Invariant differential operators on the tangent space of some symmetric spaces
Annales de l'Institut Fourier, Volume 49 (1999) no. 6, p. 1711-1741

Let 𝔤 be a complex, semisimple Lie algebra, with an involutive automorphism ϑ and set 𝔨= Ker (ϑ-I), 𝔭= Ker (ϑ+I). We consider the differential operators, 𝒟(𝔭) K , on 𝔭 that are invariant under the action of the adjoint group K of 𝔨. Write τ:𝔨 Der 𝒪(𝔭) for the differential of this action. Then we prove, for the class of symmetric pairs (𝔤,𝔨) considered by Sekiguchi, that d𝒟(𝔭):d𝒪 ( 𝔭 ) K =0=𝒟(𝔭)τ(𝔨). An immediate consequence of this equality is the following result of Sekiguchi: Let (𝔤 0 ,𝔨 0 ) be a real form of one of these symmetric pairs (𝔤,𝔨), and suppose that T is a K 0 -invariant eigendistribution on 𝔭 0 that is supported on the singular set. Then, T=0. In the diagonal case (𝔤,𝔨)=(𝔤 𝔤 ,𝔤 ) this is a well-known result due to Harish-Chandra.

Soient 𝔤 une algèbre de Lie semi-simple et ϑ une involution de 𝔤. Si 𝔨= Ker (ϑ-I) et 𝔭= Ker (ϑ+I), nous étudions les opérateurs différentiels, 𝒟(𝔭) K , sur 𝔭 qui sont invariants sous l’action du groupe adjoint K de 𝔨. Soit τ:𝔨 Der 𝒪(𝔭) la différentielle de cette action. Nous démontrons que, pour une classe d’espaces symétriques (𝔤,𝔨) considérée par Sekiguchi, on a d𝒟(𝔭):d𝒪 ( 𝔭 ) K =0=𝒟(𝔭)τ(𝔨). Une conséquence immédiate de cette égalité est le résultat suivant de Sekiguchi : Soient (𝔤 0 ,𝔨 0 ) une forme réelle de l’un de ces espaces symétriques (𝔤,𝔨), et T une distribution K 0 -invariante sur 𝔭 0 à support dans l’ensemble des éléments singuliers; alors, T=0. Dans le cas diagonal (𝔤,𝔨)=(𝔤 𝔤 ,𝔤 ) 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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {6},
     year = {1999},
     pages = {1711-1741},
     doi = {10.5802/aif.1736},
     zbl = {0943.22015},
     mrnumber = {2001b:16025},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_1999__49_6_1711_0/

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