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

Koufany, Khalid; Ørsted, Bent

Annales de l'Institut Fourier, Tome 46 (1996) no. 3, pp. 689-722.

Nous étudions la série discrète holomorphe de $\mathrm{SU}(2,2)$ et son prolongement analytique par restriction à la forme réelle non-compacte $\mathrm{S}\left(\mathrm{U}(1,1)\times \mathrm{U}(1,1)\right)$. Nous construisons pour cela une transformation de Cayley entre le sous semi-groupe de Ol’shanskiĭ de $\mathrm{GL}(2,ℂ)$ ayant $\mathrm{U}(1,1)$ comme frontière de Šilov et un ouvert dense du domaine tube de $\mathrm{SU}(2,2)$. Nous déterminons ainsi la décomposition de la restriction à $\mathrm{S}\left(\mathrm{U}(1,1)\times \mathrm{U}(1,1)\right)$ d’une représentation de la série discrète holomorphe. En particulier nous montrons que la représentation de $\mathrm{S}\left(\mathrm{U}(1,1)\times \mathrm{U}(1,1)\right)$ 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 $\mathrm{SU}(2,2)$. Ceci infirme un résultat de Gindikin.

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

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