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Resonances and residue operators for symmetric spaces of rank one

✍ Scribed by J. Hilgert; A. Pasquale

Publisher
Elsevier Science
Year
2009
Tongue
English
Weight
195 KB
Volume
91
Category
Article
ISSN
0021-7824

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✦ Synopsis

Let X = G/K be a rank-one Riemannian symmetric space of the noncompact type and letbe the Laplace-Beltrami operator on X. We show that the resolvent operator R(z) of can be meromorphically continued across the spectrum and explicitly determine the poles, i.e. the resonances. Further we describe the residue operators in terms of finite-dimensional spherical representations of G. The result answers a question posed by M. Zworski in [M. Zworski, What are the residues of the resolvent of the Laplacian on non-compact symmetric spaces? Seminar held at the IRTG Summer School 2006, Schloss Reisensburg, 2006. Available at http://math.berkeley.edu/~zworski/reisensburg.pdf]. The rank of the residue operators is derived from a restricted root version of the Weyl dimension formula for spherical highest weight representations which we prove for arbitrary symmetric spaces of the noncompact type.

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