No abstract

The minimal representation π of the indefinite orthogonal group O(m + 1, 2) is realized on the Hilbert space of square integrable functions on R m with respect to the measure |x| −1 dx 1 · · · dx m . This article gives an explicit integral formula for the holomorphic extension of π to a holomorphic semigroup of O(m + 3, C) by means of the Bessel function. Taking its 'boundary value', we also find the integral kernel of the 'inversion operator' corresponding to the inversion element on the Minkowski space R m,1 . (2000) : Primary 22E30; Secondary 22E45, 33C10 35J10, 43A80, 43A85, 47D05, 51B20. Mathematics Subject Classifications

The minimal representation π of O(p, q) (p + q: even) is realized on the Hilbert space of square integrable functions on the conical subvariety of R p+q−2 . This model presents a close resemblance of the Schrödinger model of the Segal-Shale-Weil representation of the metaplectic group. We shall give explicit integral formulas for the 'inversion' together with the analytic continuation to a certain semigroup of O(p + 2, C) of the minimal representation of O(p, 2) by using Bessel functions.

No abstract

We develop a theory of 'special functions' associated to a certain fourth order differential operator Dµ,ν on R depending on two parameters µ, ν. For integers µ, ν ≥ −1 with µ + ν ∈ 2N0 this operator extends to a self-adjoint operator on L 2 (R+, x µ+ν+1 dx) with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, L 2 -norms, integral representations and various recurrence relations.This fourth order differential operator Dµ,ν arises as the radial part of the Casimir action in the Schrödinger model of the minimal representation of the group O(p, q), and our 'special functions' give K-finite vectors.2000 Mathematics Subject Classification: Primary 33C45; Secondary 22E46, 34A05, 42C15.