Soit $G$ le groupe $\text{GL}\,\left( N,\,F \right)$, où $F$ est un corps localement compact et non archimédien. En utilisant la théorie des types simples de Bushnell et Kutzko, ainsi qu'une idée originale d'Henniart, nous construisons des pseudo-coefficients explicites pour les représentations de la série discrète de $G$. Comme application, nous en déduisons des formules inédites pour la valeur du charactère d'Harish- Chandra de certaines telles représentations en certainséléments elliptiques réguliers.

Let $h_{\vee }^{\infty }\,\left( \text{B} \right)$ and $h_{\vee }^{\infty }\,\left( \text{B} \right)$ be the spaces of harmonic functions in the unit disk and multidimensional unit ball admitting a two-sided radial majorant $v\left( r \right)$. We consider functions $v$ that fulfill a doubling condition. In the two-dimensional case let

$$u\left( r{{e}^{i\theta }},\xi \right)\,=\,\sum\limits_{j=0}^{\infty }{\left( {{a}_{j0}}{{\xi }_{j0}}{{r}^{j}}\,\cos \,j\theta \,+\,{{a}_{j1}}{{\xi }_{j1}}{{r}^{j}}\,\sin \,j\theta \right)}$$

where $\xi \,=\,\left\{ {{\xi }_{ji}} \right\}$ is a sequence of random subnormal variables and ${{a}_{ji}}$ are real. In higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients ${{a}_{ji}}$ that imply that $u$ is in $h_{\vee }^{\infty }\,\left( \text{B} \right)$ almost surely. Our estimate improves previous results by Bennett, Stegenga, and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces that generalizes results by Anderson, Clunie, and Pommerenke and by Guo and Liu.

A subset $X$ of a Polish group $G$ is called Haar null if there exist a Borel set $B\,\supset \,X$ and Borel probability measure $\mu$ on $G$ such that $\mu \left( g\,Bh \right)\,=\,0$ for every $g,\,h\,\in \,G$. We prove that there exist a set $X\,\subset \,\text{R}$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu \left( X\,+\,t \right)\,=\,0$ for every $t\,\in \,\text{R}$. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.)

This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C\,\subset \,G$ such that for every non-meagre set $X\,\subset \,\text{G}$ there exists a $t\in \text{G}$ such that $C\,\cap \,\left( X\,+\,t \right)$ is relatively non-meagre in $C$. This essentially generalizes results of Bartoszyński and Burke–Miller.

Let $W$ be an infinite Coxeter group. We initiate the study of the set $E$ of limit points of “normalized” roots (representing the directions of the roots) of $\text{W}$. We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form $B$ associated with a geometric representation, and we illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of $W$ on $E$, and then we exhibit a countable subset of $E$, formed by limit points for the dihedral reflection subgroups of $W$. We explain how this subset is built fromthe intersection with $Q$ of the lines passing through two positive roots, and finally we establish that it is dense in $E$.

Due to work of $\text{W}$. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on ${{\mathbb{H}}^{3}}$ is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in ${{\text{H}}^{2}}$ where $\text{E}$. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

A new characterization of the uniform convexity of Banach space is obtained in the sense of the Bishop–Phelps–Bollobás theorem. It is also proved that the couple of Banach spaces $\left( X,Y \right)$ has the Bishop–Phelps–Bollobás property for every Banach space $Y$ when $X$ is uniformly convex. As a corollary, we show that the Bishop–Phelps–Bollobás theorem holds for bilinear forms on ${{\ell }_{p}}\,\times \,{{\ell }_{q}}$$\left( 1\,<p,q\,<\,\infty \right)$.

We study the image of the model subspace ${{K}_{\theta }}$ under the composition operator ${{C}_{\varphi }}$, where $\varphi $ and $\theta $ are inner functions, and find the smallest model subspace which contains the linear manifold ${{C}_{\varphi }}{{K}_{\theta }}$. Then we characterize the case when ${{C}_{\varphi }}$ maps ${{K}_{\theta }}$ into itself. This case leads to the study of the inner functions $\varphi $ and $\psi $ such that the composition $\psi \,\text{o}\,\varphi $ is a divisor of $\psi $ in the family of inner functions.

We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of ${{\mathbb{S}}^{n}}\,\times \,\mathbb{R}$, extending the classification of umbilical surfaces in ${{\mathbb{S}}^{2}}\times \mathbb{R}$ by Souam and Toubiana as well as the local description of umbilical hypersurfaces in ${{\mathbb{S}}^{n}}\,\times \,\mathbb{R}$ by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic ${{\mathbb{S}}^{1}}\times \mathbb{R}$ or ${{\mathbb{S}}^{2}}\times \mathbb{R}$, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of ${{\mathbb{S}}^{n}}\times \mathbb{R}$ and ${{\mathbb{H}}^{n}}\times \mathbb{R}$. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor $\mathbb{R}$ is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in ${{\mathbb{S}}^{n}}\times \mathbb{R}$ and ${{\mathbb{H}}^{n}}\times \mathbb{R}$ having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of ${{\mathbb{S}}^{n}}\times \mathbb{R}$ and ${{\mathbb{H}}^{n}}\times \mathbb{R}$.

For parabolic linear operators $L$ of second order in divergence form, we prove that the solvability of initial ${{L}^{p}}$ Dirichlet problems for the whole range $1\,<\,p\,<\,\infty $ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of $L$ satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of $p\,>\,1$, the initial ${{L}^{p}}$ Dirichlet problem associated with $Lu\,=\,0$ over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations.

We prove that the two-variable $\text{BMW}$ algebra embeds into an algebra constructed from the $\text{HOMFLY-PT}$ polynomial. We also prove that the $\mathfrak{s}{{\mathfrak{O}}_{2N}}-\text{BMW}$ algebra embeds in the $q$-Schur algebra of type $A$. We use these results to suggest a schema providing categorifications of the $\mathfrak{s}{{\mathfrak{D}}_{2N}}-\text{BMW}$ algebra.