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Laguerre polynomials, restriction principle, and holomorphic representations of SL(2). (English) Zbl 0993.22012

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

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

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:
22E46Semisimple Lie groups and their representations
43A85Analysis on homogeneous spaces
33C45Orthogonal polynomials and functions of hypergeometric type
33C80Connections of hypergeometric functions with groups and algebras