Symmetric spaces and the Kashiwara Vergne method

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SYMMETRIC SPACES AND THE KASHIWARA-VERGNE METHOD Reykjavik, August 15, 2007 François Rouvière (Université de Nice, France) til Sigur?ar Helgasonar me? ?ökkum, vinsemd og vir?ingu Introduction Let G=K denote a Riemannian symmetric space of the non-compact type. The action of the G-invariant di?erential operators on G=K on the radial functions on G=K is isomorphic with the action of certain di?erential operators with constant coe¢ cients. The isomorphism in question is used in Harish- Chandra?s work on the Fourier analysis on G and is related to the Radon transform on G=K. For the case when G is complex a more direct isomorphism of this type is given, again as a consequence of results of Harish-Chandra. This is a quote from S. Helgason [8] Fundamental solutions of invariant di?erential opera- tors on symmetric spaces (1964), one of the ?rst mathematical papers I studied. Those two results were fascinating to me, they still are and they motivated my everlasting interest in invariant di?erential operators as well as Radon transforms. Yet I was dreaming simpler proofs could be given, without relying on Harish-Chandra?s deep study of semisimple Lie groups... Then, in the fall of 1977, came a preprint by Kashiwara and Vergne [11] The Campbell- Hausdor? formula and invariant hyperfunctions, showing that similar results could be obtained - for solvable Lie groups at least - only by means of elementary (but very clever) computations with the exponential mapping and the Campbell-Hausdor? formula