Group Theory
Algebras of invariant differential operators on a class of multiplicity free spaces
Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1343-1346.

Let G be a connected reductive algebraic group and let G=[G,G] be its derived subgroup. Let (G,V) be a multiplicity free representation with a one-dimensional quotient (see definition below). We prove that the algebra D(V)G of G-invariant differential operators with polynomial coefficients on V, is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith (1990) as a class of algebras similar to U(sl2). Our result generalizes the case of the Weil representation, where the associative algebra generated by Q(x) and Q() (Q being a non-degenerate quadratic form on V) is a quotient of U(sl2). Other structure results are obtained when (G,V) is a regular prehomogeneous vector space of commutative parabolic type.

Soit G un groupe algébrique réductif connexe et soit G=[G,G] son groupe dérivé. Soit (G,V) un espace sans multiplicités ayant un quotient unidimensionel (voir la définition ci-dessous). Nous montrons que l'algèbre D(V)G des opérateurs différentiels à coefficients poynomiaux G-invariants sur V, est isomorphe à un quotient d'une algèbre de Smith sur son centre. Sur C cette classe d'algèbres, avait été introduite par S.P. Smith (1990) comme une classe d'algèbres semblables à U(sl2). Notre résultat généralise le cas de la représentation de Weil, où l'algèbre associative engendrée par Q(x) et Q() (Q étant une forme quadratique non dégénérée sur V), est un quotient de U(sl2). D'autres résultats de structure sont obtenus lorsque (G,V) est un espace préhomogène parabolique commutatif régulier.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2009.10.015
Rubenthaler, Hubert 1

1 Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7, rue René-Descartes, 67084 Strasbourg cedex, France
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Rubenthaler, Hubert. Algebras of invariant differential operators on a class of multiplicity free spaces. Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1343-1346. doi : 10.1016/j.crma.2009.10.015. http://www.numdam.org/articles/10.1016/j.crma.2009.10.015/

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