Hamiltonian spaces and L-functions

Duration: 1 hour 3 mins

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Description:	Sakellaridis, Y Friday 28th May 2021 - 16:45 to 17:45


Created:	2021-06-01 10:03
Collection:	New connections in number theory and physics
Publisher:	Isaac Newton Institute for Mathematical Sciences
Copyright:	Sakellaridis, Y
Language:	eng (English)


Abstract:	I will talk about a small part of ongoing work with David Ben-Zvi and Akshay Venkatesh, which aims to understand the relations between periods of automorphic forms and L-functions in the context of geometric Langlands duality. With this motivation in the background, I will present a down-to-earth result about the structure of cotangent bundles of spherical varieties, extending results of Friedrich Knop, and confirming an observation that we first heard from Vincent Lafforgue. The result is also relevant to work of Gaiotto on S-duality of boundary conditions (or rather, its semiclassical limit). Namely, Knop has proven that the cotangent bundle of a spherical variety is generically — up to codimension one — very simple: it is a torsor for a certain abelian group scheme over an appropriate cover of its moment image. Our result shows, under certain assumptions on the variety, that — up to codimension two — the space has more structure, modelled on a certain "toric embedding" of this abelian group scheme. The extra divisors determining this embedding are closely related to the L-function associated to the spherical variety. If time permits, I will also describe the quantized version of this picture, expressing certain p-adic integrals in terms of L-functions, including recent work with Jonathan Wang on the case of singular affine spherical varieties.

Abstract:

I will talk about a small part of ongoing work with David Ben-Zvi and Akshay Venkatesh, which aims to understand the relations between periods of automorphic forms and L-functions in the context of geometric Langlands duality.

With this motivation in the background, I will present a down-to-earth result about the structure of cotangent bundles of spherical varieties, extending results of Friedrich Knop, and confirming an observation that we first heard from Vincent Lafforgue. The result is also relevant to work of Gaiotto on S-duality of boundary conditions (or rather, its semiclassical limit).

Namely, Knop has proven that the cotangent bundle of a spherical variety is generically — up to codimension one — very simple: it is a torsor for a certain abelian group scheme over an appropriate cover of its moment image. Our result shows, under certain assumptions on the variety, that — up to codimension two — the space has more structure, modelled on a certain "toric embedding" of this abelian group scheme. The extra divisors determining this embedding are closely related to the L-function associated to the spherical variety.

If time permits, I will also describe the quantized version of this picture, expressing certain p-adic integrals in terms of L-functions, including recent work with Jonathan Wang on the case of singular affine spherical varieties.

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