Penghui Li 李鹏辉

Eisenstein series via factorization homology of Hecke categories
with Quoc P. Ho

The Jordan--Chevalley decomposition for $G$-bundles on elliptic curves
with Dragos Fratila and Sam Gunningham
Abstract: We study the moduli stack of degree 0 semistable G-bundles on an irreducible curve E of arithmetic genus 1, where G is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups $H$ of G (the E-pseudo-Levi subgroups), where each stratum is computed in terms of H-bundles together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan--Chevalley theorem for such bundles equipped with a framing at a fixed basepoint. In the case where E has a single cusp (respectively, node), this gives a new proof of the Jordan--Chevalley theorem for the Lie algebra g (respectively, group \(G \) ). We also provide a Tannakian description of these moduli stacks and use it to show that if E is an ordinary elliptic curve, the collection of framed unipotent bundles on E is equivariantly isomorphic to the unipotent cone in G. Finally, we classify the E-pseudo-Levi subgroups using the Borel--de Siebenthal algorithm and compute some explicit examples.

A colimit of traces of reflection groups
Proceedings of the AMS, Volume 147, Number 11 (Nov. 2019)
Abstract: Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti Geometric Langlands Conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space.

Derived categories of character sheaves
Under review, pending revisions.
Abstract: We give a block decomposition of the dg category of character sheaves on a simple and simply-connected complex reductive group \(G\), similar to the one in generalized Springer correspondence. As a corollary, we identify the category of character sheaves on \(G \) as the category of quasi-coherent sheaves on an explicitly defined derived stack \(\widehat{G}\).

Uniformization of semistable bundles on elliptic curves
with David Nadler
Advances in Mathematics, Vol. 380 (Mar. 2021)
Abstract: Let \(G\) be a connected reductive complex algebraic group, and \(E\) a complex elliptic curve. Let \(G_E\) denote the connected component of the trivial bundle in the stack of semistable \(G\)-bundles on \(E\). We introduce a complex analytic uniformization of \(G_E\) by adjoint quotients of reductive subgroups of the loop group of \(G\). This can be viewed as a nonabelian version of the classical complex analytic uniformization \( E \simeq \mathbb{C}^*/q^{\mathbb{Z}}\). We similarly construct a complex analytic uniformization of \(G\) itself via the exponential map, providing a nonabelian version of the standard isomorphism \(\mathbb{C}^* \simeq \mathbb{C}/\mathbb{Z}\), and a complex analytic uniformization of \(G_E\) generalizing the standard presentation \(E \simeq \mathbb{C}/(\mathbb{Z} \oplus \mathbb{Z} \tau )\). Finally, we apply these results to the study of sheaves with nilpotent singular support. As an application to Betti geometric Langlands conjecture in genus 1, we define a functor from \(Sh_\mathcal{N}(G_E)\) (the semistable part of the automorphic category) to \( IndCoh_{\check{\mathcal{N}}}(Locsys_{\check{G}} (E)) \) (the spectral category).