We establish the geometric Langlands correspondence for rank-one groups over the projective line with three points of tame ramification.

In this paper, we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment.

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form

$$Au(x)=\int_{{\open R}^n}\int_{{\open R}^n}e^{{\rm i}(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)\,{\rm d}y\,{\rm d}\xi,$$

$\tau :{\open R}^n\to {\open R}^n$

$\tau (x)=0$

$\tau (x)=x$

$\tau (x)={x}/{2}$

${\open R}^n$

A space X is said to be Lipschitz 1-connected if every Lipschitz loop 𝛾 : S1 → X bounds a O (Lip(𝛾))-Lipschitz disk f : D2 → X. A Lipschitz 1-connected space admits a quadratic isoperimetric inequality, but it is unknown whether the converse is true. Cornulier and Tessera showed that certain solvable Lie groups have quadratic isoperimetric inequalities, and we extend their result to show that these groups are Lipschitz 1-connected.

We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

In this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.

We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_{1}\subseteq G_{2}\subseteq \cdots \,$ of topological groups $G_{n}$ n such that $G_{n}$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on $G_{n}$, for each $n\in \mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each $G_{n}$ whenever $\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$ is an identity neighbourhood in $G$ for all identity neighbourhoods $V_{n}\subseteq G_{n}$. If, moreover, each $G_{n}$ is complete, then $G$ is complete. We also show that the weak direct product $\oplus _{j\in J}G_{j}$ is complete for each family $(G_{j})_{j\in J}$ of complete Lie groups $G_{j}$. As a consequence, every strict direct limit $G=\cup _{n\in \mathbb{N}}G_{n}$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\text{Diff}_{c}(M)$ of a paracompact finite-dimensional smooth manifold $M$ and the test function group $C_{c}^{k}(M,H)$, for each $k\in \mathbb{N}_{0}\cup \{\infty \}$ and complete Lie group $H$ modelled on a complete locally convex space.

A problem in representation theory of $p$-adic groups is the computation of the Casselman basis of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials that are deformations of the Kazhdan–Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{-1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.

We investigate continuous transitive actions of semitopological groups on spaces, as well as separately continuous transitive actions of topological groups.

In this paper, we study the warped structures of Finsler metrics. We obtain the differential equation that characterizes Finsler warped product metrics with vanishing Douglas curvature. By solving this equation, we obtain all Finsler warped product Douglas metrics. Some new Douglas Finsler metrics of this type are produced by using known spherically symmetric Douglas metrics.

We consider the following question: for which metrizable separable spaces $X$ does the free abelian topological group $A(X\oplus X)$ isomorphically embed into $A(X)$. While for many natural spaces $X$ such an embedding exists, our main result shows that if $X$ is a Cook continuum or $X$ is a rigid Bernstein set, then $A(X\oplus X)$ does not embed into $A(X)$ as a topological subgroup. The analogous statement is true for the free boolean group $B(X)$.

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of $t$-designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal $t$-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

We consider generalised metrisability and cardinal invariants in quasitopological groups. We construct examples to show that some equalities of cardinal invariants in topological groups cannot be extended to quasitopological groups.

We combine the ideas of a Harish-Chandra–Howe local character expansion, which can be centred at an arbitrary semisimple element, and a Kim–Murnaghan asymptotic expansion, which so far has been considered only around the identity. We show that, for most smooth, irreducible representations (those containing a good, minimal K-type), Kim–Murnaghan-type asymptotic expansions are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple elements. We then give an explicit, inductive recipe for computing the coefficients in an asymptotic expansion for a tame supercuspidal representation. The only additional information needed in the inductive step is a fourth root of unity, which we expect to be useful in proving stability and endoscopic-transfer identities.

Let $X$ be a solenoid, i.e. a compact, finite-dimensional, connected abelian group with normalized Haar measure $\unicode[STIX]{x1D707}$, and let $\unicode[STIX]{x1D6E4}\rightarrow \operatorname{Aff}(X)$ be an action of a countable discrete group $\unicode[STIX]{x1D6E4}$ by continuous affine transformations of $X$. We show that the probability measure preserving action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ does not have the spectral gap property if and only if there exists a $p_{\text{a}}(\unicode[STIX]{x1D6E4})$-invariant proper subsolenoid $Y$ of $X$ such that the image of $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually solvable group, where $p_{\text{a}}:\operatorname{Aff}(X)\rightarrow \operatorname{Aut}(X)$ is the canonical projection. When $\unicode[STIX]{x1D6E4}$ is finitely generated or when $X$ is the $a$-adic solenoid for an integer $a\geq 1$, the subsolenoid $Y$ can be chosen so that the image $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually abelian group. In particular, an action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ by affine transformations on a solenoid $X$ has the spectral gap property if and only if $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is strongly ergodic.

If $G\ncong \operatorname{Alt}(\mathbb{N})$ is an inductive limit of finite alternating groups, then the indecomposable characters of $G$ are precisely the associated characters of the ergodic invariant random subgroups of $G$.

We prove an inverse theorem for the Gowers $U^{2}$-norm for maps $G\rightarrow {\mathcal{M}}$ from a countable, discrete, amenable group $G$ into a von Neumann algebra ${\mathcal{M}}$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert \cdot \Vert$. We use this result to prove a stability result for unitary-valued $\unicode[STIX]{x1D700}$-representations $G\rightarrow {\mathcal{U}}({\mathcal{M}})$ with respect to $\Vert \cdot \Vert$.

Wreath products of nondiscrete locally compact groups are usually not locally compact groups, nor even topological groups. As a substitute introduce a natural extension of the wreath product construction to the setting of locally compact groups. Applying this construction, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without any open compact normal subgroup.

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form $\mathbb{R}^{n}\times K$, where $K$ is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group $G$ which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form $G\times D$, where $D$ is a discrete group.