Comptes Rendus
Group Theory
Algebras of invariant differential operators on a class of multiplicity free spaces
[Algèbres d'opérateurs différentiels invariants associés à certains espaces sans multiplicités]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1343-1346.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.10.015

Hubert Rubenthaler 1

1 Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7, rue René-Descartes, 67084 Strasbourg cedex, France
@article{CRMATH_2009__347_23-24_1343_0,
     author = {Hubert Rubenthaler},
     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},
}
TY  - JOUR
AU  - Hubert Rubenthaler
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
DO  - 10.1016/j.crma.2009.10.015
LA  - en
ID  - CRMATH_2009__347_23-24_1343_0
ER  - 
%0 Journal Article
%A Hubert Rubenthaler
%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
%R 10.1016/j.crma.2009.10.015
%G en
%F CRMATH_2009__347_23-24_1343_0
Hubert Rubenthaler. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.10.015/

[1] C. Benson; G. Ratcliff On multiplicity free actions, Representations of Real and p-Adic Groups, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 2, University Press, World Scientific, Singapore, 2004, pp. 221-304

[2] R. Howe; T. Umeda The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann., Volume 290 (1991) no. 3, pp. 565-619

[3] J.I. Igusa On Lie algebras generated by two differential operators, Manifolds and Lie Groups, Progr. Math., vol. 14, Birkhäuser, Boston, Mass., 1981, pp. 187-195

[4] F. Knop Some remarks on multiplicity free spaces (A. Broer; A. Daigneault; G. Sabidussi, eds.), Representation Theory and Algebraic Geometry, Nato ASI Series C, vol. 514, Kluwer, Dordrecht, 1998, pp. 301-317

[5] A. Leahy A classification of multiplicity free representations, J. Lie Theory, Volume 8 (1998), pp. 367-391

[6] T. Levasseur Radial components, prehomogeneous vector spaces, and rational Cherednik algebras, Int. Math. Res. Not. IMRN, Volume 3 (2009), pp. 462-511

[7] T. Nomura Algebraically independent generators of invariant differential operators on a symmetric cone, J. Reine Angew. Math., Volume 400 (1989), pp. 122-133

[8] H. Rubenthaler Invariant differential operators and an infinite dimensional Howe-type correspondence, part I: Structure of the associated algebras of differential operators | arXiv

[9] S.P. Smith A class of algebras similar to the enveloping algebra of sl(2), Trans. Amer. Math. Soc., Volume 322 (1990) no. 1, pp. 285-314

[10] Z. Yan Invariant differential operators and holomorphic function spaces, J. Lie Theory, Volume 10 (2000) no. 1, pp. 1-31

Cité par Sources :

Commentaires - Politique