Positive traces and analytic Langlands correspondence

Abstract

I will describe my results with co-authors in two directions.

The first direction is the problem of classification of positive traces on quantized Coulomb branches. In the most general setting, this problem generalizes the classical problem of describing irreducible unitary representations of real reductive Lie groups. We consider the case of Kleinian singularities of type A and provide a complete classification of positive traces.

The second direction is analytic Langlands correspondence, which is the following. Let X be a smooth irreducible projective curve over C, G be a complex reductive group. On one side of this conjectural correspondence there are G superscript v -opers on X satisfying a certain topological condition ( real opers), where G superscript v is Langlands dual group. On the other side there is joint spectrum of certain operators on L²(Bun subscript G), called Hecke operators, where Bun subscript G is the variety of stable parabolic G-bundles on X and L²(Bun subscript G) is a Hilbert space of square-integrable half-densities. We prove most of the main conjectures of analytic Langlands correspondence in the case when G=PGL₂(C) and X either a genus one curve with points or X is P¹ with higher structures at points.