SYMMETRIC SPACES AND THE KASHIWARA-VERGNE METHOD

Abstract : Abstract The Kashiwara-Vergne method reduces the proof of a deep result in analysis on a Lie group (transferring convolution of invariant distributions from the group to its Lie algebra, by means of the exponential mapping) to checking two formal Lie brackets identities linked to the Campbell-Hausdor§ formula. First, we expound this method and a proof, for quadratic or solvable Lie algebras, of the Kashiwara-Vergne conjecture allowing to apply the method to those cases. We then extend it to a general symmetric space S = G=H . This leads to introduce a function e ( X;Y ) of two tangent vectors X;Y at the origin of S , allowing to make explicit, in the exponential chart, G -invariant di§erential operators of S , the structure of the algebra of all such operators, and an expansion of mean value operators and spherical functions. For Riemannian symmetric spaces of the noncompact type, otherwise well-known from the work of Harish-Chandra and Helgason, we compare this approach with the classical one. For rank one spaces (the hyperbolic spaces), we give an explicit formula for e ( X;Y ) . Finally, we explain a construction of e for a general symmetric space by means of Lie series linked to the Campbell-Hausdor§ formula, in the spirit of the original Kashiwara-Vergne method. Proved from this construction, the main properties of e thus link the fundamental tools of H -invariant analysis on a symmetric space to its inÖnitesimal structure. The results extend to line bundles over symmetric spaces
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François Rouvière. SYMMETRIC SPACES AND THE KASHIWARA-VERGNE METHOD. 2014, ⟨10.1007/978-3-319-09773-2⟩. ⟨hal-01331776⟩

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