On algebraic spaces with an action of G_m

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.

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