On algebraic spaces with an action of G_m

TL;DR

This paper studies algebraic spaces with G_m actions, introducing a new construction related to the graph closure of the action, with applications to automorphic forms and a new proof of Braden's theorem.

Contribution

It defines a novel algebraic space associated with G_m actions on algebraic spaces, providing new tools for geometric automorphic forms and a new proof of Braden's theorem.

Findings

Introduces a new algebraic space related to G_m actions.

Provides a new proof of Braden's theorem.

Applies to automorphic forms in geometric representation theory.

Abstract

Let Z be an algebraic space of finite type over a field, equipped with an action of the multiplicative group $G_m$. In this situation we define and study a certain algebraic space equipped with an unramified morphism to $A^1\times Z\times Z$, where $A^1$ is the affine line. (If Z is affine and smooth this is just the closure of the graph of the action map $G_m\times Z\to Z$.) In articles joint with D.Gaitsgory we use this set-up to prove a new result in the geometric theory of automorphic forms and to give a new proof of a very important theorem of T. Braden.