HECKE OPERATORS AND ANALYTIC LANGLANDS CORRESPONDENCE FOR CURVES OVER LOCAL FIELDS

Pavel Etingof, Edward Frenkel, David Kazhdan

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We construct analogues of the Hecke operators for the moduli space of G-bundles on a curve X over a local field F with parabolic structures at finitely many points. We conjecture that they define commuting compact normal operators on the Hilbert space of half-densities on this moduli space. In the case F = ℂ, we also conjecture that their joint spectrum is in a natural bijection with the set of LG-opers on X with real monodromy. This may be viewed as an analytic version of the Langlands correspondence for complex curves. Furthermore, we conjecture an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators. Assuming the compactness conjecture, this formula follows from a certain system of differential equations satisfied by the Hecke operators, which we prove here for G = PGLn.

Original languageEnglish
Pages (from-to)2015-2071
Number of pages57
JournalDuke Mathematical Journal
Volume172
Issue number11
DOIs
StatePublished - 2023

Bibliographical note

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© 2023 Duke University Press. All rights reserved.

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