La formule des traces relative de Jacquet–Rallis (pour les groupes unitaires ou les groupes linéaires généraux) est une identité entre des périodes des représentations automorphes et des distributions géométriques. Selon Jacquet et Rallis, une comparaison de ces deux formules des traces relatives devrait aboutir à une démonstration des conjectures de Gan–Gross–Prasad et Ichino–Ikeda pour les groupes unitaires. Les termes géométriques des groupes unitaires ou des groupes linéaires sont indexés par les points rationnels d'un espace quotient commun. Nous établissons que ces termes géométriques peuvent être vus comme des fonctionnelles sur des espaces d'intégrales orbitales semi-simples régulières locales. En outre, nous montrons que point par point ces distributions sont en fait égales, via l'identification des espaces d'intégrales orbitales locales donnée par le transfert et le lemme fondamental (essentiellement connus dans cette situation). Cela donne leur comparaison et cela clôt la partie géométrique du programme de Jacquet–Rallis. Notre résultat principal est donc un analogue de la stabilisation de la partie géométrique de la formule des traces due à Langlands, Kottwitz et Arthur.

Suppose G is an amenable locally compact group with lattice subgroup $\Gamma $ . Grosvenor [‘A relation between invariant means on Lie groups and invariant means on their discrete subgroups’, Trans. Amer. Math. Soc.288(2) (1985), 813–825] showed that there is a natural affine injection $\iota : {\text {LIM}}(\Gamma )\to {\text {TLIM}}(G)$ and that $\iota $ is a surjection essentially in the case $G={\mathbb R}^d$ , $\Gamma ={\mathbb Z}^d$ . In the present paper it is shown that $\iota $ is a surjection if and only if $G/\Gamma $ is compact.

Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.

Given a G-flow X, let $\mathrm{Aut}(G, X)$ , or simply $\mathrm{Aut}(X)$ , denote the group of homeomorphisms of X which commute with the G action. We show that for any pair of countable groups G and H with G infinite, there is a minimal, free, Cantor G-flow X so that H embeds into $\mathrm{Aut}(X)$ . This generalizes results of [2, 7].

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, g is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

Let ${\mathbf {G}}$ be a semisimple algebraic group over a number field K, $\mathcal {S}$ a finite set of places of K, $K_{\mathcal {S}}$ the direct product of the completions $K_{v}, v \in \mathcal {S}$ , and ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map. We describe the closures of the locally divergent orbits ${T\pi (g)}$ where T is a maximal $K_{\mathcal {S}}$ -split torus in G. If $\# S = 2$ then the closure $ \overline{T\pi (g)}$ is a finite union of T-orbits stratified in terms of parabolic subgroups of ${\mathbf {G}} \times {\mathbf {G}}$ and, consequently, $\overline{T\pi (g)}$ is homogeneous (i.e. $\overline{T\pi (g)}= H\pi (g)$ for a subgroup H of G) if and only if ${T\pi (g)}$ is closed. On the other hand, if $\# \mathcal {S}> 2$ and K is not a $\mathrm {CM}$ -field then $\overline {T\pi (g)}$ is homogeneous for ${\mathbf {G}} = \mathbf {SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ${\mathbf {G}} \neq \mathbf {SL}_{n}$ . As an application, we prove that $\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$ for the class of non-rational locally K-decomposable homogeneous forms $f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$ .

Let G be a semisimple real algebraic group defined over ${\mathbb {Q}}$ , $\Gamma $ be an arithmetic subgroup of G, and T be a maximal ${\mathbb {R}}$ -split torus. A trajectory in $G/\Gamma $ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which accounts for the divergence. If this is the case, the divergent trajectory is called obvious. Given a closed cone in T, we study the existence of non-obvious divergent trajectories under its action in $G\kern-1pt{/}\kern-1pt\Gamma $ . We get a sufficient condition for the existence of a non-obvious divergence trajectory in the general case, and a full classification under the assumption that $\mathrm {rank}_{{\mathbb {Q}}}G=\mathrm {rank}_{{\mathbb {R}}}G=2$ .

Let $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity.

First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type.

Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

We consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric Pöschl–Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fernández provides large classes of Pöschl-Teller partner potentials to which our analysis applies.

A subset X of a Polish group G is Haar null if there exists a Borel probability measure μ and a Borel set B containing X such that μ(gBh) = 0 for every g, h ∈ G. A set X is Haar meager if there exists a compact metric space K, a continuous function f : K → G and a Borel set B containing X such that f−1(gBh) is meager in K for every g, h ∈ G. We calculate (in ZFC) the four cardinal invariants (add, cov, non, cof) of these two σ-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb {Z}^\omega$. In fact, most results work for separable Banach spaces as well, and many results work for Polish groups admitting a two-sided invariant metric. This answers a question of the first named author and Vidnyánszky.

We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $\operatorname {SL}(2, \mathbb {R})$ -action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $\operatorname {SL}(2, \mathbb {R})$ -action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the ‘weak convergence’ of push-forwards of horocycle measures under the geodesic flow and a short proof of weaker versions of theorems of Chaika and Eskin on Birkhoff genericity and Oseledets regularity in almost all directions for the Teichmüller geodesic flow.

A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute towards Haar measure, often exponentially fast and locally uniformly in the starting position. In addition, many classical limit theorems are shown to hold. In the infinite volume case, we prove recurrence and a ratio limit theorem for symmetric spread out random walks on homogeneous spaces of at most quadratic growth. This settles one direction in a long-standing conjecture.

We describe the connected components of the space $\text {Hom}(\Gamma ,SU(2))$ of homomorphisms for a discrete nilpotent group $\Gamma$. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $\mathbb {RP}^{3}$. We give explicit calculations when $\Gamma$ is a finitely generated free nilpotent group. In the second part of the paper, we study the filtration $B_{\text {com}} SU(2)=B(2,SU(2))\subset \cdots \subset B(q,SU(2))\subset \cdots$ of the classifying space $BSU(2)$ (introduced by Adem, Cohen and Torres-Giese), showing that for every $q\geq 2$, the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for $SO(3)$ and $U(2)$ as well.

The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group $\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.

In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $\pi $ be a minimal or next-to-minimal automorphic representation of G. We prove that any $\eta \in \pi $ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $\operatorname {GL}_n$ . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $D_5$ and $E_8$ with a view toward applications to scattering amplitudes in string theory.

We show local rigidity of hyperbolic triangle groups generated by reflections in pairs of n-dimensional subspaces of $\mathbb {R}^{2n}$ obtained by composition of the geometric representation in $\mathsf {PGL}(2,\mathbb {R})$ with the diagonal embeddings into $\mathsf {PGL}(2n,\mathbb {R})$ and $\mathsf {PSp}^\pm (2n,\mathbb {R})$ .

We generalize Uhlenbeck’s generator theorem of ${\mathcal{L}}^{-}\operatorname{U}_{n}$ to the full rational loop group ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{C}$ and its subgroups ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{R}$, ${\mathcal{L}}^{-}\operatorname{U}_{p,q}$: they are all generated by just simple projective loops. Recall that Terng–Uhlenbeck studied the dressing actions of such projective loops as generalized Bäcklund transformations for integrable systems. Our result makes a nice supplement: any rational dressing is the composition of these Bäcklund transformations. This conclusion is surprising in the sense that Lie theory suggests the indispensable role of nilpotent loops in the case of noncompact reality conditions, and nilpotent dressings appear quite complicated and mysterious. The sacrifice is to introduce some extra fake singularities. So we also propose a set of generators if fake singularities are forbidden. A very geometric and physical construction of $\operatorname{U}_{p,q}$ is obtained as a by-product, generalizing the classical construction of unitary groups.