Automorphic functions on moduli spaces of bundles on curves over local fields: a survey

Abstract

This paper is the written version of D. Kazhdan's plenary talk at ICM 2022. It is dedicated to an exposition of recent results and (mostly) conjectures attempting to construct an analog of the theory of automorphic functions on moduli spaces of bundles on curves over local fields (both archimedian and nonarchimedian). The talk is based on joint works of D. Kazhdan with A.Braverman, P.Etingof, E.Frenkel and A.Polishchuk.

An analytic version of the Langlands correspondence for complex curves

“…This makes the operator H x,λ at least densely defined on H(s, τ, α), although it is not obvious that it lands in H(s, τ, α). We have conjectured in [EFK2,EFK3] (see also [BK2]) that in fact it does, and moreover it extends to a bounded operator which is norm-continuous in x (see Conjecture 2.16 below).…”

“…Remark 2.19. Another, more geometric definition of the Schwartz space based on the geometry of the stack of G-bundles on X, which is conjecturally equivalent to the above, is proposed in the non-archimedean case in [BK2], see also [BKP1], [BKP2]. A similar definition is expected to exist in the archimedean case.…”

“…Remark 2.37. At least for ℓ = 1, one should be able to extend this theory to the case of admissible representations V i (not necessarily tempered, or even unitarizable) by working in the Schwartz space context instead of L 2 space (for example, using the approach of [BK2]). For instance, in this context the extension to complementary series from Remark 2.36 should be much more straightforward -we don't need to worry about positive inner products and can just do analytic continuation with respect to the Casimir eigenvalues 1 2 (λ j + 1) 2 (which are no longer required to be real).…”