Equivariant Beilinson's Theorem and Extensions of Equivariant Perverse Sheaves

Abstract: Beilinson's Theorem is an important theorem in algebraic geometry that says there is a triangulated equivalence of categories $D^bc(X;\overline{\mathbb{Q}}{\ell}) \simeq D^b(\mathbf{Perv}(X;\overline{\mathbb{\Q}}{\ell}))$ fixing the category of perverse sheaves for any variety $X$ over an algebraically closed field $K$ with $\ell$ a positive integer prime distinct from the characteristic of $K$. While a result of fundamental importance, as it allows the computation of extensions between perverse sheaves in the derived category on $X$ to be performed in their own bounded derived category, the equivariant version of this result has been elusive and known in general only for complex varieties with actions by finite complex algebraic groups. In this talk I'll discuss a general proof for an equivariant version of Beilinson's Theorem, i.e.,a triangulated equivalence $D^b_G(X;\overline{\mathbb{Q}}{\ell}) \simeq D^bG(\mathbf{Perv}(X;\overline{\mathbb{Q}}{\ell}))$ which fixes the category of equivariant perverse sheaves valid for any variety equipped with an action by a smooth algebraic group $G$ over a field $K$. Afterwards I'll give a short discussion as well about what this means for equivariant extensions between equivariant perverse sheaves.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the topic

Comments: I'll try to keep the category theory in this talk to as gentle a level as possible and at as understandable a level as possible as well, so please ask questions as we go!

Canadian Rockies Representation Theory

Series comments: Topics include, but are not limited to, geometric and categorical aspects of the Langlands Programme. Please write to Jose Cruz for zoom instructions.