Related papers: A `relative' local Langlands correspondence

Let F be a p-adic field and n a positive integer. The local Langlands conjecture asserts the existence of a bijection between irreducible admissible representations of GL(n,F) and n-dimensional admissible representations of the Weil-Deligne…

We calculate the asymptotic behavior of the dimension of the fixed vectors of $\pi$ with respect to compact open subgroups $1+ M_n(\mathfrak{p}^N)\subset\mathrm{GL}_n(F)$ for $\pi$ an admissible representation of $\mathrm{GL}_n(F)$, and $F$…

We consider from a geometric point of view the conjectural fundamental lemma of Langlands and Shelstad for unitary groups over a local field of positive characteristic. We introduce projective algebraic varieties over the finite residue…

Let k be an algebraically closed field of characteristic >2, F=k((t)) and Mp(F) denote the metaplectic extension of Sp_{2d}(F). In this paper we propose a geometric analog of the Weil representation of Mp(F). This is a category of certain…

This paper performs the following steps toward the proof of GLC in the de Rham setting: (i) We deduce GLC for G=GL_n; (ii) We prove that the Langlands functor L_G constructed in [GLC1], when restricted to the cuspidal category, is…

Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p>0. Using Poincar\'e series, we globalize supercuspidal representations of H(F) in such a way that we have control over ramification at all…

By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation $\pi$ of…

We prove the local Gross-Prasad conjecture for generic L-packets of representations of special orthogonal groups. The proof uses the same result for tempered L-packets proved in a preceding paper, and irreducibility results for the induced…

These lecture notes give an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory. We start with a motivated introduction to the Langlands Program, including its geometric…

It is expected that, under mild conditions, local Langlands correspondence preserves depths of representations. In this article, we formulate a conjectural geometrisation of this expectation. We prove half of this conjecture by showing that…

Let $\Pi$ be an irreducible unitary completion of a locally algebraic ${\rm GL}_2(\qp)$-representation. We describe those first-order deformations of $\Pi$ which are themselves completions of a locally algebraic representation. This answers…

V. Drinfeld proposed conjectures on geometric Langlands correspondence and its quantum deformation. We refine these conjectures and propose their relationship with algebraic conformal field theory.

Interpolating Langlands Quantum Groups of Rank 1 -- We study a certain family, parameterized by an positive integer g, of double deformations of the envelopping algebra U(sl2), in the spirit of arXiv:0809.4453. We prove that each of these…

Let V be a finite-dimensional superspace and G a simple (or a ``close'' to simple) matrix Lie superalgebra, i.e., a Lie subsuperalgebra in GL(V). Under the classical invariant theory for G we mean the description of G-invariant elements of…

In this article, we consider the links between parabolic induction and the local Langlands correspondence. We enunciate a conjecture about the (enhanced) Langlands parameters of supercuspidal representation of split reductives $p$-adics…

In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and…

Given a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a connected $p$-adic reductive group $G/L$, and a smooth irreducible representation $\pi$ of $G(L)$, Fargues-Scholze recently attached a semisimple Weil parameter to such $\pi$,…

Let G be a connected reductive group. In this paper we are studying the invariant theory of symplectic G-modules. Our main result is that the invariant moment map is equidimensional. We deduce that the categorical quotient is a fibration…

We prove a version of quantum geometric Langlands conjecture in characteristic $p$. Namely, we construct an equivalence of certain localizations of derived categories of twisted crystalline $\mathcal D$-modules on the stack of rank $N$…

Let $K$ be a field and $G$ be a group of its automorphisms endowed with the compact-open topology. There are many situations, where it is natural to study the category $Sm_K(G)$ of smooth (i.e. with open stabilizers) $K$-semilinear…