TY - JOUR
T1 - HECKE OPERATORS AND ANALYTIC LANGLANDS CORRESPONDENCE FOR CURVES OVER LOCAL FIELDS
AU - Etingof, Pavel
AU - Frenkel, Edward
AU - Kazhdan, David
N1 - Publisher Copyright:
© 2023 Duke University Press. All rights reserved.
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85171545219&partnerID=8YFLogxK
U2 - 10.1215/00127094-2022-0068
DO - 10.1215/00127094-2022-0068
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AN - SCOPUS:85171545219
SN - 0012-7094
VL - 172
SP - 2015
EP - 2071
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 11
ER -