Abstract
We give an elementary unified approach to rank one symmetric spaces of the noncompact type, including proofs of their basic properties and of their classification, with the development of a formalism to facilitate future computations.
Our approach is based on the theory of Lie groups of H-type. An algebraic condition of H-type algebras, called J2,is crucial in the description of the symmetric spaces. The classification of H-type algebras satisfying J2 leads to a very simple description of the rank one symmetric spaces of the noncompact type.
We also prove Kostant’s double transitive theorem; we describe explicitly the Riemannian metric of the space and the standard decompositions of its isometry group.
Examples of the use of our theory include the description of the Poisson kernel and the admissible domains for convergence of Poisson integrals to the boundary.
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References
Corlette, K. Hausdorff dimension of limit sets I,Invent. Math.,102, 521–542, (1990).
Cowling, M. Unitary and uniformly bounded representations of some simple Lie groups, inHarmonic Analysis and Group Representations, 49–128; C.I.M.E. II ciclo 1980. Liguori, Napoli, 1982.
Cowling, M. and Haagerup, U. Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one,Invent. Math.,96, 507–549, (1989).
Cowling, M., Dooley, A.H., Korányi, A., and Ricci, F.H-type groups and Iwasawa decompositions,Adv. Math.,87, 1–41, (1991).
Damek, E. The geometry of a semi-direct extension of a Heisenberg type nilpotent group,Coll. Math.,53, 255–268, (1987).
Damek, E. and Ricci, F. Harmonic analysis on solvable extension ofH -type groups,J. Geom. Analysis,2, 213–248, (1992).
Dooley, A.H. and Ricci, F. The contraction ofK to −NM, J. Funct Anal.,63, 344–368, (1985).
Faraut, J. Analyse harmonique sur les espaces riemanniens symétriques de rang un, CIMPA Ecole d’Eté, 1980.
Gindikin, S. and Karpelevič, S. Plancherel measure of Riemannian symmetric spaces of non-positive curvature,Dokl. Akad. Nauk SSSR,145, 252–255, (1962).
Helgason, S. A duality for symmetric spaces with applications to group representations,Adv. Math.,5, 1–54, (1970).
Helgason, S.Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, Academic Press, New York, 1978.
Helgason, S.Groups and Geometric Analysis, Pure and Applied Mathematics, Academic Press, New York, 1984.
Heintze, E. On homogeneous manifolds of negative curvature,Math. Annalen,214, 23–34, (1974).
Johnson, K.D. Composition series and intertwining operators for the spherical principal series II,Trans. Am. Math. Soc.,215, 269–283, (1976).
Johnson, K.D. and Wallach, N. Composition series and intertwining operators for the spherical principal series I,Trans. Am. Math. Soc.,229, 131–173, (1977).
Kaplan, A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms,Trans. Am. Math. Soc.,258, 147–153, (1980).
Kaplan, A. and Ricci, F. Harmonic analysis on groups of Heisenberg type. InHarmonic Analysis, 416–435; Lecture Notes in Math., Springer-Verlag,992, 1983.
Korányi, A. Harmonic functions on Hermitian hyperbolic space,Trans. Am. Math. Soc.,135, 509–516, (1969).
Korányi, A. Boundary behaviour of Poisson integrals on symmetric spaces,Trans. Am. Math. Soc.,190, 393–409, (1969).
Korányi, A. Geometric properties of Heisenberg-type groups,Adv. Math.,56, 28–38, (1985).
Kostant, B. On the existence and irreducibility of certain series of representations, inLie Groups and their Representations, Gelfand, I.M., Ed., John Wiley & Sons, New York, 1975.
Mostow, G.D.Strong Rigidity of Locally Symmetric Spaces, Princeton University Press, Princeton, NJ, 1973.
Mostow, G.D. A remark on quasiconformal mappings on Carnot groups,Michigan Math. J.,41, 31–37, (1994).
Riehm, C. The automorphism group of a composition of quadratic forms,Trans. Am. Math. Soc.,269, 403–414, (1982).
Rudin, W.Function Theory in the Unit Ball in C n, Springer-Verlag, New York, 1980.
Schiffmann, G. Travaux de Kostant sur la série principale. InAnalyse harmonique sur les groupes de Lie II, Lecture Notes in Math.,739, 460–510, Springer-Verlag, New York, 1979.
Takahashi, R. Quelques résultats sur l’analyse harmonique dans l’espace symétrique non compact de rang un du type exceptionnel. InAnalyse harmonique sur les groupes de Lie II, Lecture Notes in Math.,739, 511–567. Springer-Verlag, New York, 1979.
Takeuchi, M. Cell decompositions and Morse inequalities on certain symmetric spaces,J. Fac. Sci. Univ. Tokyo, Sec. I,12, 81–192, (1965).
Wallach, N.Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, New York, 1973.
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Cowling, M., Dooley, A., Korányi, A. et al. An approach to symmetric spaces of rank one via groups of Heisenberg type. J Geom Anal 8, 199–237 (1998). https://doi.org/10.1007/BF02921641
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DOI: https://doi.org/10.1007/BF02921641