Unitary Representations and Conical Symplectic Singularities

In this talk, I will explain a connection between two seemingly unrelated types of objects: unitary representations of real reductive groups and conical symplectic singularities. To each conical symplectic singularity X, I will attach a complex reductive algebraic group G_X which controls the "representation theory" of X. I will then use this new invariant to formulate a conjectural classification of the unitary dual of a complex reductive group. I will present some examples to support this conjecture. This talk is based on ongoing joint work with Ivan Losev and Mitya Matvieievskyi.