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Invariant differential operators on symmetric spaces. I: Orbit methods. (Opérateurs différentiels invariants sur les espaces symétriques. I: Méthodes des orbites.) (French) Zbl 0803.43003
Let $$G/K$$ be a symmetric space with symmetry $$\sigma$$, and let $${\mathfrak g}= {\mathfrak k}+ {\mathfrak p}$$ be the corresponding decomposition of the Lie algebra $${\mathfrak g}$$ of $$G$$. The present paper is motivated by a study of the algebra $${\mathcal A}$$ of $$G$$-invariant differential operators on the line bundle of half-densities on $$G/K$$.
The author obtains here three main results. First, assuming $$G$$ is a linear algebraic group with rational involution $$\sigma$$, he shows that every strongly regular linear form on $${\mathfrak p}$$ admits a polarization which satisfies Pukanszky’s condition. Second, given a linear form $$f$$ on $${\mathfrak p}$$, he constructs (under certain assumptions) a distribution vector given by a Kirillov type formula on the orbit $$K.f$$. This extends to $${\mathfrak g}$$ previous results by Benoist for nilpotent $${\mathfrak g}$$. Third, he shows how such distribution vectors can provide characters of the algebra $${\mathcal A}$$. These three results lay the foundations for the second part of the article [cf. the following review Zbl 0803.43004], where they will be combined to construct a generalized Harish-Chandra homomorphism for $${\mathcal A}$$.
Reviewer: F.Rouvière (Nice)

##### MSC:
 43A85 Harmonic analysis on homogeneous spaces 53C35 Differential geometry of symmetric spaces 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 32C38 Sheaves of differential operators and their modules, $$D$$-modules
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