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hal-00574585, version 1

## Invariant differential operators on a class of multiplicity free spaces

Hubert Rubenthaler () 1

(08/03/2011)

Résumé : If $Q$ is a non degenerate quadratic form on ${\bb C}^n$, it is well known that the differential operators $X=Q(x)$, $Y=Q(\partial)$, and $H=E+\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebra isomorphic to ${\go sl}_{2}$. Therefore the associative algebra they generate is a quotient of the universal enveloping algebra ${\cal U}({\go sl}_{2})$. This fact is in some sense the foundation of the metaplectic representation. The present paper is devoted to a generalization where $Q(x)$ is replaced by $\Delta_{0}(x)$ which is a relative invariant of a multiplicity free representation $(G,V)$ with a one dimensional quotient (see definition below). Over these spaces we study various algebras of differential operators. In particular if $G'=[G,G]$ is the derived group of the reductive group $G$, we prove that the algebra $D(V)^{G'}$ of $G'$-invariant differential operators with polynomial coefficients on $V$, is a quotient of a Smith algebra over its center. Over ${\bb C}$ this class of algebras was introduced by S.P. Smith as a class of algebras similar to ${\cal U}({\go s}{\go l}_{2})$. This allows us to describe by generators and relations the structure of $D(V)^{G'}$. As a corollary we obtain that various "algebras of radial components" are quotients of ordinary Smith algebras over ${\bb C}$. We also give the complete classification of the multiplicity free spaces $(G,V)$ with a one dimensional quotient, and pay particular attention to the subclass of prehomogeneous vector spaces of commutative parabolic type, for which further results are obtained.

• 1 :  Institut de Recherche Mathématique Avancée (IRMA)
• CNRS : UMR7501 – Université de Strasbourg
• Domaine : Mathématiques/Théorie des représentations
Mathématiques/Anneaux et algèbres
• Mots-clés : invariant differential operators – multiplicity free spaces – Smith algebras – radial components

• hal-00574585, version 1
• oai:hal.archives-ouvertes.fr:hal-00574585
• Contributeur :
• Soumis le : Mardi 8 Mars 2011, 14:16:07
• Dernière modification le : Mercredi 9 Mars 2011, 08:31:15