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.

In this paper, it is shown that every compact Hausdorff $K$-space has countable tightness. This result gives a positive answer to a problem posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl.104 (2000), 181–190]. We show that a semitopological group $G$ that is a $K$-space is first countable if and only if $G$ is of point-countable type. It is proved that if a topological group $G$ is a $K$-space and has a locally paracompact remainder in some Hausdorff compactification, then $G$ is metrisable.

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 prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group.

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.

We extend our previous work in collaboration with Ngô Bao Châu and give a fixed point formula for the elliptic part of moduli spaces of $G$-shtukas with arbitrary modifications. Our formula is similar to the fixed point formula of Kottwitz for certain Shimura varieties. Our method is inspired by that of Kottwitz and simpler than that of Lafforgue for the fixed point formula of the moduli space of Drinfeld $\text{GL} (r)$-shtukas.

This paper deals with the Schrödinger equation $i{\partial }_{s} u(\mathbf{z} , t; s)- \mathcal{L} u(\mathbf{z} , t; s)= 0, $ where $ \mathcal{L} $ is the sub-Laplacian on the Heisenberg group. Assume that the initial data $f$ satisfies $\vert f(\mathbf{z} , t)\vert \lesssim {q}_{\alpha } (\mathbf{z} , t), $ where ${q}_{s} $ is the heat kernel associated to $ \mathcal{L} . $ If in addition $\vert u(\mathbf{z} , t; {s}_{0} )\vert \lesssim {q}_{\beta } (\mathbf{z} , t), $ for some ${s}_{0} \in \mathbb{R} \setminus \{ 0\} , $ then we prove that $u(\mathbf{z} , t; s)= 0$ for all $s\in \mathbb{R} $ whenever $\alpha \beta \lt { s}_{0}^{2} . $ This result holds true in the more general context of $H$-type groups. We also prove an analogous result for the Grushin operator on ${ \mathbb{R} }^{n+ 1} . $

Let $G$ be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces to the invariant vectors coming from webs. In the case of $G= \mathrm{SL} (3)$, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is $\mathrm{CAT} (0)$, is explained by the fact that affine buildings are $\mathrm{CAT} (0)$.

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture. We prove this conjecture in the minuscule case.

We study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.

We construct dense Borel measurable subgroups of Lie groups of intermediate Hausdorff dimension. In particular, we generalize the Erdős–Volkmann construction [Additive Gruppen mit vorgegebener Hausdorffscher Dimension, J. Reine Angew. Math. 221 (1966), 203–208], showing that any nilpotent $\sigma $-compact Lie group $N$ admits dense Borel subgroups of arbitrary dimension between zero and $\dim N$. In algebraic groups defined over a finite extension of the rationals, using diophantine properties of algebraic numbers, we are also able to construct dense subgroups of arbitrary dimension, but the general case remains open. In particular, we raise the following question: does there exist a measurable proper subgroup of $ \mathbb{R} $ of positive Hausdorff dimension which is stable under multiplication by a transcendental number? Subgroups of nilpotent $p$-adic analytic groups are also discussed.

Let $G$ be a connected, reductive algebraic group over a number field $F$ and let $E$ be an algebraic representation of ${G}_{\infty } $. In this paper we describe the Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ of $G$ below a certain degree ${q}_{ \mathsf{res} } $ in terms of Franke’s filtration of the space of automorphic forms. This entails a description of the map ${H}^{q} ({\mathfrak{m}}_{G} , K, \Pi \otimes E)\rightarrow { H}_{\mathrm{Eis} }^{q} (G, E)$, $q\lt {q}_{ \mathsf{res} } $, for all automorphic representations $\Pi $ of $G( \mathbb{A} )$ appearing in the residual spectrum. Moreover, we show that below an easily computable degree ${q}_{ \mathsf{max} } $, the space of Eisenstein cohomology ${ H}_{\mathrm{Eis} }^{q} (G, E)$ is isomorphic to the cohomology of the space of square-integrable, residual automorphic forms. We discuss some more consequences of our result and apply it, in order to derive a result on the residual Eisenstein cohomology of inner forms of ${\mathrm{GL} }_{n} $ and the split classical groups of type ${B}_{n} $, ${C}_{n} $, ${D}_{n} $.

In this paper, we consider the $\mathrm{SL} (2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\mathrm{GL} (2)$: one due to Harder–Langlands–Rapoport about non-vanishing of period integrals on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$ of cuspidal automorphic representations on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{E} )$ where $E$ is a quadratic extension of a number field $F$, and the other due to Waldspurger involving toric periods of automorphic forms on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$. In both these cases, now involving $\mathrm{SL} (2)$, we analyze period integrals on global$L$-packets; we prove that under certain conditions, a global automorphic $L$-packet which at each place of a number field has a distinguished representation, contains globally distinguished representations, and further, an automorphic representation which is locally distinguished is globally distinguished.