Related papers: Spectral decomposition of genuine cusp forms over …

We discuss recent developments in the Langlands program for function fields, and in the geometric Langlands program. In particular we explain a canonical decomposition of the space of cuspidal automorphic forms for any reductive group G…

In \cite{lafforgue2012chtoucas}, Vicent Lafforgue attaches a semisimple Langlands parameter (or, what amounts to the same thing, a $\hat{G}$-pseudocharacter) to every cuspidal automorphic representation of a reductive group $G$ over the…

We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include…

For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the "automorphic to Galois"…

This is a translation in English of version 5 of the article arXiv:1404.3998, which is itself an introduction to arXiv:1209.5352. We explain all the ideas of the proof of the following theorem. For any reductive group G over a global…

We show that a compact representation of a semisimple Lie group has an orthogonal decomposition into finite length representations. This generalises and simplifies a number of more special spectral theorems in the literature. We apply it to…

In this paper we give a geometric version of the Satake isomorphism. Given a connected complex reductive algebraic group, we show that the category of representations of its Langlands dual is naturally equivalent to a certain category of…

Let $k$ be an algebraically closed field of characteristic $p$. Denote by $W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified finite extension of $W(k)[1/p]$ and $\mathcal{O}$ the its ring of integers. For a connected…

In this paper we extend the twisted Satake equivalence established in arXiv:0809.3738 for almost simple groups to the case of split reductive groups.

We show that the cotangent bundle $T^*(G/K)$ of a quasi-split symmetric space $G/K$ is isomorphic to the dual variety of the loop symmetric space for the Langlands dual group, providing instances of the relative Langlands duality for…

We elaborate an explicit version of the relative trace formula on $\PGL(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral…

In one article, the author has defined an L-group associated to a cover of a quasisplit reductive group over a local or global field. In another article, Wee Teck Gan and Fan Gao define (following an unpublished letter of the author) an…

We introduce the space of parameters for the metaplectic Langlands theory as *factorization gerbes* on the affine Grassmannian, and develop metaplectic Langlands duality in the incarnation of the metaplectic geometric Satake functor. We…

I give another proof of the geometric Satake equivalence from Mirkovic-Vilonen over a separably closed field. Using Galois descent, I obtain a canonical construction of the Galois form of the full $L$-group.

Suppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We…

These notes present an application of the geometric Satake equivalence to the description of characters of indecomposable tilting modules for reductive algebraic groups over fields of positive characteristic, obtained in joint work with G.…

A class of conformal deformations of Rindler-like spaces is analyzed. We study the spectral properties of the Laplace operators associated with $p-$forms and acting in these spaces and in their spatial sections. The spectral density of…

We construct the geometric Satake equivalence for quasi-split reductive groups over nonarchimedean local fields, using \'etale Artin-Tate motives with $\mathbb{Z}[\frac{1}{p}]$-coefficients. We consider local fields of both equal and mixed…

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…

Let H be any reductive p-adic group. We introduce a notion of cuspidality for enhanced Langlands parameters for H, which conjecturally puts supercuspidal H-representations in bijection with such L-parameters. We also define a cuspidal…