Invariant differential operators on the tangent space of some symmetric spaces

Levasseur, Thierry; Stafford, J. Toby

Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1711-1741.

Soient $𝔤$ une algèbre de Lie semi-simple et $\vartheta$ une involution de $𝔤$. Si $𝔨=\mathrm{Ker}(\vartheta -I)$ et $𝔭=\mathrm{Ker}(\vartheta +I)$, nous étudions les opérateurs différentiels, $𝒟(𝔭{)}^K$, sur $𝔭$ qui sont invariants sous l’action du groupe adjoint $K$ de $𝔨$. Soit $\tau :𝔨\to \mathrm{Der}𝒪\left(𝔭\right)$ la différentielle de cette action. Nous démontrons que, pour une classe d’espaces symétriques $(𝔤,𝔨)$ considérée par Sekiguchi, on a $\left\{d\in 𝒟\left(𝔭\right):d\left(𝒪(𝔭{)}^K\right)=0\right\}=𝒟\left(𝔭\right)\tau \left(𝔨\right)$. 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 $(𝔤,𝔨)=({𝔤}^{\prime }\oplus {𝔤}^{\prime },{𝔤}^{\prime })$ ce résultat bien connu est dû à Harish-Chandra.

Let $𝔤$ be a complex, semisimple Lie algebra, with an involutive automorphism $\vartheta$ and set $𝔨=\mathrm{Ker}(\vartheta -I)$, $𝔭=\mathrm{Ker}(\vartheta +I)$. We consider the differential operators, $𝒟(𝔭{)}^K$, on $𝔭$ that are invariant under the action of the adjoint group $K$ of $𝔨$. Write $\tau :𝔨\to \mathrm{Der}𝒪\left(𝔭\right)$ for the differential of this action. Then we prove, for the class of symmetric pairs $(𝔤,𝔨)$ considered by Sekiguchi, that $\left\{d\in 𝒟\left(𝔭\right):d\left(𝒪(𝔭{)}^K\right)=0\right\}=𝒟\left(𝔭\right)\tau \left(𝔨\right)$. 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 $(𝔤,𝔨)=({𝔤}^{\prime }\oplus {𝔤}^{\prime },{𝔤}^{\prime })$ this is a well-known result due to Harish-Chandra.

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