La théorie de M. Sato et T. Shintani associe à toute forme réelle d’un espace préhomogène irréductible régulier dont le groupe est réductif, une fonction zêta qui vérifie une équation fonctionnelle remarquable. Dans cet article, nous classifions les formes réelles infinitésimales des espaces préhomogènes irréductibles de type parabolique. Cette classification est obtenue en termes de diagrammes de Satake à poids.

Generalizations of the classical Schwarzian derivative of complex analysis have been proposed by Osgood and Stowe [12, 13], Carne [5], and Ahlfors [3]. We present another generalization of the Schwarzian derivative over vector spaces.

Let $M_n$ be the multiplicative semigroup of all $n\times n$ complex matrices, and let $U_n$ and $G{L}_n$ be the $n$–degree unitary group and general linear group over complex number field, respectively. We characterize group homomorphisms from $U_n$ to $G{L}_m$ when $n>m\ge 1$ or $n=m\ge 3$, and thereby determine multiplicative homomorphisms from $U_n$ to $M_m$ when $n>m\ge 1$ or $n=m\ge 3$. This generalize Hochwald’s result in [Lin. Alg. Appl.  212/213:339-351(1994)]: if $f:U_n\to M_n$ is a spectrum–preserving multiplicative homomorphism, then there exists a matrix $R$ in $G{L}_n$ such that $f\left(A\right)=RAR$ for...

Soit $G$ un groupe algébrique semi-simple complexe, $U$ un sous-groupe unipotent maximal de $G$, $T$ un tore maximal de $G$ normalisant $U$. Si $V$ est un $G$-module rationnel de dimension finie, alors $G$ opère sur l’algèbre $\mathbf{C}\left[V\right]$ des fonctions polynomiales sur $V$; la structure de $G$-module de $\mathbf{C}\left[V\right]$ est décrite par la $T$-algèbre $\mathbf{C}[V{]}^U$ des $U$-invariants de $\mathbf{C}\left[V\right]$. Cette algèbre est de type fini et multigraduée (par le degré de $\mathbf{C}\left[V\right]$ et le poids par rapport à $T$). On donne une formule intégrale pour la série de Poincaré de cette algèbre graduée....

We consider problems in invariant theory related to the classification of four vector subspaces of an $n$-dimensional complex vector space. We use castling techniques to quickly recover results of Howe and Huang on invariants. We further obtain information about principal isotropy groups, equidimensionality and the modules of covariants.

An affine Hecke algebras can be realized as an equivariant $K$-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the corresponding Lie algebra. In this paper we will show that the two-sided ideals are in fact the two-sided ideals of the affine Hecke algebra defined through two-sided cells of the corresponding affine Weyl group after the two-sided ideals are tensored by $\mathbb{Q}$. This...

In this paper we study dynamical properties of linear actions by free groups via the induced action on projective space. This point of view allows us to introduce techniques from Thermodynamic Formalism. In particular, we obtain estimates on the growth of orbits and their limiting distribution on projective space.

Let $H\subset GL\left(V\right)$ be a connected complex reductive group where $V$ is a finite-dimensional complex vector space. Let $v\in V$ and let $G=\{g\in \mathrm{GL}(V)\mid gHv=Hv\}$. Following Raïs we say that the orbit $Hv$ is characteristic for $H$ if the identity component of $G$ is $H$. If $H$ is semisimple, we say that $Hv$ is semi-characteristic for $H$ if the identity component of $G$ is an extension of $H$ by a torus. We classify the $H$-orbits which are not (semi)-characteristic in many cases.