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Invariant analysis and contractions of symmetric spaces. I. (English) Zbl 0697.53049
The main problem of this paper is the following: is it possible to transform an invariant differential operator on a homogeneous space into a constant coefficient differential operator on some vector space? The author considers the case of a simply connected symmetric space \(M=G/H\) with the algebra \({\mathbb{D}}(M)\) of all G-invariant linear differential operators on M. For such spaces a function e(x,y) of two tangent vectors at the origin of M, obtained from the corresponding infinitesimal structure of Lie triple system is defined and studied. The study is based on the idea of contractions of M into its tangent space.
Reviewer: A.Fleischer

MSC:
53C35 Differential geometry of symmetric spaces
43A85 Harmonic analysis on homogeneous spaces
58J70 Invariance and symmetry properties for PDEs on manifolds
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17A40 Ternary compositions
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References:
[1] J.-L. Clerc , Une formule asymptotique du type Mehler-Heine pour les zonales d’un espace riemannien symétrique , Studia Math. 57 (1976) 27-32. · Zbl 0335.43010
[2] A.H. Dooley , Contractions of Lie groups and applications to analysis , Topics in modern harmonic analysis, Roma 1983. · Zbl 0551.22006
[3] A.H. Dooley and J.W. Rice , On contractions of semi-simple Lie groups , Trans. Amer. Math. Soc. 289 (1985) 185-202. · Zbl 0546.22017
[4] M. Duflo , Opérateurs différentiels bi-invariants sur un groupe de Lie , Ann. Scient. Ec. Norm. Sup. 10 (1977) 265-288. · Zbl 0353.22009
[5] M. Flensted-Jensen , Analysis on non-Riemannian symmetric spaces , Reg. Conf. Series in Maths. no. 61, Amer. Math. Soc. 1986. · Zbl 0589.43008
[6] V. Guillemin and S. Sternberg , Symplectic techniques in Physics , Cambridge University Press, Cambridge 1984. · Zbl 0576.58012
[7] P. Harinck , Fonctions généralisées sphériques sur GC/GR, Thèse , Université Paris7, 1988.
[8] Harish-Chandra , Invariant eigendistributions on a semi-simple Lie group , Trans. Amer. Math. Soc. 119 (1965) 457-508. · Zbl 0199.46402
[9] S. Helgason , Fundamental solutions of invariant differential operators on symmetric spaces , Amer. J. Maths. 86 (1964) 565-601. · Zbl 0178.17001
[10] S. Helgason , Differential geometry, Lie groups and symmetric spaces , Academic Press, New-York 1978. · Zbl 0451.53038
[11] S. Helgason , Groups and geometric analysis , Academic Press, Orlando 1984. · Zbl 0543.58001
[12] M. Kashiwara and M. Vergne , The Campbell-Hausdorff formula and invariant hyperfunctions , Invent. Math. 47 (1978) 249-272. · Zbl 0404.22012
[13] S. Kobayashi and K. Nomizu , Foundations of differential geometry , vol. II, Interscience Publishers, New-York 1969. · Zbl 0175.48504
[14] G. Lion , Résolubilité d’opérateurs différentiels invariants sur un nilespace homogène , prépublication, 1986.
[15] O. Loos , Symmetric spaces , vol. I, Benjamin, New-York 1969. · Zbl 0175.48601
[16] F. Rouviere , Espaces symétriques et méthode de Kashiwara-Vergne , Ann. Scient. Ec. Norm. Sup. 19 (1986) 553-581. · Zbl 0612.43012
[17] R.C. Thompson , Proof of a conjectured exponential formula , Linear and Multilinear Algebra 19 (1986) 187-197. · Zbl 0596.15025
[18] R.C. Thompson , Special cases of a matrix exponential formula , preprint, 1987. · Zbl 0655.15024
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