Tempered sheaves

Let G be a compact Lie group. Equivariant elliptic cohomology Ell_G is a G-equivariant cohomology theory which a generalization of the equivariant ordinary (Borel) cohomology (height zero) and the equivariant K-theory (height one) to higher heights. The first proposed construction for the rationalized equivariant elliptic cohomology theory was given by I. Grojnowski, and recently, J. Lurie (for finite groups) and D. Gepner and L. Meier (for all compact Lie groups) proposed an alternative (non-rationalized) construction for equivariant elliptic cohomology. Furthermore, J. Lurie defined the category of local tempered systems LocSys(X/G) associated with the stacky quotient of a G-space X by G. This category happened to be useful in proving of many structural properties of Ell_G.

In my talk, I will define the category of tempered sheaves Shv(X/G) which extends the category of local tempered systems and I will explain the six-functor formalism in this context. The category Shv(X/G) allows us to define the equivariant elliptic cohomology with compact support and to show its expected properties. If time permits, I will also explain the applications of the category of tempered sheaves to geometric representation theory. Joint work in progress with I. Perunov and A. Prikhodko.