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The residues of the resolvent on Damek-Ricci spaces

Author(s): R. J. Miatello; C. E. Will
Journal: Proc. Amer. Math. Soc. 128 (2000), 1221-1229.
MSC (1991): Primary 22E30
Posted: October 18, 1999
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Abstract: We determine the poles and residues of the resolvent kernel of the Laplacian on a Damek-Ricci space $S.$ We show that all poles are simple and the residues define convolution operators of finite rank. This generalizes a result of Guillopé-Zworski for the real hyperbolic $n$-space. If $S$ corresponds to a symmetric space of negative curvature $G/K$, the image of each residue is a ${\frak g}_c$-module with a specific highest weight. We compute the dimension by the Weyl dimension formula.

References:

[1]
ANKER J.P., DAMEK E., YACOUB CH., Spherical analysis on harmonic $AN$ groups, Annali della Scuola Normale Sup. di Pisa 23 (1996), 643-679. MR 99a:22014

[2]
DAMEK E., RICCI F., Harmonic analysis on solvable extensions of H-type groups, Journal of Geometric Analysis 2 (1992), 213-248. MR 93d:43006

[3]
GUILLOPÉ L., ZWORSKI M., Polynomial bounds for the number of resonances for some complete spaces of constant negative curvature near infinity, Asymptotic Analysis 11 (1995), 1-22.

[4]
HELGASON S., Groups and Geometric Analysis, Pure and Applied Math 113, Academic Press, 1984. MR 86c:22017

[5]
MIATELLO R.J., On the Plancherel measure for linear Lie groups of rank one, Manuscripta Math. 29 (1979), 249-276. MR 80h:22021

[6]
MIATELLO R.J., WALLACH N.R., The resolvent of the Laplacian on negatively curved locally symmetric spaces of finite volume, Jour. Diff. Geometry 36 (1992), 663-698. MR 93i:58160

[7]
WALLACH N. R., Real Reductive Groups I, Pure and Applied Math 132, Academic Press, 1988. MR 89i:22029

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Additional Information:

R. J. Miatello
Affiliation: Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
Email: miatello@mate.uncor.edu

C. E. Will
Affiliation: Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
Email: cwill@mate.uncor.edu

DOI: 10.1090/S0002-9939-99-05498-2
PII: S 0002-9939(99)05498-2
Received by editor(s): May 27, 1998
Posted: October 18, 1999
Additional Notes: This research was partially supported by Conicet, Conicor, SecytUNC (Córdoba), and I.C.T.P. (Trieste)
Communicated by: Roe Goodman
Copyright of article: Copyright 2000, American Mathematical Society

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