Alexander Polishchuk - Schwartz spaces associated with the stack of rank 2 bundles on a curve over a local field

Abstract

This talk is based on joint works with Alexander Braverman and David Kazhdan. The motivation for this project comes from the Analytic Langlands program, which is an analog of the usual Langlands correspondence involving a curve over a local field. We study the natural Schwartz space of half-densities associated with the stack of rank 2 bundles on such a curve. The long term goal is to understand the action of Hecke operators on this space. For this it is useful to understand the behavior of elements of this space near stable bundles. Our first result is that a Schwartz half-density on the stack of rank 2 bundles defines a smooth (locally constant in the non-archimedean case) half-density on the open subscheme of very stable bundles, i.e., bundles without nonzero nilpotent Higgs fields. We then formulate a set of algebro-geometric conjectures (that we can check for low genus) that imply boundedness of these half-densities near stable bundles.

Talk

Motivation

In the classical Langlands correspondence we consider a curve $C/mathbbF?1$

In the Analytic Langlands correspondence we consider a curve $C/F$ with $F$ being a local field (like C or R). We consider Kecke cossepondences.

We talk about some Symbol

Ich dachte mir gerade, wenn ich einen Vortrag nicht verstehe, sollte ich darüber nachdenken, warum ich den Vortrag nicht verstehe.

Gertsgory-Kazddon-Sakellaritz habenein Symbol eingeführt, das ich nicht wieder erkennen kann $?(\mathcal{X}(F))$. Let $X$ be a Algebraische Varietät over the field $F$. We assume $F$ to be archimedean.

Let $L$ be an algebraicline bundle (whatever that is) over the variety $X$.

We do some equations and stuff. We continue with a group $G$ acting on the algebraic variety $X$. We consider the quotient $X/G$ and look at something called a $G$-equivariant line bundle on $X$.

We take $\mathcal{X}$ to be a union of open substack ${\mathcal{X}}_i$ of finite type. (I dont know what a substack is and what finite type means for a substack)

We do some stuff and than look at something called a very stable bundle. A vectur bundle von $C$ is called very stable if every map $\varphi :V\otimes {\omega }_C$ (whatever $\omega$ is) is nilpotent implies $\varphi +0$. Similarly we also look at stable bundles but I don’t know the definition of that.

*Anotherthing I have been thinking about is how long should I try to listen until I do something else. If I do something else immediately I won’t learn anything. But then all of the mathematician start reading something else after a while so it does seem to have some use. I figured the smartest thing to listen to half and do something else the other half (assuming I get lost until then) *

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