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Abstract: The metaplectic geometric Langlands program can be understood in two ways: as a twisted analogue of the geometric Langlands program, and as a geometric analogue of the Langlands program for covering groups, namely the metaplectic Langlands program. One of the key structures in the geometric Langlands program is the spectral action, which roughly says that the spectral side acts on the automorphic side, allowing one to spectrally decompose the automorphic category over the moduli of \check{G}-local systems. This can be viewed as a geometric incarnation of the automorphic-to-Galois direction.

One may hope that the spectral action carries over to the metaplectic setting, and this expectation was formulated by Gaitsgory and Lysenko under the name “metaplectic vanishing conjecture.” In this talk, I will discuss my recent work on constructing the spectral action in the Betti metaplectic context. I will focus on what changes in the metaplectic setting and on the categorical framework for studying factorizable actions of local systems of symmetric monoidal categories.

Abstract: Classical Harish-Chandra bimodules are related to many objects in representation theory such as infinite dimensional representations of complex groups viewed as real groups, category O and character sheaves. The notion of Harish-Chandra bimodules for quantum groups was introduced in the works about topological quantum field theory of surfaces. I will mention some results about quantum Harish-Chandra bimodules at odd order roots of unity, connect them to affine Soergel bimodules and coherent sheaves on Steinberg variety. 

Abstract: In joint work with Gurbir Dhillon, we prove a universal monodromic extension of Bezrukavnikov's two realizations of the affine Hecke category over the integers. I will discuss some ingredients that go into to the proof of the Arkhipov-Bezrukavnikov equivalence in this setting. These include analysis of multi-graded rings, deformation between characteristic zero and positive characteristic, and the tilting property of Whittaker averaged central sheaves. The proof of the latter is based on ideas of Faergeman-Raskin and Bezrukavnikov-Morton-Ferguson. 

Abstract: I will report on an incomplete project, whose goal is to geometrically realize representations of critical level affine Lie algebras at all central characters. To do so, we study these categories of representations from the point of view of the local geometric Langlands program. As an accidental consequence, we will prove some new results in the local Langlands program, both in geometry and in arithmetic. No knowledge of affine Lie algebras or of Langlands will be assumed. Some parts are joint work with a number of other mathematicians, including Dhillon, Eteve, Gaitsgory, Raskin and Varshavsky 

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