Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm {GL}_2$. Here, we use ‘hyperendoscopy’ techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$, eventually getting an analog of the Eichler–Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard and Taïbi to compute dimensions of spaces of level-$1$ quaternionic representations. On the way, we prove a Jacquet–Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form $G_2^c$.

The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being ‘regular’. A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series.

We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.

We consider orthogonally invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {C})$. Astérisque 287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

We construct pairs of residually finite groups with isomorphic profinite completions such that one has non-vanishing and the other has vanishing real second bounded cohomology. The examples are lattices in different higher-rank simple Lie groups. Using Galois cohomology, we actually show that $\operatorname {SO}^0(n,2)$ for $n \ge 6$ and the exceptional groups $E_{6(-14)}$ and $E_{7(-25)}$ constitute the complete list of higher-rank Lie groups admitting such examples.

We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archimedean local field of characteristic zero.

Let $G= N\rtimes H$ be a locally compact group which is a semi-direct product of a closed normal subgroup N and a closed subgroup H. The Bohr compactification ${\rm Bohr}(G)$ and the profinite completion ${\rm Prof}(G)$ of G are, respectively, isomorphic to semi-direct products $Q_1 \rtimes {\rm Bohr}(H)$ and $Q_2 \rtimes {\rm Prof}(H)$ for appropriate quotients $Q_1$ of ${\rm Bohr}(N)$ and $Q_2$ of ${\rm Prof}(N).$ We give a precise description of $Q_1$ and $Q_2$ in terms of the action of H on appropriate subsets of the dual space of N. In the case where N is abelian, we have ${\rm Bohr}(G)\cong A \rtimes {\rm Bohr}(H)$ and ${\rm Prof}(G)\cong B \rtimes {\rm Prof}(H),$ where A (respectively B) is the dual group of the group of unitary characters of N with finite H-orbits (respectively with finite image). Necessary and sufficient conditions are deduced for G to be maximally almost periodic or residually finite. We apply the results to the case where $G= \Lambda\wr H$ is a wreath product of discrete groups; we show in particular that, in case H is infinite, ${\rm Bohr}(\Lambda\wr H)$ is isomorphic to ${\rm Bohr}(\Lambda^{\rm Ab}\wr H)$ and ${\rm Prof}(\Lambda\wr H)$ is isomorphic to ${\rm Prof}(\Lambda^{\rm Ab} \wr H),$ where $\Lambda^{\rm Ab}=\Lambda/ [\Lambda, \Lambda]$ is the abelianisation of $\Lambda.$ As examples, we compute ${\rm Bohr}(G)$ and ${\rm Prof}(G)$ when G is a lamplighter group and when G is the Heisenberg group over a unital commutative ring.

We study the Eisenstein series associated to the full rank cusps in a complete hyperbolic manifold. We show that given a Kleinian group $\Gamma <{\operatorname{\mathrm{Isom}}}^+(\mathbb H^{n+1})$, each full rank cusp corresponds to a cohomology class in $H^{n}(\Gamma , V)$, where V is either the trivial coefficient or the adjoint representation. Moreover, by computing the intertwining operator, we show that different cusps give rise to linearly independent classes.

In a series of three earlier papers, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction problems were formulated in terms of a pair $W \subset V$ of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces. In this paper, we consider a twisted variant of these conjectures in one particular case: that of a pair of skew-Hermitian spaces $W = V$.

A. Mark and J. Paupert [Presentations for cusped arithmetic hyperbolic lattices, 2018, arXiv:1709.06691.] presented a method to compute a presentation for any cusped complex hyperbolic lattice. In this note, we will use their method to give a presentation for the Eisenstein-Picard modular group in three complex dimensions.

We prove Wiener Tauberian theorem type results for various spaces of radial functions, which are Banach algebras on a real-rank-one semisimple Lie group G. These are natural generalizations of the Wiener Tauberian theorem for the commutative Banach algebra of the integrable radial functions on G.

In this note, assuming the nonvanishing result of explicit theta correspondence for the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$, we show that, for metapletic–orthogonal dual pair over $\mathbb {R}$ and the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$, the theta correspondence is compatible with tempered condition by directly estimating the matrix coefficients, without using the classification theorem.

Sarnak’s density conjecture is an explicit bound on the multiplicities of nontempered representations in a sequence of cocompact congruence arithmetic lattices in a semisimple Lie group, which is motivated by the work of Sarnak and Xue ([58]). The goal of this work is to discuss similar hypotheses, their interrelation and their applications. We mainly focus on two properties – the spectral spherical density hypothesis and the geometric Weak injective radius property. Our results are strongest in the p-adic case, where we show that the two properties are equivalent, and both imply Sarnak’s general density hypothesis. One possible application is that either the spherical density hypothesis or the Weak injective radius property imply Sarnak’s optimal lifting property ([57]). Conjecturally, all those properties should hold in great generality. We hope that this work will motivate their proofs in new cases.

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0< p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher.

We state a conjecture that relates the derived category of smooth representations of a $p$-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of ${\rm GL}_n$ by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and of Emerton and Helm.

We construct an action of the affine Hecke category on the principal block $\mathrm {Rep}_0(G_1T)$ of $G_1T$-modules where G is a connected reductive group over an algebraically closed field of characteristic $p> 0$, T a maximal torus of G and $G_1$ the Frobenius kernel of G. To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.

We study the arithmeticity of $\mathbb {C}$-Fuchsian subgroups of some nonarithmetic lattices constructed by Deraux et al. [‘New non-arithmetic complex hyperbolic lattices’, Invent. Math. 203 (2016), 681–771]. Our results give an answer to a question raised by Wells [Hybrid Subgroups of Complex Hyperbolic Isometries, Doctoral thesis, Arizona State University, 2019].

Let $\alpha $ be a $C^{\infty }$ volume-preserving action on a closed n-manifold M by a lattice $\Gamma $ in $\mathrm {SL}(n,\mathbb {R})$, $n\ge 3$. Assume that there is an element $\gamma \in \Gamma $ such that $\alpha (\gamma )$ admits a dominated splitting. We prove that the manifold M is diffeomorphic to the torus ${{\mathbb T}^{n}={\mathbb R}^{n}/{\mathbb Z}^{n}}$ and $\alpha $ is smoothly conjugate to an affine action. Anosov diffeomorphisms and partial hyperbolic diffeomorphisms admit a dominated splitting. We obtained a topological global rigidity when $\alpha $ is $C^{1}$. We also prove similar theorems for actions on $2n$-manifolds by lattices in $\textrm {Sp}(2n,{\mathbb R})$ with $n\ge 2$ and $\mathrm {SO}(n,n)$ with $n\ge 5$.

We give cases in which nearby cycles commute with pushforward from sheaves on the moduli stack of shtukas to a product of curves over a finite field. The proof systematically uses the property that taking nearby cycles of Satake sheaves on the Beilinson–Drinfeld Grassmannian with parahoric reduction is a central functor together with a ‘Zorro's lemma’ argument similar to that of Xue [Smoothness of cohomology sheaves of stacks of shtukas, Preprint (2020), arXiv:2012.12833]. As an application, for automorphic forms at the parahoric level, we characterize the image of tame inertia under the Langlands correspondence in terms of two-sided cells.

We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and Bowen–Margulis–Sullivan densities of geodesic planes, with degree given by the modified critical exponents.

We establish (some directions of) a Ledrappier correspondence between Hölder cocycles, Patterson–Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson–Sullivan measures for $\vartheta $-Anosov representations over a local field and show that these are parameterized by the $\vartheta $-critical hypersurface of the representation. We use these Patterson–Sullivan measures to establish a dichotomy concerning directions in the interior of the $\vartheta $-limit cone of the representation in question: if ${\mathsf {u}}$ is such a half-line, then the subset of ${\mathsf {u}}$-conical limit points has either total mass if $|\vartheta |\leq 2$ or zero mass if $|\vartheta |\geq 4.$ The case $|\vartheta |=3$ remains unsettled.

In this paper, we give an explicit computable algorithm for the Zelevinsky–Aubert duals of irreducible representations of $p$-adic symplectic and odd special orthogonal groups. To do this, we establish explicit formulas for certain derivatives and socles. We also give a combinatorial criterion for the irreducibility of certain parabolically induced representations.