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Symmetric Spaces and the Kashiwara-Vergne Method (Paperback, 2014 ed.)
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Symmetric Spaces and the Kashiwara-Vergne Method (Paperback, 2014 ed.)
Series: Lecture Notes in Mathematics, 2115
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Gathering and updating results scattered in journal articles over
thirty years, this self-contained monograph gives a comprehensive
introduction to the subject. Its goal is to: - motivate and explain
the method for general Lie groups, reducing the proof of deep
results in invariant analysis to the verification of two formal Lie
bracket identities related to the Campbell-Hausdorff formula (the
"Kashiwara-Vergne conjecture"); - give a detailed proof of the
conjecture for quadratic and solvable Lie algebras, which is
relatively elementary; - extend the method to symmetric spaces;
here an obstruction appears, embodied in a single remarkable object
called an "e-function"; - explain the role of this function in
invariant analysis on symmetric spaces, its relation to invariant
differential operators, mean value operators and spherical
functions; - give an explicit e-function for rank one spaces (the
hyperbolic spaces); - construct an e-function for general symmetric
spaces, in the spirit of Kashiwara and Vergne's original work for
Lie groups. The book includes a complete rewriting of several
articles by the author, updated and improved following Alekseev,
Meinrenken and Torossian's recent proofs of the conjecture. The
chapters are largely independent of each other. Some open problems
are suggested to encourage future research. It is aimed at graduate
students and researchers with a basic knowledge of Lie theory.
General
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