Relative Langlands seminar, Fall 2023

Overview

This is a seminar about relative Langlands duality (also known as "periods--L-functions duality"). The main reference is the paper [BSV] of Ben-Zvi--Sakellaridis--Venkatesh. It is being organized by Benjamin Gammage and Sanath Devalapurkar. Email one of us if you'd like to be added to the mailing list.

The scheduled time for the seminar to meet is 1pm Tuesdays in Room SC232.

Schedule

Date	Speaker	Topic	Notes
12 September	Benjamin Gammage	Introductory talk & organization	Notes
19 September	Grant Barkley	Spherical varieties	Notes
26 September	Wyatt Reeves	Hyperspherical varieties & Whittaker reduction	Notes
3 October	Zeyu Wang	The dual group of a spherical variety	Notes
10 October	Ivan Karpov	The local conjecture
17 October	Siddharth Mahendraker	Abelian examples	Notes
24 October	Sanath Devalapurkar	Overview
31 October🎃	Sanath Devalapurkar	Examples in rank 1
7 November	Haoshuo Fu	The Plancherel/BFN algebra	Notes
14 November	Siddharth Mahendrakar	Number theory
21 November	[No seminar]	[No seminar -- Thanksgiving week]
28 November	Charles Fu	The global conjecture

References

Spherical varieties

[P] Perrin, Sanya lectures, https://doi.org/10.1007/s10114-017-7163-6
[G] Gandini, Embeddings of spherical homogeneous spaces, https://doi.org/10.1007/s10114-018-7162-2
[B] Brion, Lecture notes on spherical varieties, http://www-fourier.univ-grenoble-alpes.fr/~mbrion/notes.html
[LV] Luna-Vust, Plongements d’espaces homogènes, https://doi.org/10.1007/BF02564633
[K1] Knop, The Luna-Vust theory of spherical embeddings, https://users.mccme.ru/valya/Knop%20The%20Luna-Vust%20theory%20of%20spherical%20embeddings.pdf

Hyperspherical varieties

[BSV], §§3.4–3.7

The dual group

[K2] Knop, On the set of orbits for a Borel subgroup, https://doi.org/10.1007/BF02566009
[K3] Knop, The asymptotic behavior of invariant collective motion, https://doi.org/10.1007/BF01231563
[GN] Gaitsgory-Nadler, Spherical varieties and Langlands duality, https://doi.org/10.17323/1609-4514-2010-10-1-65-137
[BSV] §4.2
[S] Sakellaridis, Functorial transfer between relative trace formulas in rank one, https://doi.org/10.1215/00127094-2020-0046, §2.1
[W] Wang, notes 1 2 3
[SV] Sakellaridis-Venkatesh, Periods and harmonic analysis on spherical varieties, https://doi.org/10.24033/ast.1040

Abelian groups (geometric Tate's thesis)

[HR] Hilburn-Raskin, Tate's thesis in the de Rham setting, https://doi.org/10.1090/jams/1010
[GH] Gammage-Hilburn, Betti Tate's thesis and the trace of perverse schobers, https://doi.org/10.48550/arXiv.2210.06548

Examples in rank 1

[BF] Braverman-Finkelberg, A quasi-coherent description of the category D-mod(Gr_GL(n)), https://doi.org/10.1007/978-3-030-82007-7_5
[BFGT] Braverman-Finkelberg-Ginzburg-Travkin, Mirabolic Satake equivalence and supergroups, https://doi.org/10.1112/S0010437X21007387
[S] Sakellaridis, Functorial transfer between relative trace formulas in rank one, https://doi.org/10.1215/00127094-2020-0046

The Plancherel algebra

[BSV] §8
[BFN1] Braverman-Finkelberg-Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, II, https://doi.org/10.4310/ATMP.2018.v22.n5.a1
[BFN2] Braverman-Finkelberg-Nakajima, Ring objects in the equivariant derived Satake category arising from Coulomb branches (with an appendix by Gus Lonergan), https://doi.org/10.4310/ATMP.2019.v23.n2.a1

The global conjecture

[BSV] §10