Function spaces on the Olśhanskiĭsemigroup and the Gel'fand-Gindikin program
Annales de l'Institut Fourier, Volume 46 (1996) no. 3, p. 689-722
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.
Nous étudions la série discrète holomorphe de SU (2,2) et son prolongement analytique par restriction à la forme réelle non-compacte SU ( 1 , 1 ) × U ( 1 , 1 ). Nous construisons pour cela une transformation de Cayley entre le sous semi-groupe de Ol’shanskiĭ de GL (2,ℂ) ayant U(1,1) comme frontière de Šilov et un ouvert dense du domaine tube de SU (2,2). Nous déterminons ainsi la décomposition de la restriction à SU ( 1 , 1 ) × U ( 1 , 1 ) d’une représentation de la série discrète holomorphe. En particulier nous montrons que la représentation de SU ( 1 , 1 ) × U ( 1 , 1 ) 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 SU (2,2). Ceci infirme un résultat de 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 = {https://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, Volume 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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