Let G be a connected reductive algebraic group and let be its derived subgroup. Let be a multiplicity free representation with a one-dimensional quotient (see definition below). We prove that the algebra of -invariant differential operators with polynomial coefficients on V, is a quotient of a so-called Smith algebra over its center. Over this class of algebras was introduced by S.P. Smith (1990) as a class of algebras similar to . Our result generalizes the case of the Weil representation, where the associative algebra generated by and (Q being a non-degenerate quadratic form on V) is a quotient of . Other structure results are obtained when is a regular prehomogeneous vector space of commutative parabolic type.
Soit G un groupe algébrique réductif connexe et soit son groupe dérivé. Soit un espace sans multiplicités ayant un quotient unidimensionel (voir la définition ci-dessous). Nous montrons que l'algèbre des opérateurs différentiels à coefficients poynomiaux -invariants sur V, est isomorphe à un quotient d'une algèbre de Smith sur son centre. Sur cette classe d'algèbres, avait été introduite par S.P. Smith (1990) comme une classe d'algèbres semblables à . Notre résultat généralise le cas de la représentation de Weil, où l'algèbre associative engendrée par et (Q étant une forme quadratique non dégénérée sur V), est un quotient de . D'autres résultats de structure sont obtenus lorsque est un espace préhomogène parabolique commutatif régulier.
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@article{CRMATH_2009__347_23-24_1343_0, author = {Rubenthaler, Hubert}, title = {Algebras of invariant differential operators on a class of multiplicity free spaces}, journal = {Comptes Rendus. Math\'ematique}, pages = {1343--1346}, publisher = {Elsevier}, volume = {347}, number = {23-24}, year = {2009}, doi = {10.1016/j.crma.2009.10.015}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2009.10.015/} }
TY - JOUR AU - Rubenthaler, Hubert TI - Algebras of invariant differential operators on a class of multiplicity free spaces JO - Comptes Rendus. Mathématique PY - 2009 SP - 1343 EP - 1346 VL - 347 IS - 23-24 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.10.015/ DO - 10.1016/j.crma.2009.10.015 LA - en ID - CRMATH_2009__347_23-24_1343_0 ER -
%0 Journal Article %A Rubenthaler, Hubert %T Algebras of invariant differential operators on a class of multiplicity free spaces %J Comptes Rendus. Mathématique %D 2009 %P 1343-1346 %V 347 %N 23-24 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.10.015/ %R 10.1016/j.crma.2009.10.015 %G en %F CRMATH_2009__347_23-24_1343_0
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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