Nous étudions la série discrète holomorphe de et son prolongement analytique par restriction à la forme réelle non-compacte . Nous construisons pour cela une transformation de Cayley entre le sous semi-groupe de Ol’shanskiĭ de ayant comme frontière de Šilov et un ouvert dense du domaine tube de . Nous déterminons ainsi la décomposition de la restriction à d’une représentation de la série discrète holomorphe. En particulier nous montrons que la représentation de dans l’espace de Hardy construit par Gel’fand et Gindikin n’est pas obtenue par restriction d’une représentation de la série discrète holomorphe de . Ceci infirme un résultat de Gindikin.
For the scalar holomorphic discrete series representations of and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside . We construct a Cayley transform between the Ol’shanskiĭ semigroup having as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for . This allows calculating the composition series in terms of harmonic analysis on . In particular we show that the Ol’shanskiĭ Hardy space for is different from the classical Hardy space for ; this provides a counterexample to a statement in a paper by Gindikin.
@article{AIF_1996__46_3_689_0,
author = {Koufany, Khalid and \O rsted, Bent},
title = {Function spaces on the Ol\'shanski\u\i semigroup and the Gel'fand-Gindikin program},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {46},
number = {3},
year = {1996},
pages = {689-722},
doi = {10.5802/aif.1528},
zbl = {0853.22010},
language = {en},
url = {aif.centre-mersenne.org/item/AIF_1996__46_3_689_0/}
}
Koufany, Khalid; Ørsted, Bent. Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program. Annales de l'Institut Fourier, Tome 46 (1996) no. 3, pp. 689-722. doi : 10.5802/aif.1528. https://aif.centre-mersenne.org/item/AIF_1996__46_3_689_0/
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