Summer School on the Langlands Program

Collection Summer School on the Langlands Program

Organizer(s) Pierre-Henri Chaudouard, Wee Teck Gan, Tasho Kaletha, Yiannis Sakellaridis
Date(s) 11/07/2022 - 29/07/2022
linked URL
00:00:00 / 00:00:00
10 58

Arthur's Conjectures and the Orbit Method for Real Reductive Groups

By Lucas Mason-Brown

The most fundamental unsolved problem in the representation theory of Lie groups is the Problem of the Unitary Dual: given a reductive Lie group G, this problem asks for a parameterization of the set of irreducible unitary G-representations. There are two big "philosophies" for approaching this problem. The Orbit Method of Kostant and Kirillov seeks to parameterize irreducible unitary representations in terms of finite covers of co-adjoint G-orbits. Arthur's conjectures suggest a parameterization in terms of certain combinatorial gadgets (i.e. Arthur parameters) related to the Langlands dual group G^{\vee} of G. In this talk, I will define these correspondences precisely in the case of complex groups. I will also define a natural duality map from Arthur parameters (for G^{\vee}) to co-adjoint covers (for G) which, in a certain precise sense, intertwines these correspondences. This talk is partially based on joint work with Ivan Losev and Dmitryo Matvieievskyi.

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