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Rouvière, François

Espaces symétriques et méthode de Kashiwara-Vergne. (Symmetric spaces and the Kashiwara-Vergne method). (French) Zbl 0612.43012

Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 553-581 (1986).
The author studies a class of symmetric spaces G/H for which the exponential mapping transforms convolution of invariant distributions into convolution on the tangent space at the origin. This holds, e.g., if G is solvable. On such spaces, invariant differential operators have an invariant fundamental solution, and harmonic analysis reduces to the usual Fourier transform on the tangent space.
Reviewer: P.Godin

Cited in 12 Documents

MSC:

43A85 Harmonic analysis on homogeneous spaces
58J70 Invariance and symmetry properties for PDEs on manifolds
53C35 Differential geometry of symmetric spaces

Keywords:

Campbell-Hausdorff formula; symmetric spaces; invariant distributions; invariant differential operators
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References:

[1] Y. BENOIST , Sur l’algèbre des opérateurs différentiels invariants sur un espace symétrique nilpotent (C.R. Acad. Sc., Paris, vol. 295, 1982 , p. 59-62). MR 84k:17009 | Zbl 0501.17005 · Zbl 0501.17005
[2] Y. BENOIST , Analyse harmonique sur les espaces symétriques nilpotents (J. Funct. Anal., vol. 59, 1984 , p. 211-253). MR 86i:22014 | Zbl 0555.43012 · Zbl 0555.43012 · doi:10.1016/0022-1236(84)90073-9
[3] M. DUFLO , Opérateurs différentiels bi-invariants sur un groupe de Lie , (Ann. E.N.S., vol. 10, 1977 , p. 265-288). Numdam | MR 56 #3188 | Zbl 0353.22009 · Zbl 0353.22009
[4] M. DUFLO , Opérateurs différentiels invariants sur un espace symétrique (C.R. Acad. Sc., Paris, vol. 289, 1979 , p. 135-137). MR 82e:58104 | Zbl 0419.43012 · Zbl 0419.43012
[5] J. J. DUISTERMAAT , J. A. C. KOLK et V. S. VARADARAJAN , Spectra of compact locally symmetric manifolds of negative curvature (Invent. Math., vol. 52, 1979 , p. 27-93). MR 82a:58050a | Zbl 0434.58019 · Zbl 0434.58019 · doi:10.1007/BF01389856
[6] S. HELGASON , Fundamental solutions of invariant differential operators on symmetric spaces (Amer. J. Math., vol. 86, 1964 , p. 565-601). MR 29 #2323 | Zbl 0178.17001 · Zbl 0178.17001 · doi:10.2307/2373024
[7] S. HELGASON , Differential geometry, Lie groups, and symmetric spaces , Academic Press, New York, 1978 . MR 80k:53081 | Zbl 0451.53038 · Zbl 0451.53038
[8] S. HELGASON , Groups and geometric analysis , Acad. Press, New York, 1984 . MR 86c:22017 | Zbl 0543.58001 · Zbl 0543.58001
[9] M. KASHIWARA et M. VERGNE , The Campbell-Hausdorff formula and invariant hyperfunctions (Invent. Math., vol. 47, 1978 , p. 249-272). MR 58 #11232 | Zbl 0404.22012 · Zbl 0404.22012 · doi:10.1007/BF01579213
[10] G. LION , Résolubilité d’opérateurs différentiels semi-invariants sur l’espace d’une représentation induite (C.R. Acad. Sc., Paris, vol. 300, 1985 , p. 535-538). MR 86f:58164 | Zbl 0587.58053 · Zbl 0587.58053
[11] O. LOOS , Symmetric spaces-1 ; Benjamin, New York, 1969 . MR 39 #365a | Zbl 0175.48601 · Zbl 0175.48601
[12] M. RAÏS , Solutions élémentaires des opérateurs différentiels bi-invariants sur un groupe de Lie nilpotent (C.R. Acad. Sc., Paris, vol. 273, 1971 , p. 495-498). MR 44 #6908 | Zbl 0236.46047 · Zbl 0236.46047
[13] F. ROUVIÈRE , Démonstration de la conjecture de Kashiwara-Vergne pour l’algèbre sl(2) (C.R. Acad. Sc., Paris, vol. 292, 1981 , p. 657-660). MR 82m:17001 | Zbl 0467.22011 · Zbl 0467.22011
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