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Laguerre polynomials, restriction principle, and holomorphic representations of SL$\left(2ℝ\right)$. (English) Zbl 0993.22012

For $\alpha >0$, let ${H}_{\alpha }$ be the Bergman space of holomorphic functions in the upper half-plane $ℋ$ square-integrable with respect to the measure ${\left(\text{Im}z\right)}^{\alpha -1}dz$, where $dz$ is the Lebesgue measure; the spaces ${H}_{\alpha }$ can, in fact, be “analytically continued” to all $\alpha >-1$. The Cayley transform induces an isomorphism $c:{H}_{\alpha }\left(𝒟\right)\to {H}_{\alpha }$, $cf\left(z\right):=f\left(\frac{z-i}{z+i}\right){\left(z+i\right)}^{-\alpha -1}$, between ${H}_{\alpha }$ and the similar Bergman space on the unit disc $𝒟$ with respect to the measure const${·\left(1-|z|}^{2}{\right)}^{\alpha -1}$.

The authors consider the restriction map $R$ on ${H}_{\alpha }$ given by $Rf\left(t\right):=f\left(it\right)$, $t>0$. It turns out that $R$ is a densely defined, closed, and injective operator from ${H}_{\alpha }$ into ${L}^{2}\left({ℝ}^{+},{t}^{\alpha }dt\right)$ with dense range; hence, the partial isometry component $U$ in the polar decomposition ${R}^{*}=U\sqrt{R{R}^{*}}$ is unitary. They identify $U$ explicitly, and show that ${U}^{*}c\left({z}^{n}\right)=\text{const}·{e}^{-t}{L}_{n}^{\alpha }\left(2t\right)$, where ${L}_{n}^{\alpha }$ are the Laguerre polynomials. This fact can then be used to obtain various recurrence formulas for ${L}_{n}^{\alpha }$.

The main idea underlying all these developments is the so-called “restriction principle” of B.Ørsted and the third author. For more information and other applications of this principle, see the papers of B. Ørsted and G. Zhang [Indiana Univ. Math. J. 43, 551-583 (1994; Zbl 0805.46053)] and G. Ólafsson and B. Ørsted [in: Lie theory and its applications in physics (Clausthal, 1995; Zbl 0916.22006)].

##### MSC:
 22E46 Semisimple Lie groups and their representations 43A85 Analysis on homogeneous spaces 33C45 Orthogonal polynomials and functions of hypergeometric type 33C80 Connections of hypergeometric functions with groups and algebras