Soit $G$ le groupe des points définis sur un corps $p$-adique d'un groupe réductif connexe. A l'aide des caractères virtuels supertempérés de $G$, on prouve (conjectures de Clozel) que toute représentation irréductible tempérée de $G$ est irréductiblement induite d'une essentielle d'un sousgroupe de Lévi de $G$.

In this paper we estimate the $\left( {{L}^{p}}-{{L}^{2}} \right)$-norm of the complex harmonic projectors $\pi \ell {\ell }',\,1\le p\le 2$, uniformly with respect to the indexes $\ell,{\ell}'$. We provide sharp estimates both for the projectors ${{\pi }_{\ell{\ell}'}}$, when $\ell,{\ell}'$ belong to a proper angular sector in $\mathbb{N}\,\times \,\mathbb{N}$, and for the projectors ${{\pi }_{\ell0}}$ and ${{\pi }_{0\ell}}$. The proof is based on an extension of a complex interpolation argument by C. Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the ${{L}^{1}}$ norm on the unit sphere of ${{\mathbb{R}}^{2n}}$.

The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group $K={{O}_{p}}\left( C \right)\times {{O}_{q}}\left( C \right)$ acts on the space ${{M}_{p,\,q}}\,\text{of}\,p\,\times \,q$ complex matrices by $\left( a,b \right)\cdot x=ax{{b}^{-1}}$, and so does its identity component ${{K}^{0}}=\text{S}{{\text{O}}_{p}}\left( \text{C} \right)\times \text{S}{{\text{O}}_{\text{q}}}\left( \text{C} \right)$. A $K$-orbit (or ${{K}^{0}}$-orbit) in ${{M}_{p,q}}$ is said to be nilpotent if its closure contains the zero matrix. The closure, $\bar{\mathcal{O}}$, of a nilpotent $K$-orbit (resp. ${{K}^{0}}$-orbit) $\mathcal{O}$ in ${{M}_{p,q}}$ is a union of $\mathcal{O}$ and some nilpotent $K$-orbits (resp. ${{K}^{0}}$-orbits) of smaller dimensions. The description of the closure of nilpotent $K$-orbits has been known for some time, but not so for the nilpotent ${{K}^{0}}$-orbits. A conjecture describing the closure of nilpotent ${{K}^{0}}$-orbits was proposed in $[11]$ and verified when $\min \left( p,\,q \right)\le 7$. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to $\mathcal{O}$ and determination of the basic relative invariants of these spaces.

The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra $\mathfrak{s}\mathfrak{o}\left( p,\,q \right)$ under the adjoint action of the identity component of the real orthogonal group $\text{O}\left( p,\,q \right)$.

For given multiplicative function $f$, with $\left| f\left( n \right) \right|\le 1$ for all $n$, we are interested in how fast its mean value $\left( 1/x \right)\sum{_{n\le x}f\left( n \right)}$ converges. Halász showed that this depends on the minimum $M\,\left( \text{over}\,y\,\in \,\mathbb{R} \right)\,\text{of}\,\sum{_{p\le x}\left( 1-\operatorname{Re}\left( f\left( p \right){{p}^{-iy}} \right) \right)}/p$, and subsequent authors gave the upper bound $\ll \left( 1+M \right){{e}^{-M}}$. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Halász-Montgomery lemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of $y$ that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the $k$-th powers mod $p$.

A model subspace ${{K}_{\Theta }}$ of the Hardy space ${{H}^{2}}={{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ for the upper half plane ${{\mathbb{C}}_{+}}$ is ${{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)\ominus \Theta {{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ where $\Theta $ is an inner function in ${{\mathbb{C}}_{+}}$. A function $\omega :\,\mathbb{R}\mapsto [0,\,\infty )$ is called an admissible majorant for ${{K}_{\Theta }}$ if there exists an $f\,\in \,{{K}_{\Theta }},\,f\,\not{\equiv }\,0,\,|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on $\mathbb{R}$. For some (mainly meromorphic) $\Theta $'s some parts of Adm $\Theta $ (the set of all admissible majorants for ${{K}_{\Theta }}$) are explicitly described. These descriptions depend on the rate of growth of arg $\Theta $ along $\mathbb{R}$. This paper is about slowly growing arguments (slower than $x$). Our results exhibit the dependence of Adm $B$ on the geometry of the zeros of the Blaschke product $B$. A complete description of Adm $B$ is obtained for $B$'s with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.

This paper is a continuation of $[6]$. We consider the model subspaces ${{K}_{\Theta }}={{H}^{2}}\ominus \Theta {{H}^{2}}$ of the Hardy space ${{H}^{2}}$ generated by an inner function $\Theta $ in the upper half plane. Our main object is the class of admissible majorants for ${{K}_{\Theta }}$, denoted by Adm $\Theta $ and consisting of all functions $\omega $ defined on $\mathbb{R}$ such that there exists an $f\ne 0,f\in {{K}_{\Theta }}$ satisfying $|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on $\mathbb{R}$. Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any ${{K}_{\Theta }}$ generated by a meromorphic inner function. In contrast with $[6]$, we consider the generating functions $\Theta $ such that the unit vector $\Theta \left( x \right)$ winds up fast as $x$ grows from $-\infty \,\text{to}\,\infty $. In particular, we consider $\Theta \,=\,B$ where $B$ is a Blaschke product with “horizontal” zeros, i.e., almost uniformly distributed in a strip parallel to and separated from $\mathbb{R}$. It is shown, among other things, that for any such $B$, any even $\omega $ decreasing on $\left( 0,\,\infty \right)$ with a finite logarithmic integral is in Adm $B$ (unlike the “vertical” case treated in $[6]$), thus generalizing (with a new proof) a classical result related to Adm $\exp \left( i\sigma z \right),\,\sigma \,>\,0$. Some oscillating $\omega $'s in Adm $B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm $\exp \left( i\sigma z \right),\,\sigma \,>\,0$, and to de Branges’ space $H\left( E \right)$.

We completely determine the ideal structures of the crossed products of Cuntz algebras by quasi-free actions of abelian groups and give another proof of A. Kishimoto's result on the simplicity of such crossed products. We also give a necessary and sufficient condition that our algebras become primitive, and compute the Connes spectra and $K$-groups of our algebras.