Let be a complex, semisimple Lie algebra, with an involutive automorphism and set , . We consider the differential operators, , on that are invariant under the action of the adjoint group of . Write for the differential of this action. Then we prove, for the class of symmetric pairs considered by Sekiguchi, that . An immediate consequence of this equality is the following result of Sekiguchi: Let be a real form of one of these symmetric pairs , and suppose that is a -invariant eigendistribution on that is supported on the singular set. Then, . In the diagonal case this is a well-known result due to Harish-Chandra.
Soient une algèbre de Lie semi-simple et une involution de . Si et , nous étudions les opérateurs différentiels, , sur qui sont invariants sous l’action du groupe adjoint de . Soit la différentielle de cette action. Nous démontrons que, pour une classe d’espaces symétriques considérée par Sekiguchi, on a . Une conséquence immédiate de cette égalité est le résultat suivant de Sekiguchi : Soient une forme réelle de l’un de ces espaces symétriques , et une distribution -invariante sur à support dans l’ensemble des éléments singuliers; alors, . Dans le cas diagonal ce résultat bien connu est dû à Harish-Chandra.
@article{AIF_1999__49_6_1711_0, author = {Levasseur, Thierry and Stafford, J. Toby}, title = {Invariant differential operators on the tangent space of some symmetric spaces}, journal = {Annales de l'Institut Fourier}, pages = {1711--1741}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {49}, number = {6}, year = {1999}, doi = {10.5802/aif.1736}, zbl = {0943.22015}, mrnumber = {2001b:16025}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1736/} }
TY - JOUR AU - Levasseur, Thierry AU - Stafford, J. Toby TI - Invariant differential operators on the tangent space of some symmetric spaces JO - Annales de l'Institut Fourier PY - 1999 SP - 1711 EP - 1741 VL - 49 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1736/ DO - 10.5802/aif.1736 LA - en ID - AIF_1999__49_6_1711_0 ER -
%0 Journal Article %A Levasseur, Thierry %A Stafford, J. Toby %T Invariant differential operators on the tangent space of some symmetric spaces %J Annales de l'Institut Fourier %D 1999 %P 1711-1741 %V 49 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1736/ %R 10.5802/aif.1736 %G en %F AIF_1999__49_6_1711_0
Levasseur, Thierry; Stafford, J. Toby. Invariant differential operators on the tangent space of some symmetric spaces. Annales de l'Institut Fourier, Volume 49 (1999) no. 6, pp. 1711-1741. doi : 10.5802/aif.1736. https://aif.centre-mersenne.org/articles/10.5802/aif.1736/
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