We present topics in the Langlands Program to graduate students and a wider mathematically mature audience. study both global local aspects characteristic zero as well p p ...

In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in classical theory. Gaitsgory--Lurie proposed conjectural substitute, later termed fundamental local equivalence. With few exceptions, we prove this conjecture and its extension to affine flag variety by using what amount Soergel module techniques.

Representation theory. Let us try to study a given object X through vector spaces V associated naturally to it. The symmetries S of X can form a group or more generally some kind of an algebra usually related to a group (Lie algebra, quantum group,...). The Vector space V inherits the symmetries of X, and they act on V by linear operators. This is what one means by “V is a representation of the...

Let F be a totally real number field. We define global L-packets for GSp(2) over F which should correspond to the elliptic tempered admissible homomorphisms from the conjectural Langlands group of F to the L-group of GSp(2) which are reducible, or irreducible and induced from a totally real quadratic extension of F . We prove that the elements of these global L-packets occur in the space of cus...

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...

This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n, A) with a Siegel modular form of degree n for the full modular group Γn = Sp(2n, Z), generalizing the well-known procedure for n = 1. This will show that the so-called “standard” and “spin” L-functions associated with such forms are obtained as Langlands L-functions. The theory of Euler produc...

We introduce a new approach to the representation theory of reductive p-adic groups G, based on the Geometric Invariant Theory (GIT) of Moy-Prasad quotients. Stable functionals on these quotients are used to give a new construction of supercuspidal representations of G having small positive depth, called epipelagic. With some restrictions on p, we classify the stable and semistable functionals ...

Let Mm be the formal scheme which represents the functor of deformations of a one-dimensional formal module over F̄p equipped with a level-m-structure. By work of Boyer (in mixed characterististic) and Harris and Taylor, the l-adic étale cohomology of the generic fibre Mm of Mm realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld...