arXiv subjects
Publications and source records attributed to Colette Moeglin.
We check Sakellaridis-Venkatesh conjectures giving a description of the discrete spectrum of a spherical variety $\mathscr{X}=G/H$ in the Langlands-Arthur formalism when $G$ is a classical real group and $\mathscr{X}$ is a symmetric space. Then, we compute explicitly the representations in the relevant Arthur paquets which appear in the discrete spectrum, and we establish some multiplicity one results.
Soit $π$ un module de plus haut poids unitaire du groupe $G=Sp(2n,\mathbb R)$. On s'intéresse aux paquets d'Arthur contenant $π$. Lorsque le plus haut poids est scalaire, on détermine les paramètres de ces paquets, on établit la propriété de multiplicité un de $π$ dans le paquet, et l'on calcule le caractère $ρ_π$ (du groupe des composantes connexes du centralisateur du paramètre dans le groupe dual) associé à $π$ et qui joue un grand rôle dans la théorie d'Arthur. On fait de même pour certains modules de plus haut poids unitaires unipotents $σ_{n,k}$. Let $π$ be an irreducible unitary highest weight module for $G=Sp(2,\mathbb R)$. We would like to determine the Arthur packets containing $π$. When the highest weight is scalar, we determine the Arthur parameter of these packets, we establish the multiplicity one property of $π$ in the packet and we compute the character $ρ_π$ (of the group of connected components of the centralizer of $ψ$ in the dual group) associated to $π$ which plays an important role in Arthur's theory. We also deal with the case of some unipotent unitary highest weight modules $σ_{n,k}$.
This article is part of a project which consists of investigating Arthur packets for real classical groups. Our goal is to give an explicit description of these packets and to establish the multiplicity one property (which is known to hold for $p$-adic and complex groups). The main result in this paper is a construction of packets from unipotent packets on $c$-Levi factors using cohomological induction. An important tool used in the argument is a statement of commutativity between cohomological induction and spectral endoscopic transfer.
We give an explicit construction of Arthur packets for real unitary groups by cohomological and parabolic induction and following an idea communicated to us by P. Trapa, we show that they satisfy the multiplicity one property. In particular, we show the irreducibility of some parabolically induced representations for unitary groups, and use this to give the proof of analogous statements made in our work on Arthur packets of classical groups. Nous donnons une construction explicite des paquets d'Arthur des groupes unitaires réels par induction cohomologique et induction parabolique et en suivant une idée communiquée par P. Trapa, nous établissons la propriété de multiplicité un de ceux-ci. Nous montrons en particulier des résultats d'irréductibilité de certaines induites paraboliques pour les groupes unitaires, ce qui nous permet de compléter les démonstrations d'énoncés analogues annoncés dans nos travaux sur les paquets d'Arthur des groupes classiques.
Nous étendons aux groupes orthogonaux et unitaires non quasi-déployés sur un corps local des résultats de J. Arthur et de la première auteure établis dans le cas quasi-déployé. En particulier, nous obtenons une classification de Langlands complète pour les représentations tempérées dans le cas $p$-adique. Nous en déduisons en utilisant l'involution d'Aubert-Schneider-Stuhler un résultat de multiplicité un dans les paquets unipotents, et par des méthodes globales, le même résultat pour les paquets unipotents dans le cas archimédien. We extend to non quasi-split orthogonal and unitary groups over a local field some results of J. Arthur and the first author established in the quasi-split case. In particular, we obtain a full Langlands classification for tempered representations in the $p$-adic case. Using Aubert-Schneider-Stuhler involution, we deduce from this a multiplicity one result for unipotent packets, and by global methods, the same result for unipotent packets in the archimedean case.
This article is part of a project which aims to describe as explicitly as possible the Arthur packets of classical real groups and to prove a multiplicity one result for them. Let $G$ be a symplectic or special orthogonal real group, and $ψ: W_{\mathbb R}\times \mathbf{SL}_2(\mathbb C)\rightarrow {}^LG$ be an Arthur parameter for $G$. Let $A(ψ)$ the component group of the centralizer of $ψ$ in $\hat G$. Attached to $ψ$ is a finite length unitary representation $π^A(ψ)$ of $G\times A(ψ)$, which is characterized by the endoscopic identities (ordinary and twisted) it satisfies. In [arXiv:1703.07226] we gave a description of the irreducible components of $π^A(ψ)$ when the parameter $ψ$ is "very regular, with good parity". In the present paper, we use translation of infinitesimal character to describe $π^A(ψ)$ in the general good parity case from the representation $π^A(ψ_+)$ attached to a very regular, with good parity, parameter $ψ_+$ obtained from $ψ$ by a simple shift.
Nous décrivons explicitement les paquets d'Arthur des groupes classiques complexes, ainsi que leur paramétrisation interne par les caractères du groupe des composantes connexes du centralisateur de leur paramètre. Nous montrons d'abord qu'ils sont obtenus par induction parabolique préservant l'irréductibilité à partir des paquets unipotents de "bonne parité". Pour ceux-ci, nous montrons qu'ils coïncident avec les paquets définis par Barbasch-Vogan. Nous utilisons des résultats profonds de Barbasch entrant dans sa classification du dual unitaire de ces groupes. We describe explicitly Arthur packets for complex classical groups, as well as their internal parametrization by the group of characters of the component group of the stabilizer of their parameter. We first show that they are obtained by parabolic induction preserving irreducibility from unipotent packets of "good parity". For these, we show that they coincide with the packets defined by Barbasch and Vogan. We use deep results of Barbasch entering his classification of the unitary dual of these groups.
Let $G=\mathbf{G}(\mathbb{R})$ be the group of real points of a quasi-split connected reductive algebraic group defined over $\mathbb{R}$. Assume furthermore that $G$ is a classical group (symplectic, special orthogonal or unitary). We show that the packets of irreducible unitary cohomological representations defined by Adams and Johnson in 1987 coincide with the ones defined recently by J. Arthur in his work on the classification of the discrete automorphic spectrum of classical groups (C.-P. Mok for unitary groups). For this, we compute the endoscopic transfer of the stable distributions on $G$ supported by these packets to twisted $\mathbf{GL}_N$ in terms of standard modules and show that it coincides with the twisted trace prescribed by Arthur.
We show here that the fundamental lemma for twisted endoscopy, now proved for the unit elements in the spherical Hecke algebras, implies the fundamental lemma for all elements of these Hecke algebras. The proof, whose idea is due to Arthur, uses the transfer, which is known as a consequence of the fundamental lemma for the units.
We prove the Noether-Lefschetz conjecture on the moduli space of quasi-polarized K3 surfaces. This is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles, i.e. sub-arithmetic manifolds of the same type. For compact manifolds this was proved in \cite{BMM11}, here we extend the results of \cite{BMM11} to non-compact manifolds. This allows us to apply our results to the moduli spaces of quasi-polarized K3 surfaces.
We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree $n$ of compact congruence $p$-dimensional hyperbolic manifolds "of simple type" as long as $n$ is strictly smaller than $\frac{p}{3}$. We also prove that for connected Shimura varieties associated to $\OO (p,2)$ the Hodge conjecture is true for classes of degree $< \frac{p+1}{3}$. The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by \cite{ArthurBook}. As such our results are conditional on the hypothesis made in this book, whose proofs have only appear on preprint form so far; see the second paragraph of subsection \ref{org2} below.
It is the last paper of a long series. We present the spectral side of the twisted trace formula and its stable version. Using the method of Arthur, we finish the proof of the stabilization of the two sides of this twisted trace formula.
This paper is one of a series whose goal is to stabilize the twisted Arthur-Selberg's trace formula. Here we define the objects appearing in the geometric side of the twisted trace formula. We define also the similar stable and endoscopic objects. We state the principal theorems concerning the stabilization of this geometric side. The proofs will be given in forthcoming papers.
Let $S$ be a closed Shimura variety uniformized by the complex $n$-ball. The Hodge conjecture predicts that every Hodge class in $H^{2k} (S, \Q)$, $k=0, \ldots, n$, is algebraic. We show that this holds for all degree $k$ away from the neighborhood $]n/3, 2n/3[$ of the middle degree. We also address the Tate conjecture and the generalized form of the Hodge conjecture and extend most of our results to Shimura varieties associated to unitary groups of any signature. The proofs make use of the recent endoscopic classification of automorphic representations of classical groups by \cite{ArthurBook,Mok}. As such our results are conditional on the stabilization of the trace formula for the (disconnected) groups $\GL (N) \rtimes \langle θ\rangle$ associated to base change. Unfortunately, at present the stabilization of the trace formula has been proved only for the case of {\it connected} groups. The extension needed is part of work in progress by the Paris-Marseille team of automorphic form researchers. For more detail, see the second paragraph of subsection \ref{org2} below.
In this paper we gives the Langlands parameters of Langlands' packets of discrete series using the twisted endoscopy as explained by Arthur; this holds for orthogonal, symplectic, unitary and G-Spin groups and gives the most simple proof available. We have assume that the groups are quasi-split but this is just for simplicity. The proof explaines first what is the classification from the representation's theory point of view; this gives the Langlands' packets purely in terms of representation theory. And then using the theory of L-function of Shahidi and the doubling method of Rallis and Piatetskii-Shapiro, we translate this result in term of the $L$-group. Only the first part differs at some places of Arthur's point of view and gives more results about reducibility points of induced representations. We hope that this paper will make very clear how fruitful is the doubling method.
Let $π$ be a square integrable representation of a classical group and let $ρ$ be a cuspidal representation of a general linear group. We can define in two different ways an L-function $L(ρ\times π,s)$: first we can use the Langlands parametrization at each places which is now available, thanks to Arthur's work, and secondly we can transfer $π$ to a general linear group, using the twisted endoscopy as established by Arthur. In this paper, we compare the two definitions and we prove, as expected, that the first one has less poles that the second one. Assuming that $π$ is cuspidal, we link the poles of the first L-function to the poles of the Eisensteins series and when $ρ$ is a quadratic character and when the groupe is a special orthogonal group, we also link theses poles with the theta lifts. We have some hypothesis at the archimedean places.
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 representations of whose the elements of the L-packets are Langlands quotients.
We have shown in a preceeding paper how to normalize intertwining operators for classical groups using the twisted endoscopy lifting. In this paper, we prove that the image of such an operator in the cases interesting in the theory of Eisenstein Series, is either 0 or an irreducible representation. As a consequence we compute explicitly the points where Eisenstein Series for square integrable representations are not holomorphic under some hypothesis at the archimedean places: at that places we mainly assume that the infinitesimal character is integral and regular.