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hal-00238390, version 1

## Invariant differential operators and an infinite dimensional Howe-type correspondence. Part I: Structure of the associated algebras of differential operators.

Hubert Rubenthaler () 1

(01/2008)

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 the study of the case where $Q(x)$ is replaced by $\Delta_{0}(x)$, where $\Delta_{0}(x)$ is the relative invariant of a prehomogeneous vector space of commutative parabolic type (${\go g},V$), or equivalently where $\Delta_{0}$ is the "determinant" function of a simple Jordan algebra $V$ over ${\bb C}$. In this Part I we show several structure results for the associative algebra generated by $X=\Delta_{0}(x)$, $Y=\Delta_{0}(\partial)$. Our main result shows that if we consider this algebra as an algebra over a certain commutative ring ${\bf A}$ of invariant differential operators it is isomorphic to the quotient of what we call a generalized Smith algebra $S(f, {\bf A}, n)$ where $f\in {\bf A}[t]$. The Smith algebras (over ${\bb C}$) were introduced by P. Smith as "natural" generalizations of ${\cal U}({\go sl}_{2})$.

• 1 :  Institut de Recherche Mathématique Avancée (IRMA)
• CNRS : UMR7501 – Université Louis Pasteur - Strasbourg I
• Domaine : Mathématiques/Théorie des représentations
• Mots-clés : Howe correspondence
• Commentaire : 39 pages

• hal-00238390, version 1
• oai:hal.archives-ouvertes.fr:hal-00238390
• Contributeur :
• Soumis le : Lundi 4 Février 2008, 14:49:30
• Dernière modification le : Mercredi 1 Octobre 2008, 14:57:16