The geometric Langlands correspondence for function fields over finite fields has been proved by Frenkel, Gaitsgory, Vilonen in [FGV]. The aim of this article is to write translation for curves over C and prove the correspondence in some special cases and some new cases, outside the frame of the usual geometric Langlands correspondence. 1 The geometric version of Langlands conjecture We start b...

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseu...

Abstract We use the theta correspondence to study equivalence between Godement–Jacquet and Jacquet–Langlands L -functions for ${\mathrm {GL}}(2)$ . show that resulting comparison is in fact an expression of exotic symmetric monoidal structure on category -modules. Moreover, this enables us construct abelian abstractly automorphic representations, whose irreducible objects are usual representati...

We show that the completed Hecke algebra of $p$ -adic modular forms is isomorphic to continuous automorphic for units quaternion ramified at and $\infty$ . This gives an affirmative answer a question posed by Serre in 1987 letter Tate. The proof geometric, lifts mod argument due Serre: we evaluate identifying quaternionic double-coset with fiber Hodge–Tate period map, extend functions off using...

1.1 Whittaker and Bessel models are of importance in the theory of automorphic representations of GSp4. This paper is the first in a series of two, where we study some phenomena corresponding to these models in the geometric Langlands program. The theory of Whittaker functors for GLn is an essential technical tool in Gaitsgory’s proof of the Vanishing Conjecture appearing in the geometric Langl...

This paper is intended as an introduction to the gauge theory approach [15] to the geometric Langlands correspondence. But, rather than a conventional overview, which I have attempted elsewhere [25, 26], the focus here is on understanding a very particular result, which I learned of from [13]. (Another standard reference on closely related matters is [17].) This introduction is devoted to descr...

We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to affine Kac–Moody algebras. Our experience with the Gaudin models associated to finite-dimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model assoc...

Twenty-five years ago R. Langlands proposed [L] a “fantastic generalization” of Artin-Hasse reciprocity law in the classical class field theory. He conjectured the existence of a correspondence between automorphic irreducible infinite-dimensional representations of a reductive groupG over a global number field on the one hand, and (roughly speaking) finite dimensionsional representations of the...

‎In the spirit of the Langlands proposal on  Beyond Endoscopy ‎we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$-functions‎. ‎It is well-known that the poles of the $L$-functions have deep impact to the Langlands functoriality‎. ‎Our discussion also‎ ‎includes the meaning of the central value of the tensor product $L$-functions in terms of the Langl...