Canonical and Boundary Representations on Rank One Para-Hermitian Spaces

Anatoli A. Artemov

G. R. Derzhavin Tambov State University, Tambov, Russia.

DOI: 10.4236/am.2013.411A3006                

Abstract

This work studies the canonical representations (Berezin representations) for para-Hermitian symmetric spaces of rank one. These spaces are exhausted up to the covering by spaces G/H with G = SL(n,R),H = GL(n-1,R) . For Hermitian symmetric spaces G/K, canonical representations were introduced by Berezin and Vershik-Gelfand-Graev. They are unitary with respect to some invariant non-local inner product (the Berezin form). We consider canonical representations in a wider sense: we give up the condition of unitarity and let these representations act on spaces of distributions. For our spaces G/H, the canonical representations turn out to be tensor products of representations of maximal degenerate series and contragredient representations. We decompose the canonical representations into irreducible constituents and decompose boundary representations.

Keywords

Para-Hermitian Symmetric Spaces; Overgroups; Canonical Representations; Boundary Representations; Poisson and Fourier Transforms

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. M. Vershik, I. M. Gelfand and M. I. Graev, “Representations of the Group Where Is a Ring of Functions,” Uspekhi Matematicheskikh Nauk, Vol. 28, No. 5, 1973, pp. 83-128.
[2]	F. A. Berezin, “Quantization in Complex Symmetric Spaces,” Izvestia Akademii nauk SSSR, Vol. 39, No. 2, 1975, pp. 363-402.
[3]	V. F. Molchanov, “Quantization on Para-Hermitian Symmetric Spaces,” American Mathematical Society Translations, Vol. 175, Series 2, 1996, pp. 81-95.
[4]	G. van Dijk and V. F. Molchanov, “Tensor Products of Maximal Degenerate Series Representations of the Group ,” Journal de Mathématiques Pures et Appliquées, Vol. 78, No. 1, 1999, pp. 99-119. http://dx.doi.org/10.1016/S0021-7824(99)80011-7
[5]	G. van Dijk and V. F. Molchanov, “The Berezin form for Rank One Para-Hermitian Symmetric Spaces,” Journal de Mathématiques Pures et Appliquées, Vol. 77, No. 8, 1998, pp. 747-799. http://dx.doi.org/10.1016/S0021-7824(98)80008-1
[6]	V. F. Molchanov, “Canonical and Boundary Representations on a Hyperboloid of One Sheet,” Acta Applicandae Mathematicae, Vol. 81, No. 1-3, 2004, pp. 191-214.
[7]	V. F. Molchanov, “Canonical Representations on the TwoSheeted Hyperboloid,” Indagationes Mathematicae, Vol. 16, No. 3-4, 2005, pp. 609-630.
[8]	A. A. Artemov, “Canonical and Boundary Representations on a Sphere with an Action of the Generalized Lorentz Group,” Monography, The Publishing House of TSU, Tambov, 2010, 235 p.