Analytic Langlands correspondence for $\operatorname{PGL}_2(\mathbb{C})$ 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 $L^2(\operatorname{Bun}_G)$, called Hecke operators, where $\operatorname{Bun}_G$ is the variety of stable $G$-bundles on $X$ and $L^2(\operatorname{Bun}_G)$ 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 $G=\operatorname{PGL}_2$ and $X=\mathbb{P}^1$ with parabolic structures. We investigate the case of $G=\operatorname{PGL}_2$ 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.