This paper deals with the geometric local theta correspondence at the Iwahori level for dual reductive pairs of type II over a non-Archimedean field $F$ of characteristic $p\neq 2$ in the framework of the geometric Langlands program. First we construct and study the geometric version of the invariants of the Weil representation of the Iwahori-Hecke algebras. In the particular case of $(\mathbf{GL}_{1},\mathbf{GL}_{m})$ we give a complete geometric description of the corresponding category. The second part of the paper deals with geometric local Langlands functoriality at the Iwahori level in a general setting. Given two reductive connected groups $G$ and $H$ over $F$, and a morphism ${\check{G}}\times \text{SL}_{2}\rightarrow \check{H}$ of Langlands dual groups, we construct a bimodule over the affine extended Hecke algebras of $H$ and $G$ that should realize the geometric local Arthur–Langlands functoriality at the Iwahori level. Then, we propose a conjecture describing the geometric local theta correspondence at the Iwahori level constructed in the first part in terms of this bimodule, and we prove our conjecture for pairs $(\mathbf{GL}_{1},\mathbf{GL}_{m})$.

A finitely generated subgroup ${\rm\Gamma}$ of a real Lie group $G$ is said to be Diophantine if there is ${\it\beta}>0$ such that non-trivial elements in the word ball $B_{{\rm\Gamma}}(n)$ centered at $1\in {\rm\Gamma}$ never approach the identity of $G$ closer than $|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group $G$ is said to be Diophantine if for every $k\geqslant 1$ a random $k$-tuple in $G$ generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most $5$, or derived length at most $2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

In this paper, we first deduce a formula of S-curvature of homogeneous Finsler spaces in terms of Killing vector fields. Then we prove that a homogeneous Finsler space has isotropic S-curvature if and only if it has vanishing S-curvature. In the special case that the homogeneous Finsler space is a Randers space, we give an explicit formula which coincides with the previous formula obtained by the second author using other methods.

Suppose that $G$ is a connected reductive algebraic group defined over $\mathbf{R}$, $G(\mathbf{R})$ is its group of real points, ${\it\theta}$ is an automorphism of $G$, and ${\it\omega}$ is a quasicharacter of $G(\mathbf{R})$. Kottwitz and Shelstad defined endoscopic data associated to $(G,{\it\theta},{\it\omega})$, and conjectured a matching of orbital integrals between functions on $G(\mathbf{R})$ and its endoscopic groups. This matching has been proved by Shelstad, and it yields a dual map on stable distributions. We express the values of this dual map on stable tempered characters as a linear combination of twisted characters, under some additional hypotheses on $G$ and ${\it\theta}$.

We show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated ‘Latin square conjecture’ due to Alon and Tarsi or, more precisely, on the validity of an equivalent ‘column Latin square conjecture’ due to Huang and Rota.

Let $(G,G^{\prime })$ be a type I irreducible reductive dual pair in Sp$(W_{\mathbb{R}})$. We assume that $(G,G^{\prime })$ is in the stable range where $G$ is the smaller member. Let $K$ and $K^{\prime }$ be maximal compact subgroups of $G$ and $G^{\prime }$ respectively. Let $\mathfrak{g}=\mathfrak{k}\bigoplus \mathfrak{p}$ and $\mathfrak{g}^{\prime }=\mathfrak{k}^{\prime }\bigoplus \mathfrak{p}^{\prime }$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G^{\prime }$ respectively. Let $\widetilde{K}$ and $\widetilde{K}^{\prime }$ be the inverse images of $K$ and $K^{\prime }$ in the metaplectic double cover $\widetilde{\text{Sp}}(W_{\mathbb{R}})$ of Sp$(W_{\mathbb{R}})$. Let ${\it\rho}$ be a genuine irreducible $(\mathfrak{g},\widetilde{K})$-module. Our first main result is that if ${\it\rho}$ is unitarizable, then except for one special case, the full local theta lift ${\it\rho}^{\prime }={\rm\Theta}({\it\rho})$ is equal to the local theta lift ${\it\theta}({\it\rho})$. Thus excluding the special case, the full theta lift ${\it\rho}^{\prime }$ is an irreducible and unitarizable $(\mathfrak{g}^{\prime },\widetilde{K}^{\prime })$-module. Our second main result is that the associated variety and the associated cycle of ${\it\rho}^{\prime }$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation ${\it\rho}^{\ast }$ respectively. Finally we obtain some interesting $(\mathfrak{g},\widetilde{K})$-modules whose $\widetilde{K}$-spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent $K_{\mathbb{C}}$-orbits in $\mathfrak{p}^{\ast }$.

The Chevalley involution of a connected, reductive algebraic group over an algebraically closed field takes every semisimple element to a conjugate of its inverse, and this involution is unique up to conjugacy. In the case of the reals we prove the existence of a real Chevalley involution, which is defined over $\mathbb{R}$, takes every semisimple element of $G(\mathbb{R})$ to a $G(\mathbb{R})$-conjugate of its inverse, and is unique up to conjugacy by $G(\mathbb{R})$. We derive some consequences, including an analysis of groups for which every irreducible representation is self-dual, and a calculation of the Frobenius Schur indicator for such groups.

Soit $G$ un groupe réductif connexe déployé sur une extension finie $F$ de $\mathbb{Q}_{p}$. Nous déterminons les extensions entre séries principales continues unitaires $p$-adiques et lisses modulo $p$ de $G(F)$ dans le cas générique. Pour cela, nous calculons le delta-foncteur $\text{H}^{\bullet }\text{Ord}_{B(F)}$ des parties ordinaires dérivées d’Emerton relatif à un sous-groupe de Borel sur certaines représentations induites de $G(F)$ en utilisant une filtration de Bruhat. Ces extensions interviennent dans le programme de Langlands $p$-adique et modulo $p$.

Cet article est une contribution à la fois au calcul du nombre de fibrés de Hitchin sur une courbe projective et à l’explicitation de la partie nilpotente de la formule des traces d’Arthur-Selberg pour une fonction test très simple. Le lien entre les deux questions a été établi dans [Chaudouard, Sur le comptage des fibrés de Hitchin. À paraître aux actes de la conférence en l’honneur de Gérard Laumon]. On décompose cette partie nilpotente en une somme d’intégrales adéliques indexées par les orbites nilpotentes. Pour les orbites de type «régulières par blocs», on explicite complètement ces intégrales en termes de la fonction zêta de la courbe.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a connected real semisimple Lie group, $V$ be a finite-dimensional representation of $G$ and $\mu $ be a probability measure on $G$ whose support spans a Zariski-dense subgroup. We prove that the set of ergodic $\mu $-stationary probability measures on the projective space $\mathbb{P}(V)$ is in one-to-one correspondence with the set of compact $G$-orbits in $\mathbb{P}(V)$. When $V$ is strongly irreducible, we prove the existence of limits for the empirical measures. We prove related results over local fields as the finiteness of the set of ergodic $\mu $-stationary measures on the flag variety of $G$.

We prove a simple level-raising result for regular algebraic, conjugate self-dual automorphic forms on $\mathrm{GL}_n$ . This gives a systematic way to construct irreducible Galois representations whose residual representation is reducible.

The superbosonization identity of Littelmann, Sommers and Zirnbauer is a new tool for use in studying universality of random matrix ensembles via supersymmetry, which is applicable to non-Gaussian invariant distributions. We give a new conceptual interpretation of this formula, linking it to harmonic superanalysis of Lie supergroups and symmetric superspaces, and in particular, to a supergeneralization of the Riesz distributions. Using the super-Laplace transformation of generalized superfunctions, the theory of which we develop, we reduce the proof to computing the Gindikin gamma function of a Riemannian symmetric superspace, which we determine explicitly.

We calculate equivariant elliptic cohomology of the partial flag variety $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where $H\subseteq G$ are compact connected Lie groups of equal rank. We identify the ${\rm RO}(G)$-graded coefficients ${\mathcal{E}} ll_G^*$ as powers of Looijenga’s line bundle and prove that transfer along the map

$$\begin{equation*}\pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$

$K$

We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups $\text{GL}(n)$ and $\text{SL}(n)$ we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur’s trace formula obtained in [Finis, Lapid, and Müller, Ann. of Math. (2) 174(1) (2011), 173–195; Finis and Lapid, Ann. of Math. (2) 174(1) (2011), 197–223], which allows us to show that for $\text{GL}(n)$ and $\text{SL}(n)$ the contribution of the continuous spectrum is negligible in the limit.

Let $\mathcal{L}^{S}\left (s,\pi ,{\mathfrak{st}}\right )$ be a partial $\mathcal{L}$ -function of degree $7$ of a cuspidal automorphic representation $\pi $ of the exceptional group $G_2$ . In this paper we construct a Rankin–Selberg integral for representations having a certain Fourier coefficient.

A compact semisimple Lie algebra $\mathfrak{g}$ induces a Poisson structure $\pi _{\mathbb{S}}$ on the unit sphere $\mathbb{S}(\mathfrak{g}^*)$ in $\mathfrak{g}^*$. We compute the moduli space of Poisson structures on $\mathbb{S}(\mathfrak{g}^*)$ around $\pi _{\mathbb{S}}$. This is the first explicit computation of a Poisson moduli space in dimension greater or equal than three around a degenerate (i.e. not symplectic) Poisson structure.

In this article, we study the homomorphisms between scalar generalized Verma modules. We conjecture that any homomorphism between scalar generalized Verma modules is a composition of elementary homomorphisms. The purpose of this article is to confirm the conjecture for some parabolic subalgebras under the assumption that the infinitesimal characters are regular.

In this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

We give a classification of irreducible admissible modulo $p$ representations of a split$p$-adic reductive group in terms of supersingular representations. This is a generalization of a theorem of Herzig.