Abstract
The Langlands correspondence for complex curves is traditionally formulated in terms of sheaves rather than functions. Recently, Langlands asked whether it is possible to construct a function-theoretic version. In this paper we use the algebra of commuting global differential operators (quantum Hitchin Hamiltonians and their complex conjugates) on the moduli space of G-bundles of a complex algebraic curve to formulate a function-theoretic correspondence. We conjecture the existence of a canonical self-adjoint extension of the symmetric part of this algebra acting on an appropriate Hilbert space and link its spectrum with the set of opers for the Langlands dual group of G satisfying a certain reality condition, as predicted earlier by Teschner. We prove this conjecture for G = GL1 and in the simplest non-abelian case.
Original language | English |
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Title of host publication | Proceedings of Symposia in Pure Mathematics |
Publisher | American Mathematical Society |
Pages | 137-202 |
Number of pages | 66 |
DOIs | |
State | Published - 2021 |
Publication series
Name | Proceedings of Symposia in Pure Mathematics |
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Volume | 103.2 |
ISSN (Print) | 0082-0717 |
ISSN (Electronic) | 2324-707X |
Bibliographical note
Publisher Copyright:© 2021 by Pavel Etingof, Edward Frenkel, and David Kazhdan..