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

In this paper we estimate the \left( {{L}^{p}}-{{L}^{2}} \right)$\left( {{L}^{p}}-{{L}^{2}} \right)$-norm of the complex harmonic projectors \pi \ell {\ell }',\,1\le p\le 2$\pi \ell {\ell }',\,1\le p\le 2$, uniformly with respect to the indexes \ell,{\ell}'$\ell,{\ell}'$. We provide sharp estimates both for the projectors {{\pi }_{\ell{\ell}'}}${{\pi }_{\ell{\ell}'}}$, when \ell,{\ell}'$\ell,{\ell}'$ belong to a proper angular sector in \mathbb{N}\,\times \,\mathbb{N}$\mathbb{N}\,\times \,\mathbb{N}$, and for the projectors {{\pi }_{\ell0}}${{\pi }_{\ell0}}$ and {{\pi }_{0\ell}}${{\pi }_{0\ell}}$. 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}}${{L}^{1}}$ norm on the unit sphere of {{\mathbb{R}}^{2n}}${{\mathbb{R}}^{2n}}$.

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

For given multiplicative function f$f$, with \left| f\left( n \right) \right|\le 1$\left| f\left( n \right) \right|\le 1$ for all n$n$, we are interested in how fast its mean value \left( 1/x \right)\sum{_{n\le x}f\left( n \right)}$\left( 1/x \right)\sum{_{n\le x}f\left( n \right)}$ 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$M\,\left( \text{over}\,y\,\in \,\mathbb{R} \right)\,\text{of}\,\sum{_{p\le x}\left( 1-\operatorname{Re}\left( f\left( p \right){{p}^{-iy}} \right) \right)}/p$, and subsequent authors gave the upper bound \ll \left( 1+M \right){{e}^{-M}}$\ll \left( 1+M \right){{e}^{-M}}$. 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$y$ that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the k$k$-th powers mod p$p$.

A model subspace {{K}_{\Theta }}${{K}_{\Theta }}$ of the Hardy space {{H}^{2}}={{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)${{H}^{2}}={{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ for the upper half plane {{\mathbb{C}}_{+}}${{\mathbb{C}}_{+}}$ is {{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)\ominus \Theta {{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)${{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)\ominus \Theta {{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ where \Theta $\Theta$ is an inner function in {{\mathbb{C}}_{+}}${{\mathbb{C}}_{+}}$. A function \omega :\,\mathbb{R}\mapsto [0,\,\infty )$\omega :\,\mathbb{R}\mapsto [0,\,\infty )$ is called an admissible majorant for {{K}_{\Theta }}${{K}_{\Theta }}$ if there exists an f\,\in \,{{K}_{\Theta }},\,f\,\not{\equiv }\,0,\,|f\left( x \right)|\,\le \,\omega \left( x \right)$f\,\in \,{{K}_{\Theta }},\,f\,\not{\equiv }\,0,\,|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on \mathbb{R}$\mathbb{R}$. For some (mainly meromorphic) \Theta $\Theta$'s some parts of Adm \Theta $\Theta$ (the set of all admissible majorants for {{K}_{\Theta }}${{K}_{\Theta }}$) are explicitly described. These descriptions depend on the rate of growth of arg \Theta $\Theta$ along \mathbb{R}$\mathbb{R}$. This paper is about slowly growing arguments (slower than x$x$). Our results exhibit the dependence of Adm B$B$ on the geometry of the zeros of the Blaschke product B$B$. A complete description of Adm B$B$ is obtained for B$B$'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]$[6]$. We consider the model subspaces {{K}_{\Theta }}={{H}^{2}}\ominus \Theta {{H}^{2}}${{K}_{\Theta }}={{H}^{2}}\ominus \Theta {{H}^{2}}$ of the Hardy space {{H}^{2}}${{H}^{2}}$ generated by an inner function \Theta $\Theta$ in the upper half plane. Our main object is the class of admissible majorants for {{K}_{\Theta }}${{K}_{\Theta }}$, denoted by Adm \Theta $\Theta$ and consisting of all functions \omega $\omega$ defined on \mathbb{R}$\mathbb{R}$ such that there exists an f\ne 0,f\in {{K}_{\Theta }}$f\ne 0,f\in {{K}_{\Theta }}$ satisfying |f\left( x \right)|\,\le \,\omega \left( x \right)$|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on \mathbb{R}$\mathbb{R}$. Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any {{K}_{\Theta }}${{K}_{\Theta }}$ generated by a meromorphic inner function. In contrast with [6]$[6]$, we consider the generating functions \Theta $\Theta$ such that the unit vector \Theta \left( x \right)$\Theta \left( x \right)$ winds up fast as x$x$ grows from -\infty \,\text{to}\,\infty $-\infty \,\text{to}\,\infty$. In particular, we consider \Theta \,=\,B$\Theta \,=\,B$ where B$B$ is a Blaschke product with “horizontal” zeros, i.e., almost uniformly distributed in a strip parallel to and separated from \mathbb{R}$\mathbb{R}$. It is shown, among other things, that for any such B$B$, any even \omega $\omega$ decreasing on \left( 0,\,\infty \right)$\left( 0,\,\infty \right)$ with a finite logarithmic integral is in Adm B$B$ (unlike the “vertical” case treated in [6]$[6]$), thus generalizing (with a new proof) a classical result related to Adm \exp \left( i\sigma z \right),\,\sigma \,>\,0$\exp \left( i\sigma z \right),\,\sigma \,>\,0$. Some oscillating \omega $\omega$'s in Adm B$B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm \exp \left( i\sigma z \right),\,\sigma \,>\,0$\exp \left( i\sigma z \right),\,\sigma \,>\,0$, and to de Branges’ space H\left( E \right)$H\left( E \right)$.

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$K$-groups of our algebras.