie groups or Lie algebras. I will talk about the Lie algebra case. My recent result here is that there is a natural map from the set of (co)adjoint orbits to the set of primitive ideals that is often an embedding. To produce this map I compare commutative and noncommutative deformations of singular symplectic varieties, a spectacular class of singular algebraic varieties introduced by Beauville in 2000.
Biosketch: Ivan Losev is a Full Professor at Northeastern University. He received his Ph.D. in 2007 at Moscow State University under Ernest Vinberg. Prof. Losev works in Representation theory and its connections to Geometry and Combinatorics. His distinctions include an invited ICM talk (2010), a Sloan fellowship (2014), and an AMS fellowship (class of 2018).
Colloquium URL: https://web.math.osu.edu/colloquium/