Analytic Langlands correspondence for PGL₂(ℂ) on a genus one curve with parabolic structures

2023/11/21 by Daniil Klyuev, Klyuev, Daniil
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Functional Analysis (math.FA) #Geometry and complex manifolds #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2311.13030

openalex publication_date 2023/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Analytic Langlands correspondence was proposed by Etingof, Frenkel and Kazhdan. On one side of this correspondence there are certain operators on L²(Bun_G), called Hecke operators, where Bun_G is the variety of stable G-bundles on X and L²(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=PGL₂ and X=ℙ¹ with parabolic structures. We investigate the case of G=PGL₂ 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.

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