### GRT at home Seminar

Quoc Ho, IST, Austria

Eisenstein series via factorization homology of Hecke categories

Organizer's time: 2021-05-11 17:00 Europe/Rome

Duration: 1 hour 15 minutes

Access information: https://us02web.zoom.us/j/84259232669?pwd=aXlkZTJ1dnFzT05Kb2ZkOXk1NEp6dz09

For any reductive group $G$ and a Riemann surface $M$, the Betti Langlands Conjecture (BLC) states that we have an equivalence of categories $\mathsf{IndCoh}_{N}(\mathbf{LocSys}_G(M)) = \mathsf{D}\textrm{-}\mathsf{mod}_{N'}(\mathbf{Bun}_{G'}(M))$, where $N$ and $N'$ denote some singular support condition and $G'$ is the Langlands dual group of $G$. It is hoped that various forms of gluing allow one to prove this conjecture by building up from simpler cases. Roughly speaking, two kinds of gluing are involved, which we call parabolic (i.e. Eisenstein series) gluing and manifold gluing. Respectively, they allow one to build up the BLC for the pair $G$, $M$ from smaller groups and from smaller pieces of $M$.

Motivated by this picture and working on the spectral side, we construct an $E_2$-analog (i.e. braided monoidal) of the Hecke category and show that its factorization homology on a topological surface $M$ recovers the spectral Eisenstein series of $M$. In other words, this shows that spectral Eisenstein series themselves can be glued together from disks. Our result naturally extends previously known computations of Ben-Zvi--Francis--Nadler and Beraldo. This is joint work with Penghui Li.

Video

Submitted by: Francesco Sala