Eisenstein series via factorization homology of Hecke categories

Abstract

Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group $G$, a parabolic subgroup $P$, and a topological surface $M$, the (enhanced) spectral Eisenstein series category of $M$ is the factorization homology over $M$ of the ${\mathrm{E}}_2$-Hecke category ${\mathrm{H}}_{G,P}=\mathrm{IndCoh}({\mathrm{LS}}_{G,P}(D^2,S^1))$, where ${\mathrm{LS}}_{G,P}(D^2,S^1)$ denotes the moduli stack of $G$-local systems on a disk together with a $P$-reduction on the boundary circle. More generally, for any pair of stacks $\mathcal{Y}\to \mathcal{Z}$ satisfying some mild conditions and any map between topological spaces $N\to M$, we define $(\mathcal{Y},\mathcal{Z}{)}^{N,M}={\mathcal{Y}}^N{\times }_{{\mathcal{Z}}^N}{\mathcal{Z}}^M$ to be the space of maps from $M$ to $\mathcal{Z}$ along with a lift to $\mathcal{Y}$ of its restriction to $N$. Using the pair of pants construction, we define an ${\mathrm{E}}_n$-category ${\mathrm{H}}_n(\mathcal{Y},\mathcal{Z})={\mathrm{IndCoh}}_0\left({\left((\mathcal{Y},\mathcal{Z}{)}^{S^{n-1},D^n}\right)}_{\mathcal{Y}}^{\wedge }\right)$ and compute its factorization homology on any $d$-dimensional manifold $M$ with $d\le n$, \[ \int_M \mathrm{H}_n(\mathcal{Y}, \mathcal{Z}) \simeq \mathrm{IndCoh}_0\left(\left((\mathcal{Y}, \mathcal{Z})^{\partial (M\times D^{n-d}), M}\right)^\wedge_{\mathcal{Y}^M}\right), \] where ${\mathrm{IndCoh}}_0$ is the sheaf theory introduced by Arinkin--Gaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi--Francis--Nadler and Beraldo.

Final version

References in corpus (1)

• Collared and non-collared manifold boundaries

Cited by in corpus (1)

• Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on ${\mathbb{C}}^2$