Time: 3:00 - 4:00 PM

Speaker: Alexandra Utiralova (U. of Oregon)

Title: Representations of the general linear group in the Verlinde category

Abstract: Verlinde categories are defined as the semisimplification of the category of representations of Z/pZ in characteristic p. As shown by Coulembier, Etingof and Ostrik in arXiv:2107.02372, these categories play the role of the target category for the fiber functor for a large class of symmetric tensor categories (Frobenius exact, of moderate growth) in char p (usually played by the category of super vector spaces in char 0). Consequently, Tannakian reconstruction tells us that any category with a fiber functor to Ver_p (and hence any Frobenius exact category of moderate growth) is equivalent to the category of representations of some group scheme in Ver_p. I will talk about representations of the general linear group GL(X) for X in Ver_p and the related combinatorics with a focus on translation functors and the related categorical type A action.

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Time: 4:00 - 5:00 PM

Speaker: Dmitry Kubrak (Orsay, CNRS)

Title: Nearby cycles cosheaf and derived Satake with Z-coefficients

Abstract: 

Given a conically stratified topological space X with a stratification poset P one can identify the corresponding category of constructible sheaves with the category of functors from the so-called exit-path (infinity)-category Exit(X,P). This identification is known as "exodromy equivalence". Similarly, cosheaves can be described as functors from Enter(X,P):=Exit(X,P)^op. I will explain a formal construction which takes as an input a sufficiently nice map f:(Y,Q) --> (X,P) of stratified topological spaces and produces a lax-functor NC(f) from Enter(X,P) to Pr^L_st. NC(f) associates to a point x\in X the derived category of constructible sheaves on the preimage f^{-1}(x) and sends an enter path x ---> y to the nearby cycles functor between the categories of sheaves on f^{-1}(x) and f^{-1}(y), but it also takes care of higher coherences for compositions of several paths.

This relatively easy construction has some non-trivial applications, in particular to derived Satake equivalence. In general, for a factorizable stratified topological space over Ran(X) it endows the category constructible sheaves on the fiber with a lax-version of E_X-monoidal structure. When X=R^2, in the case of the Beilinson-Drinfeld-Hecke stack the corresponding lax-functor turns out to be strict, and one gets an honest E_3-monoidal structure on the spherical Hecke category Sph_G with Z-(or even spectral) coefficients. Applying a similar construction to a factorizable version of the Whittaker model we obtain an E_3-monoidal functor from Sph_G to the E_2-center of Ind(Perf(BG^v)), which is (roughly) the expected target of derived Satake equivalence. Finally, considering separately the Whittaker model with spectral coefficients, one gets a descent of Rep(G^v) to the sphere spectrum as an E_2-monoidal category. This is the content of joint works in progress with M.Kjærsgaard, G.Nocera and Q.Wang, and A.Prikhodko and R.Travkin.

Zoom: https://mit.zoom.us/j/91800438944. For the Passcode, please contact Pavel Etingof at etingof@math.mit.edu.