Penghui Li 李鹏辉

Research

Arxiv versions can be slightly out of date.

Publications and Preprints

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

[arXiv]

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.

[arXiv]

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.

[arXiv]

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 \(\widehat{G}\).

[arXiv]

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).

[arXiv]