Soit $F$ un corps local non archimédien, et $G$ le groupe des $F$-points d’un groupe réductif connexe quasi-déployé défini sur $F$. Dans cet article, on s’intéresse aux distributions sur $G$ invariantes par conjugaison, et à l’espace de leurs restrictions à l’algèbre de Hecke $\mathcal{H}$ des fonctions sur $G$ à support compact biinvariantes par un sous-groupe d’Iwahori $I$ donné. On montre tout d’abord que les valeurs d’une telle distribution sur $\mathcal{H}$ sont entièrement déterminées par sa restriction au sous-espace de dimension finie des éléments de $\mathcal{H}$ à support dans la réunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propriété pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendré d’une part par certaines intégrales orbitales semi-simples, d’autre part par les intégrales orbitales unipotentes, en montrant tout d’abord des résultats analogues sur les groupes finis.

This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.

The Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a sup-adjoint Gabor frame can be written as the finite union of Riesz basic sequences. Finally, we show how existing techniques can be applied to determine whether frames of translates can be written as the finite union of Riesz basic sequences. We end by giving an example of a frame of translates such that any Riesz basic subsequence must consist of highly irregular translates.

Positive solutions are obtained for the boundary value problem

$$\left\{ _{u\left( 0 \right)\,=\,u\left( 1 \right)\,=\,0}^{-{{\left( {{\left| {{u}'} \right|}^{p-2}}{u}' \right)}^{\prime }}\,=\,\lambda f\left( t,\,u \right),\,t\,\in \,\left( 0,\,1 \right),\,p\,>\,1} \right.$$

Here $f(t,u)\ge -M$, ($M$ is a positive constant) for $(t,u)\in [0,1]\times (0,\infty )$. We will show the existence of two positive solutions by using degree theory together with the upper–lower solution method.

Let $\Gamma \subset SO(3,1)$ be a lattice. The well known bending deformations, introduced by Thurston and Apanasov, can be used to construct non-trivial curves of representations of $\Gamma $ into $SO(4,1)$ when $\Gamma \backslash {{\mathbb{H}}^{3}}$ contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in ${{\text{H}}^{1}}\left( \Gamma ,\mathbb{R}_{1}^{4} \right)$. Our main result generalizes this construction of cohomology to the context of “branched” totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in ${{S}^{3}}$ which is not infinitesimally rigid in $SO(4,1)$. The first order deformations of this link complement are supported on a piecewise totally geodesic 2-complex.

We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study them in this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.

We prove convergence of diagonal multipoint Padé approximants of Stieltjes-type functions when a certain moment problem is determinate. This is used for the study of the convergence of Fourier–Padé and nonlinear Fourier–Padé approximants for such type of functions.

This is a note on the classical Waring's problem for algebraic forms. Fix integers $(n,d,r,s)$, and let $\Lambda $ be a general $r$-dimensional subspace of degree $d$ homogeneous polynomials in $n+1$ variables. Let $\mathcal{A}$ denote the variety of $s$-sided polar polyhedra of $\Lambda $. We carry out a case-by-case study of the structure of $\mathcal{A}$ for several specific values of $(n,d,r,s)$. In the first batch of examples, $\mathcal{A}$ is shown to be a rational variety. In the second batch, $\mathcal{A}$ is a finite set of which we calculate the cardinality.

On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup ${{\left( {{\mu }_{t}} \right)}_{t>0}}$. Using appropriate notions of distribution and smooth function spaces, we prove that $L$ is hypoelliptic if and only if ${{\left( {{\mu }_{t}} \right)}_{t>0}}$ is absolutely continuous with respect to Haar measure and admits a continuous density $x\mapsto {{\mu }_{t}}\left( x \right),t>0$, such that ${{\lim }_{t\to 0}}t\log {{\mu }_{t}}\left( e \right)=0$. In particular, this condition holds if and only if any Borel measure $u$ which is solution of $Lu=0$ in an open set $\Omega $ can be represented by a continuous function in $\Omega $. Examples are discussed.

Let $G$ be a finite group and $\chi $ be an irreducible character of $G$. An efficient and simple method to construct representations of finite groups is applicable whenever $G$ has a subgroup $H$ such that $\chi H$ has a linear constituent with multiplicity 1. In this paper we show (with a few exceptions) that if $G$ is a simple group or a covering group of a simple group and $\chi $ is an irreducible character of $G$ of degree less than 32, then there exists a subgroup $H$ (often a Sylow subgroup) of $G$ such that $\chi H$ has a linear constituent with multiplicity 1.

We compute some Hodge and Betti numbers of the moduli space of stable rank $r$, degree $d$ vector bundles on a smooth projective curve. We do not assume $r$ and $d$ are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank $r$, degree $d$ vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of $\text{S}{{\text{L}}_{n}}$ is one.

For monotone complete ${{C}^{*}}$-algebras $A\subset B$ with $A$ contained in $B$ as a monotone closed ${{C}^{*}}$-subalgebra, the relation $X=AsA$ gives a bijection between the set of all monotone closed linear subspaces $X$ of $B$ such that $AX+XA\subset X$ and $X{{X}^{*}}+{{X}^{*}}X\subset A$ and a set of certain partial isometries $s$ in the “normalizer” of $A$ in $B$, and similarly for the map $s\mapsto \text{Ad }s$ between the latter set and a set of certain “partial $*$-automorphisms” of $A$. We introduce natural inverse semigroup structures in the set of such $X$'s and the set of partial $*$-automorphisms of $A$, modulo a certain relation, so that the composition of these maps induces an inverse semigroup homomorphism between them. For a large enough $B$ the homomorphism becomes surjective and all the partial $*$-automorphisms of $A$ are realized via partial isometries in $B$. In particular, the inverse semigroup associated with a type $\text{I}{{\text{I}}_{1}}$ von Neumann factor, modulo the outer automorphism group, can be viewed as the fundamental group of the factor. We also consider the ${{C}^{*}}$-algebra version of these results.

Dans un travail antérieur, nous avions montré que l’induite parabolique (normalisée) d’une représentation irréductible cuspidale $\sigma $ d’un sous-groupe de Levi $M$ d’un groupe $p$-adique contient un sous-quotient de carré intégrable, si et seulement si la fonction $\mu $ de Harish-Chandra a un pôle en $\sigma $ d’ordre égal au rang parabolique de $M$. L’objet de cet article est d’interpréter ce résultat en termes de fonctorialité de Langlands.

The aim here is to define connections on a parabolic principal bundle. Some applications are given.

We show that certain ${{C}^{*}}$-algebras which have been studied by, among others, Arzumanian, Vershik, Deaconu, and Renault, in connection with a measure-preserving transformation of a measure space or a covering map of a compact space, are special cases of the endomorphism crossed product construction recently introduced by the first named author. As a consequence these algebras are given presentations in terms of generators and relations. These results come as a consequence of a general theorem on faithfulness of representations which are covariant with respect to certain circle actions. For the case of topologically free covering maps we prove a stronger result on faithfulness of representations which needs no covariance. We also give a necessary and sufficient condition for simplicity.

Connections between behaviour of real analytic functions (with no negative Maclaurin series coefficients and radius of convergence one) on the open unit interval, and to a lesser extent on arcs of the unit circle, are explored, beginning with Karamata's approach. We develop conditions under which the asymptotics of the coefficients are related to the values of the function near 1; specifically, $a(n)\sim f(1-1/n)/\alpha n$ (for some positive constant $\alpha $), where $f\left( t \right)=\sum{a}\left( n \right){{t}^{n}}$. In particular, if $F=\sum{c\left( n \right){{t}^{n}}}$ where $c(n)\ge 0$ and $\Sigma c(n)=1$, then $f$ defined as ${{\left( 1-F \right)}^{-1}}$ (the renewal or Green's function for $F$) satisfies this condition if ${F}'$ does (and a minor additional condition is satisfied). In come cases, we can show that the absolute sum of the differences of consecutive Maclaurin coefficients converges. We also investigate situations in which less precise asymptotics are available.

We extend the extension theorems to weighted Sobolev spaces $L_{w,k}^{p}\left( \mathcal{D} \right)$ on $(\varepsilon ,\delta )$ domains with doubling weight $w$ that satisfies a Poincaré inequality and such that ${{w}^{-1/p}}$ is locally ${{L}^{{{p}'}}}$. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above “special function approach” can be described by a classical Heine problem for differential equations that admit polynomial solutions.

The Casselman–Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of $p$-adic groups that are associated to unique models (i.e., multiplicity-free induced representations). We apply this method to the case of the Shalika model of $\text{G}{{\text{L}}_{n}}$, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y.Hironaka, and in effect reduce many such problems to straightforward calculations on the group.

In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line $\mathbb{R}$, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers $\mathbb{Q}$ onto itself is homeomorphic to the infinite power of $\mathbb{Q}$ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\mathbb{Q}$ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to ${{\mathbb{R}}^{n}}$ for $n\ge 2$, by linking the groups in question with Erdős space.