Algebras of invariant functions on the Shilov boundaries of Siegel domains
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- by Anthony H. Dooley and Genkai Zhang PDF
- Proc. Amer. Math. Soc. 126 (1998), 3693-3699 Request permission
Abstract:
Let $D=G/K$ be a bounded symmetric domain and $K/L$ the Shilov boundary of $D$. Let $\mathcal {N}$ be the Shilov boundary of the Siegel domain realization of $G/K$. We consider the case when $D$ is the exceptional non-tube type domain of the type $(\mathfrak {e}_{6(-14)}, \mathfrak {so}(10)\times \mathfrak {so}(2))$. We prove that $(\mathcal {N}\rtimes L, L)$ is not a Gelfand pair and thus resolve an open question of G. Carcano.References
- Chal Benson, Joe Jenkins, and Gail Ratcliff, Bounded $K$-spherical functions on Heisenberg groups, J. Funct. Anal. 105 (1992), no. 2, 409–443. MR 1160083, DOI 10.1016/0022-1236(92)90083-U
- C. Benson, J. Jenkins, R. L. Lipsman and G. Ratcliff, A geometric criterion for Gelfand pairs associated with the Heisenberg group, Pacific J. Math. 178 (1997), 1-36.
- Theodor Bröcker and Tammo tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985. MR 781344, DOI 10.1007/978-3-662-12918-0
- Giovanna Carcano, A commutativity condition for algebras of invariant functions, Boll. Un. Mat. Ital. B (7) 1 (1987), no. 4, 1091–1105 (English, with Italian summary). MR 923441
- Giovanna Carcano, Algebras of invariant functions on the Šilov boundary of generalized half-planes, Proc. Amer. Math. Soc. 111 (1991), no. 3, 743–753. MR 1039253, DOI 10.1090/S0002-9939-1991-1039253-2
- J. Faraut and A. Korányi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), no. 1, 64–89. MR 1033914, DOI 10.1016/0022-1236(90)90119-6
- Roger Howe and T\B{o}ru Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), no. 3, 565–619. MR 1116239, DOI 10.1007/BF01459261
- A. Hulanicki and F. Ricci, A Tauberian theorem and tangential convergence for bounded harmonic functions on balls in $\textbf {C}^{n}$, Invent. Math. 62 (1980/81), no. 2, 325–331. MR 595591, DOI 10.1007/BF01389163
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- O. Loos, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977.
- Ottmar Loos, Jordan pairs, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR 0444721, DOI 10.1007/BFb0080843
- Harald Upmeier, Jordan algebras in analysis, operator theory, and quantum mechanics, CBMS Regional Conference Series in Mathematics, vol. 67, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1987. MR 874756, DOI 10.1090/cbms/067
Additional Information
- Anthony H. Dooley
- Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, Australia
- Email: a.dooley@unsw.edu.au
- Genkai Zhang
- Affiliation: School of Mathematics, University of New South Wales, Sydney 2052, Australia
- Address at time of publication: Department of Mathematics, University of Karlstad, S-65188 Karlstad, Sweden
- Email: genkai.zhang@hks.se
- Received by editor(s): March 25, 1995
- Additional Notes: This research was sponsored by the Australian Research Council.
- Communicated by: J. Marshall Ash
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3693-3699
- MSC (1991): Primary 22E46, 32M15
- DOI: https://doi.org/10.1090/S0002-9939-98-05051-5
- MathSciNet review: 1625733