We study fibrations of the projective model for the symmetric space associated with $\operatorname {\mathrm {SL}}(2n,\mathbb {R})$ by codimension $2$ projective subspaces, or pencils of quadrics. In particular we show that if such a smooth fibration is equivariant with respect to a representation of a closed surface group, the representation is quasi-isometrically embedded, and even Anosov if the pencils in the image contain only nondegenerate quadrics. We use this to characterize maximal representations among representations of a closed surface group into $\operatorname {\mathrm {Sp}}(2n,\mathbb {R})$ by the existence of an equivariant continuous fibration of the associated symmetric space, satisfying an additional technical property. These fibrations extend to fibrations of the projective structures associated to maximal representations by bases of pencils of quadrics.

The set of sums of two squares plays a significant role in number theory. We establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares. The proofs rely on algebraic properties arising from the induced structures on the Stone–Čech compactification of the natural numbers.

In [24, 26] Guichard and Wienhard introduced the notion of $\Theta $-positivity, a generalization of Lusztig’s total positivity to real Lie groups that are not necessarily split.

Based on this notion, we introduce in this paper $\Theta $-positive representations of surface groups. We prove that $\Theta $-positive representations of closed surface groups are $\Theta $-Anosov. This implies that $\Theta $-positive representations are discrete and faithful and that the set of $\Theta $-positive representations is open in the representation variety.

We further establish important properties on limits of $\Theta $-positive representations, proving that the set of $\Theta $-positive representations is closed in the set of representations containing a $\Theta $-proximal element. This is used in [3] to prove the closedness of the set of $\Theta $-positive representations.

If A is in the p-Schatten class on ${\mathbb {R}}^n$, $1\leq p \leq {4n}/{(2n-1)}$, then the quantum translates of A are linearly independent. Moreover, there exists a nonzero operator in the p-Schatten class on ${\mathbb {R}}^n$, $p>{4n}/{(2n-1)}$, whose quantum translates are linearly dependent.

We consider induced representations $\operatorname {\mathrm {Ind}}_{\mathrm {P}(F)}^{\operatorname {\mathrm {G}}(F)} \pi $, where $\mathrm {P}$ is a maximal parabolic subgroup of a reductive group $\operatorname {\mathrm {G}}$ over a p-adic field F, and $(\pi , V)$ is a unitary supercuspidal representation of $\operatorname {\mathrm {M}}(F)$, $\operatorname {\mathrm {M}}$ being some Levi subgroup of $\mathrm {P}$. Imposing a certain ‘Heisenberg parabolic subgroup’ assumption on $\mathrm {P}$, we apply the method of Goldberg, Shahidi and Spallone to obtain an expression for a certain constant $R(\tilde {\pi })$, which captures the residue of a family $s \mapsto A(s, \pi , w_0)$ of intertwining operators associated to this situation, in terms of harmonic analysis on the twisted Levi subgroup $\tilde {\operatorname {\mathrm {M}}}(F) := \operatorname {\mathrm {M}}(F) w_0$. For $\operatorname {\mathrm {G}}$ absolutely almost simple and simply connected of type $G_2$ or $D_4$ (resp., $B_3$), and $\mathrm {P}$ satisfying the ‘Heisenberg’ condition, if the central character of $\pi $ is nontrivial (resp., trivial) on $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}(F)$, where $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}$ is the connected centre of $\operatorname {\mathrm {M}}$, our formula for $R(\tilde {\pi })$ can be rewritten in terms of the Langlands parameter of $\pi $, in the spirit of a prediction of Arthur. For the same collection of $\operatorname {\mathrm {G}}$ and $\mathrm {P}$, when these central character conditions are not satisfied, Arthur’s prediction combined with our formula for $R(\tilde {\pi })$ suggests a harmonic analytic formula for a product of one or two $\gamma $-factors associated to the situation.

The Unitary Dual Problem is one of the most important open problems in mathematics: classify the irreducible unitary representations of a given group. It is known for a real reductive Lie group that $A_{\mathfrak {q}}(\lambda )$ modules are unitary and that any unitarizable Harish-Chandra module of strongly regular infinitesimal character is isomorphic to an $A_{\mathfrak {q}}(\lambda )$. Thus, it is of interest to study representations of singular infinitesimal character. For a compact real form and any alcove of the form $w(-\lambda + \underline {A}_\circ ),$ where $\lambda $ is dominant (possibly singular) and $\underline {A}_\circ $ is the dominant fundamental alcove, the signature character of the canonical invariant Hermitian form on the irreducible Verma module of infinitesimal character in that alcove is the “negative” of a Hall–Littlewood polynomial summand at $q=-1$ times a version of the Weyl denominator. (Signature characters for other real forms and alcoves of other forms may also be expressed using Hall–Littlewood polynomial summands.) Such formulas give hope that the Unitary Dual Problem is tractable in the singular case.

The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^\omega /c(G)$, where $c(G)$ is the group of all convergent sequences in G. Let G be a Polish group. (1) G is a discrete countable group containing at least two elements iff $E(G)\sim _BE_0$; (2) if G is TSI uncountable non-Archimedean, then $E(G)\sim _BE_0^\omega $; (3) G is non-Archimedean iff $E(G)\le _B=^+$; (4) if H is a CLI Polish group but G is not, then $E(G)\not \le _BE(H)$; and (5) if H is a non-Archimedean Polish group but G is not, then $E(G)\not \le _BE(H)$. The notion of $\alpha $-l.m.-unbalanced Polish group for $\alpha <\omega _1$ is introduced. Let G and H be Polish groups, $0<\alpha <\omega _1$. If G is $\alpha $-l.m.-unbalanced but H is not, then $E(G)\not \le _B E(H)$. For TSI Polish groups, the existence of Borel reduction is transformed into the existence of a well-behaved continuous mapping between topological groups. As its applications, for any Polish group G, let $G_0$ be the connected component of the identity element $1_G$. Let G and H be two separable TSI Lie groups. If $E(G)\le _BE(H)$, then there exists a continuous locally injective map $S:G_0\to H_0$. Moreover, if $G_0$ and $H_0$ are abelian, S is a group homomorphism. In particular, for $c_0,e_0,c_1,e_1\in {\mathbb {N}}$, $E({\mathbb {R}}^{c_0}\times {\mathbb {T}}^{e_0})\le _BE({\mathbb {R}}^{c_1}\times {\mathbb {T}}^{e_1})$ iff $e_0\le e_1$ and $c_0+e_0\le c_1+e_1$.

Approximate lattices of Euclidean spaces, also known as Meyer sets, are aperiodic subsets with fascinating properties. In general, approximate lattices are defined as approximate subgroups of locally compact groups that are discrete and have finite co-volume. A theorem of Lagarias [Meyer’s concept of quasicrystal and quasiregular sets. Comm. Math. Phys. 179(2) (1996), 365–376] provides a criterion for discrete subsets of Euclidean spaces to be approximate lattices. It asserts that if a subset X of $\mathbb {R}^n$ is relatively dense and $X - X$ is uniformly discrete, then X is an approximate lattice. We prove two generalizations of Lagarias’ theorem: when the ambient group is amenable and when it is a higher-rank simple algebraic group over a characteristic $0$ local field. This is a natural counterpart to the recent structure results for approximate lattices in non-commutative locally compact groups. We also provide a reformulation in dynamical terms pertaining to return times of cross-sections. Our method relies on counting arguments involving the so-called periodization maps, ergodic theorems and a method of Tao regarding small doubling for finite subsets. In the case of simple algebraic groups over local fields, we moreover make use of deep superrigidity results due to Margulis and to Zimmer.

We introduce the $\ell ^1$-ideal intersection property for crossed product ${\mathrm {C}}^*$-algebras. It is implied by ${\mathrm {C}}^*$-simplicity as well as ${\mathrm {C}}^*$-uniqueness. We show that topological dynamical systems of arbitrary lattices in connected Lie groups, arbitrary linear groups over the integers in a number field and arbitrary virtually polycyclic groups have the $\ell ^1$-ideal intersection property. On the way, we extend previous results on ${\mathrm {C}}^*$-uniqueness of -groupoid algebras to the general twisted setting.

We show that each connected component of the moduli space of smooth real binary quintics is isomorphic to an open subset of an arithmetic quotient of the real hyperbolic plane. Moreover, our main result says that the induced metric on this moduli space extends to a complete real hyperbolic orbifold structure on the space of stable real binary quintics. This turns the moduli space of stable real binary quintics into the quotient of the real hyperbolic plane by an explicit non-arithmetic triangle group.

Let $\operatorname {GSpin}(V)$ (resp., $\operatorname {GPin}(V)$) be a general Spin group (resp., a general Pin group) associated with a nondegenerate quadratic space V of dimension n over an Archimedean local field F. For a nondegenerate quadratic space W of dimension $n-1$ over F, we also consider $\operatorname {GSpin}(W)$ and $\operatorname {GPin}(W)$. We prove the multiplicity-at-most-one theorem in the Archimedean case for a pair of groups ($\operatorname {GSpin}(V), \operatorname {GSpin}(W)$) and also for a pair of groups ($\operatorname {GPin}(V), \operatorname {GPin}(W)$); namely, we prove that the restriction to $\operatorname {GSpin}(W)$ (resp., $\operatorname {GPin}(W)$) of an irreducible Casselman–Wallach representation of $\operatorname {GSpin}(V)$ (resp., $\operatorname {GPin}(V)$) is multiplicity free.

Suppose $G\curvearrowright X$ is a Polish group action, H is a Polish group, and $G\times X\mathop {\overset {\psi }\longrightarrow } H$ is a cocycle that is continuous in the second variable. If $\psi $ is either Baire measurable or is $\lambda \times \mu $-measurable with respect to a Haar measure $\lambda $ on G and a fully supported $\sigma $-finite Borel measure $\mu $ on X, then $\psi $ is jointly continuous.

Let G be a locally compact topological group and $\mathcal {L}(G)$ the space of all its closed subgroups endowed with the Vietoris topology. Let $\mathcal {L}_c(G)$ be the subspace of all compact subgroups of G. Any continuous morphism $\varphi \colon G\to H$ between locally compact groups G and H functorially induces a continuous map $\varphi _*\colon \mathcal {L}_c(G)\to \mathcal {L}_c(H)$ given by $\varphi _*(L)=\varphi (L)$. The main problem addressed in this paper is that of determining the relationship between the openness of $\varphi $ and the openness of $\varphi _*$. For example, we show that if G is locally compact with compact identity component and H is locally compact and totally disconnected, then $\varphi $ is open if and only if $\varphi _*$ is open.

In this paper we study deformations of $C^*$-algebras that are given as cross-sectional $C^*$-algebras of Fell bundles $\mathcal A$ over locally compact groups G. Our deformation comes from a direct deformation of the Fell bundles $\mathcal A$ via certain parameters, such as automorphisms of the Fell bundle, group cocycles, or central group extensions of G by the circle group $\mathbb T$, and then taking cross-sectional algebras of the deformed Fell bundles. We then show that this direct deformation method is equivalent to the deformation via the dual coactions by similar parameters as studied previously in [4, 7].

We study the local theta correspondence for dual pairs of the form $\mathrm {Aut}(C)\times F_{4}$ over a p-adic field, where C is a composition algebra of dimension $2$ or $4$, by restricting the minimal representation of a group of type E. We investigate this restriction through the computation of maximal parabolic Jacquet modules and the Fourier–Jacobi functor.

As a consequence of our results, we prove a multiplicity one result for the $\mathrm {Spin}(9)$-invariant linear functionals of irreducible representations of $F_{4}$ and classify the $\mathrm {Spin}(9)$-distinguished representations.

Let $W_{\mathrm {aff}}$ be an extended affine Weyl group, $\mathbf {H}$ be the corresponding affine Hecke algebra over the ring $\mathbb {C}[\mathbf {q}^{\frac {1}{2}}, \mathbf {q}^{-\frac {1}{2}}]$, and J be Lusztig’s asymptotic Hecke algebra, viewed as a based ring with basis $\{t_w\}$. Viewing J as a subalgebra of the $(\mathbf {q}^{-\frac {1}{2}})$-adic completion of $\mathbf {H}$ via Lusztig’s map $\phi $, we use Harish-Chandra’s Plancherel formula for p-adic groups to show that the coefficient of $T_x$ in $t_w$ is a rational function of $\mathbf {q}$, with denominator depending only on the two-sided cell containing w, and dividing a power of the Poincaré polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information about the failure of the Kazhdan-Lusztig classification at roots of the Poincaré polynomial than is currently known.

Along the way, we show that upon specializing $\mathbf {q}=q>1$, the map from J to the Harish-Chandra Schwartz algebra is injective. As an application of injectivity, we give a novel criterion for an Iwahori-spherical representation to have fixed vectors under a larger parahoric subgroup in terms of its Kazhdan-Lusztig parameter.

In this paper we solve a long-standing problem which goes back to Laurent Schwartz’s work on mean periodic functions. Namely, we completely characterize those locally compact Abelian groups having spectral synthesis. So far a characterization theorem was available for discrete Abelian groups only. Here we use a kind of localization concept for the ideals of the Fourier algebra of the underlying group. We show that localizability of ideals is equivalent to synthesizability. Based on this equivalence we show that if spectral synthesis holds on a locally compact Abelian group, then it holds on each extensions of it by a locally compact Abelian group consisting of compact elements, and also on any extension to a direct sum with a copy of the integers. Then, using Schwartz’s result and Gurevich’s counterexamples, we apply the structure theory of locally compact Abelian groups to obtain our characterization theorem.

We show how finiteness properties of a group and a subgroup transfer to finiteness properties of the Schlichting completion relative to this subgroup.n Further, we provide a criterion when the dense embedding of a discrete group into the Schlichting completion relative to one of its subgroups induces an isomorphism in (continuous) cohomology. As an application, we show that the continuous cohomology of the Neretin group vanishes in all positive degrees.

The notion of strong 1-boundedness for finite von Neumann algebras was introduced in [Jun07b]. This framework provided a free probabilistic approach to study rigidity properties and classification of finite von Neumann algebras. In this paper, we prove that tracial von Neumann algebras with a finite Kazhdan set are strongly 1-bounded. This includes all property (T) von Neumann algebras with finite-dimensional center and group von Neumann algebras of property (T) groups. This result generalizes all the previous results in this direction due to Voiculescu, Ge, Ge-Shen, Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung and Shlyakhtenko. Our proofs are based on analysis of covering estimates of microstate spaces using an iteration technique in the spirit of Jung.

Let G be a locally compact, Hausdorff, second countable groupoid and A be a separable, $C_0(G^{(0)})$-nuclear, G-$C^*$-algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from A into a separable, quotient $C^*$-algebra. Along the way, we construct the Busby invariant for G-actions.