Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program

Khalid Koufany; Bent Ørsted

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 3, page 689-722
  • ISSN: 0373-0956

Abstract

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For the scalar holomorphic discrete series representations of SU ( 2 , 2 ) and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside SU ( 2 , 2 ) . We construct a Cayley transform between the Ol’shanskiĭ semigroup having U ( 1 , 1 ) as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for SU ( 2 , 2 ) . This allows calculating the composition series in terms of harmonic analysis on U ( 1 , 1 ) . In particular we show that the Ol’shanskiĭ Hardy space for U ( 1 , 1 ) is different from the classical Hardy space for U ( 2 ) ; this provides a counterexample to a statement in a paper by Gindikin.

How to cite

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Koufany, Khalid, and Ørsted, Bent. "Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program." Annales de l'institut Fourier 46.3 (1996): 689-722. <http://eudml.org/doc/75192>.

@article{Koufany1996,
abstract = {For the scalar holomorphic discrete series representations of $\{\rm SU\}(2,2)$ and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside $\{\rm SU\}(2,2)$. We construct a Cayley transform between the Ol’shanskiĭ semigroup having $\{\rm U\}(1,1)$ as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for $\{\rm SU\}(2,2)$. This allows calculating the composition series in terms of harmonic analysis on $\{\rm U\}(1,1)$. In particular we show that the Ol’shanskiĭ Hardy space for $\{\rm U\}(1,1)$ is different from the classical Hardy space for $\{\rm U\}(2)$; this provides a counterexample to a statement in a paper by Gindikin.},
author = {Koufany, Khalid, Ørsted, Bent},
journal = {Annales de l'institut Fourier},
keywords = {Cauchy-Szegö kernel; Cayley transform; composition series; Hardy space; holomorphic discrete series; Ol’shanskiĭ semigroup},
language = {eng},
number = {3},
pages = {689-722},
publisher = {Association des Annales de l'Institut Fourier},
title = {Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program},
url = {http://eudml.org/doc/75192},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Koufany, Khalid
AU - Ørsted, Bent
TI - Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 3
SP - 689
EP - 722
AB - For the scalar holomorphic discrete series representations of ${\rm SU}(2,2)$ and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside ${\rm SU}(2,2)$. We construct a Cayley transform between the Ol’shanskiĭ semigroup having ${\rm U}(1,1)$ as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for ${\rm SU}(2,2)$. This allows calculating the composition series in terms of harmonic analysis on ${\rm U}(1,1)$. In particular we show that the Ol’shanskiĭ Hardy space for ${\rm U}(1,1)$ is different from the classical Hardy space for ${\rm U}(2)$; this provides a counterexample to a statement in a paper by Gindikin.
LA - eng
KW - Cauchy-Szegö kernel; Cayley transform; composition series; Hardy space; holomorphic discrete series; Ol’shanskiĭ semigroup
UR - http://eudml.org/doc/75192
ER -

References

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