Abstract:

Graded categories play an important role in geometric representation theory (e.g. Koszul duality patterns à la Beilinson--Ginzburg--Soergel) and categorified knot invariants (e.g. HOMFLY-PT link homology). Classically, the geometric incarnations of these categories, a.k.a. mixed versions or graded lifts of the usual category of constructible sheaves, were only constructed in very special situations on a case-by-case basis and were technically subtle and complicated, due to Frobenius' non-semisimplicity. However, this is insufficient for many interesting problems in representation theory and categorified knot invariants.

I will present joint work with Penghui Li on our theory of graded sheaves, which provides a uniform construction of graded lifts, sidestepping Frobenius' non-semisimplicity by semi-simplifying the Frobenius action itself at the categorical level. Our sheaf theory comes with a six-functor formalism, a perverse t-structure in the sense of Beilinson--Bernstein--Deligne--Gabber, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello, all compatible, in a precise sense, with the six-functor formalism, perverse t-structures, and Frobenius weights on ell-adic sheaves. I will conclude the talk with an application to the categorical traces and Drinfeld centers of the graded Hecke categories, a.k.a. the homotopy categories of Soergel bimodules.