Invariant differential operators on the tangent space of some symmetric spaces

Levasseur, Thierry ; Stafford, J. Toby

Annales de l'Institut Fourier, Tome 49 (1999), p. 1711-1741 / Harvested from Numdam

Résumé

Soient ${\frak g}$ une algèbre de Lie semi-simple et $\vartheta $ une involution de ${\frak g}$. Si ${\frak k} = {\rm Ker}(\vartheta -I)$ et ${\frak p}= {\rm Ker}(\vartheta +I)$, nous étudions les opérateurs différentiels, ${\cal D}({\frak p})^K$, sur ${\frak p}$ qui sont invariants sous l’action du groupe adjoint $K$ de ${\frak k}$. Soit $\tau : {\frak k}\rightarrow {\rm Der}\,{\cal O}({\frak p})$ la différentielle de cette action. Nous démontrons que, pour une classe d’espaces symétriques $({\frak g},{\frak k})$ considérée par Sekiguchi, on a $\left\rbrace d \in {\cal D}({\frak p}) : d\bigl ({\cal O}({\frak p})^K\bigr )=0\right\lbrace = {\cal D}({\frak p})\tau ({\frak k})$. Une conséquence immédiate de cette égalité est le résultat suivant de Sekiguchi : Soient $({\frak g}_0,{\frak k}_0)$ une forme réelle de l’un de ces espaces symétriques $({\frak g},{\frak k})$, et $T$ une distribution $K_0$-invariante sur ${\frak p}_0$ à support dans l’ensemble des éléments singuliers; alors, $T=0$. Dans le cas diagonal $({\frak g},{\frak k})=({\frak g}^{\prime }\oplus {\frak g}^{\prime },{\frak g}^{\prime })$ ce résultat bien connu est dû à Harish-Chandra.

Abstract

Let ${\frak g}$ be a complex, semisimple Lie algebra, with an involutive automorphism $\vartheta $ and set ${\frak k} = {\rm Ker}(\vartheta -I)$, ${\frak p} = {\rm Ker}(\vartheta +I)$. We consider the differential operators, ${\cal D}({\frak p})^K$, on ${\frak p}$ that are invariant under the action of the adjoint group $K$ of ${\frak k}$. Write $\tau : {\frak k}\rightarrow {\rm Der}\,{\cal O}({\frak p})$ for the differential of this action. Then we prove, for the class of symmetric pairs $({\frak g},{\frak k})$ considered by Sekiguchi, that $\left\rbrace d \in {\cal D}({\frak p}) : d\bigl ({\cal O}({\frak p})^K\bigr )=0\right\lbrace = {\cal D}({\frak p})\tau ({\frak k})$. An immediate consequence of this equality is the following result of Sekiguchi: Let $({\frak g}_0,{\frak k}_0)$ be a real form of one of these symmetric pairs $({\frak g},{\frak k})$, and suppose that $T$ is a $K_0$-invariant eigendistribution on ${\frak p}_0$ that is supported on the singular set. Then, $T=0$. In the diagonal case $({\frak g},{\frak k})=({\frak g}^{\prime }\oplus {\frak g}^{\prime }, {\frak g}^{\prime })$ this is a well-known result due to Harish-Chandra.

Détail

Publié le : 1999-01-01

MR 2001b:16025 | Zbl 0943.22015

DOI : https://doi.org/10.5802/aif.1736

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