Seminar on universal centralizers, Fall 2021

Overview

This seminar is focused on a space, associated to a reductive group G, which has many names: the group scheme of regular centralizers; the Toda space; the bi-Whittaker reduction; the BFM space; or simply J. This space plays distinguished roles in representation theory and complex integrable systems, including, most famously, Ngô's proof of the Fundamental Lemma, via its universal action on Hamiltonian reductions X//G. The ring of functions on J for the Langlands dual group is equivalent to the Borel-Moore homology on the space of G-bundles on a formal sphere, as computed by Bezrukavnikov-Finkelberg-Mirković, later realized by Braverman-Finkelberg-Nakajima as a basic example of the Coulomb branch of a topologically twisted 3-dimensional gauge theory. It was also realized by Teleman as a receptacle for spectral decompositions of categories with a topological G-action, in an early example of what has come to be called 3d mirror symmetry.

The first half of the semester will be introductory, focusing on the example of G = SL(2) and learning how to describe this space in many of the guises mentioned above. In the second half of the semester, we will go deeper into the theory of J, possibly including applications to representation theory, 3d mirror symmetry, or whatever topics are of interest to participants in this seminar.

E-mail me (Ben G.) if you want on the mailing list.

The seminar meets 3pm Thursdays in SC507.

Schedule

Date Speaker Topic
Sept. 2 Benjamin Gammage Introduction and organization
Sept. 9 David Yang Toda lattice
Sept. 16 [No seminar] [No seminar]
Sept. 23 [Postponed] [Postponed]
Sept. 30 Keeley Hoek Quantum Toda lattice
Oct. 7 Sanath Devalapurkar Bi-Whittaker D-modules
Oct. 14 Grant Barkley The BFM calculation
Oct. 21 Jianqiao Xia Regular centralizers in the Fundamental Lemma
Oct. 28 Charles Fu (???)
Nov. 4 Jae Hae Lee QH*(G/B)

Possible topics for talks that haven't been claimed yet

References

[1] Bezrukavnikov-Finkelberg-Mirković, Equivariant (K-)homology of affine Grassmannian and Toda lattice
[2] Bezrukavnikov-Finkelberg, Equivariant Satake category and Kostant-Whittaker reduction
[3] Braverman-Finkelberg-Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, II
[4] Braverman-Finkelberg, Coulomb branches of 3-dimensional gauge theories and related structures
[5] Teleman, Gauge theory and mirror symmetry
[6] Ngô, Le lemme fondamental pour les algèbres de Lie
[7] Ginzburg, Nil Hecke algebras and Whittaker D-modules
[8] Lonergan, A Fourier transform for the quantum Toda lattice
[9] Kazhdan-Kostant-Sternberg, Hamiltonian group actions and dynamical systems of Calogero type
[10] Kostant, The solution to a generalized Toda lattice and representation theory
[11] Givental-Kim, Quantum cohomology of flag manifolds and Toda lattices
[12] Kostant, Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight ρ
[13] Rietsch, A mirror symmetric solution to the quantum Toda lattice
[14] Ben-Zvi–Gunningham, Symmetries of categorical representations and the quantum Ngô action
[15] Jin, Homological Mirror Symmetry for the universal centralizers I: The adjoint group case
[16] Balibanu, The partial compactification of the universal centralizer


Organized by Ben G. (that's me!)