SYMMETRIC SPACES AND THE KASHIWARA-VERGNE METHOD
Résumé
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
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
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