Eisenstein series via factorization homology of Hecke categories

Quoc P. Ho Department of Mathematics, Hong Kong University of Science and Technology Penghui Li YMSC, Tsinghua University

Representation Theory mathscidoc:2205.30002

Advances in Mathematics, 404, 108410, 2022.8
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 E2-Hecke category H_{G,P} =IndCoh(LS_{G,P}(D^2, S^1)), where 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 Y → Z satisfying some mild conditions and any map between topological spaces N → M, we define (Y, Z)^{N,M} = Y^N ×_{Z^N} {Z^M} to be the space of maps from M to Z along with a lift to Y of its restriction to N. Using the pair of pants construction, we define an E_n-category H_n(Y, Z) = IndCoh_0((Y, Z)^{S^n−1,D^n})_{Y}), and compute its factorization homology on any d-dimensional manifold M with d ≤ n, \int_M H_n(Y, Z) = IndCoh_0(Y, Z)^{∂(M×D^n−d),M}_{Y^M}, where IndCoh_0 is the sheaf theory introduced by ArinkinGaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi–Francis–Nadler and Beraldo.
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  title={Eisenstein series via factorization homology of Hecke categories},
  author={Quoc P. Ho, and Penghui Li},
  booktitle={Advances in Mathematics},
Quoc P. Ho, and Penghui Li. Eisenstein series via factorization homology of Hecke categories. 2022. Vol. 404. In Advances in Mathematics. pp.108410. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220518151655991808270.
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