The goal of this article is to explain Howe's correspondence to a reader who is not necessarily an expert on representation theory of real reductive groups, but is familiar with general concepts of harmonic analysis. We recall Howe's construction of the oscillator representation, the notion of a dual pair and a few basic and general facts concerning the correspondence.
For every closed subset C in the dual space of the Heisenberg group we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra and we show that in general for two closed subsets of the product of and is different from .
We prove that Huygens’ principle and the principle of equipartition of energy hold for the modified wave equation on odd dimensional Damek–Ricci spaces. We also prove a Paley–Wiener type theorem for the inverse of the Helgason Fourier transform on Damek–Ricci spaces.
Let be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in . We also derive several new and several old results on the topology of ....
L’indice d’une algèbre de Lie algébrique complexe est la codimension minimale de ses orbites coadjointes. Si est semi-simple, son indice, , est égal à son rang, . Le but de cet article est d’établir une formule générale pour l’indice de pour nilpotent, où est le normalisateur dans du centralisateur de . Plus précisément, on obtient le résultat suivant, conjecturé par D. Panyushev :où est le centre de . Panyushev obtient l’inégalité dans Panyushev 2003 et on montre que la maximalité...