We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$, where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

We show the thinness of $7$ of the $40$ hypergeometric groups having a maximally unipotent monodromy in $\mathrm{Sp}(6)$.

We consider self-propelled rigid bodies interacting through local body-attitude alignment modelled by stochastic differential equations. We derive a hydrodynamic model of this system at large spatio-temporal scales and particle numbers in any dimension $n \geq 3$. This goal was already achieved in dimension $n=3$ or in any dimension $n \geq 3$ for a different system involving jump processes. However, the present work corresponds to huge conceptual and technical gaps compared with earlier ones. The key difficulty is to determine an auxiliary but essential object, the generalised collision invariant. We achieve this aim by using the geometrical structure of the rotation group, namely its maximal torus, Cartan subalgebra and Weyl group as well as other concepts of representation theory and Weyl’s integration formula. The resulting hydrodynamic model appears as a hyperbolic system whose coefficients depend on the generalised collision invariant.

Let $E/F$ be a quadratic unramified extension of non-archimedean local fields and $\mathbb H$ a simply connected semisimple algebraic group defined and split over F. We establish general results (multiplicities, test vectors) on ${\mathbb H} (F)$-distinguished Iwahori-spherical representations of ${\mathbb H} (E)$. For discrete series Iwahori-spherical representations of ${\mathbb H} (E)$, we prove a numerical criterion of ${\mathbb H} (F)$-distinction. As an application, we classify the ${\mathbb H} (F)$-distinguished discrete series representations of ${\mathbb H} (E)$ corresponding to degree $1$ characters of the Iwahori-Hecke algebra.

We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$-type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.

We study the Diophantine transference principle over function fields. By adapting the approach of Beresnevich and Velani [‘An inhomogeneous transference principle and Diophantine approximation’, Proc. Lond. Math. Soc. (3) 101 (2010), 821–851] to function fields, we extend many results from homogeneous to inhomogeneous Diophantine approximation. This also yields the inhomogeneous Baker–Sprindžuk conjecture over function fields and upper bounds for the general nonextremal scenario.

Let $ G $ be a connected semisimple real algebraic group and $\Gamma <G$ be a Zariski dense discrete subgroup. Let N denote a maximal horospherical subgroup of G, and $P=MAN$ the minimal parabolic subgroup which is the normalizer of N. Let $\mathcal E$ denote the unique P-minimal subset of $\Gamma \backslash G$ and let $\mathcal E_0$ be a $P^\circ $-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary $ G/P $ and show that the following are equivalent for any $[g]\in \mathcal E_0$:

1. (1) $gP\in G/P$ is a horospherical limit point;

2. (2) $[g]NM$ is dense in $\mathcal E$;

3. (3) $[g]N$ is dense in $\mathcal E_0$.

The equivalence of items (1) and (2) is due to Dal’bo in the rank one case. We also show that unlike convex cocompact groups of rank one Lie groups, the $NM$-minimality of $\mathcal E$ does not hold in a general Anosov homogeneous space.

We give explicit presentations of the integral equivariant cohomology of the affine Grassmannians and flag varieties in type A, arising from their natural embeddings in the corresponding infinite (Sato) Grassmannian and flag variety. These presentations are compared with results obtained by Lam and Shimozono, for rational equivariant cohomology of the affine Grassmannian, and by Larson, for the integral cohomology of the moduli stack of vector bundles on .

Let $\Gamma =\langle I_{1}, I_{2}, I_{3}\rangle $ be the complex hyperbolic $(4,4,\infty )$ triangle group with $I_1I_3I_2I_3$ being unipotent. We show that the limit set of $\Gamma $ is connected and the closure of a countable union of $\mathbb {R}$-circles.

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.