Analytic Langlands correspondence for ${\mathrm{PGL}}_2(\mathbb{C})$ on a genus one curve with parabolic structures

Abstract

Analytic Langlands correspondence was proposed by Etingof, Frenkel and Kazhdan. On one side of this correspondence there are certain operators on $L^2({\mathrm{Bun}}_G)$, called Hecke operators, where ${\mathrm{Bun}}_G$ is the variety of stable $G$-bundles on $X$ and $L^2({\mathrm{Bun}}_G)$ is a Hilbert space of square-integrable half-densities. The compactness conjecture says that Hecke operators are bounded and, moreover, compact. In arXiv:2106.05243 Etingof, Frenkel and Kazhdan prove this and other conjectures in the case of $G={\mathrm{PGL}}_2$ and $X={\mathbb{P}}^1$ with parabolic structures. We investigate the case of $G={\mathrm{PGL}}_2$ and genus one curve over complex numbers with parabolic structures. We obtain an explicit formula for Hecke operators and prove the compactness conjecture in this case.

arXiv comment21 pages, v2: changed abstract