no. 6
Invariant differential operators on the tangent space of some symmetric spaces
Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 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 d∈𝒟(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 d∈𝒟(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.

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     title = {Invariant differential operators on the tangent space of some symmetric spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {1711--1741},
     year = {1999},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Levasseur, Thierry; Stafford, J. Toby. Invariant differential operators on the tangent space of some symmetric spaces. Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1711-1741. doi: 10.5802/aif.1736

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