zbMATH — the first resource for mathematics

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)

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
Full Text: DOI