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UID:1636635600@ist.ac.at
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DESCRIPTION:Speaker: Pavel Etingof\nhosted by Tamas Hausel\nAbstract: I wil
 l review an analytic approach to the geometric Langlands correspondence\, 
 following my work with E. Frenkel and D. Kazhdan\,arXiv:1908.09677\, arXiv
 :2103.01509\, arXiv:2106.05243. This approach was developed by us in the l
 ast couple of years and involves ideas from previous and ongoing works of 
 a number of mathematicians and mathematical physicists\, Kontsevich\, Lang
 lands\, Teschner\, and Gaiotto-Witten. One of the goals of this approach i
 s to understand single-valued real analytic eigenfunctions of the quantum 
 Hitchin integrable system. The main method of studying these functions is 
 realizing them as the eigenbasis for certain compact normal commuting inte
 gral operators the Hilbert space of L2 half-densities on the (complex poin
 ts of) the moduli space Bun_G of principal G-bundles on a smooth projectiv
 e curve X\, possibly with parabolic points. These operators actually make 
 sense over any local field\, and over non-archimedian fields are a replace
 ment for the quantum Hitchin system. We conjecture them to be compact and 
 prove this conjecture in the genus zero case (with parabolic points) for G
 =PGL(2). I will first discuss the simplest non-trivial example of Hecke o
 perators over local fields\, namely G=PGL(2) and genus 0 curve with 4 para
 bolic points. In this case the moduli space of semistable bundles Bun_G^{s
 s} is P^1\, and the situation is relatively well understood\; over C it is
  the theory of single-valued eigenfunctions of the Lame operator with coup
 ling parameter -1/2 (previously studied by Beukers and later in a more fun
 ctional-analytic sense in our work with Frenkel and Kazhdan). I will consi
 der the corresponding spectral theory and then explain its generalization 
 to N>4 points and conjecturally to higher genus curves.
LOCATION:https://mathseminars.org/seminar/AGNTISTA\, ISTA
ORGANIZER:birgit.oosthuizen-noczil@ist.ac.at
SUMMARY:Pavel Etingof: Hecke operators over local fields and an analytic ap
 proach to the geometric Langlands correspondence
URL:https://talks-calendar.ista.ac.at/events/3273
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