Seminar

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.