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The indefinite orthogonal group G = O(p, q) has a distinguished infinite dimensional irreducible unitary representation π for p+ q even and greater than 4, which is the “smallest” in the sense that the Gelfand–Kirillov dimension of π attains its (positive) minimum value p + q − 3 among the unitary dual of G. Moreover, π is the minimal representation if p+ q(More)
We introduce a generalization of the Fourier transform, denoted by FC , on the isotropic cone C associated to an indefinite quadratic form of signature (n1, n2) on R (n = n1 + n2: even). This transform is in some sense the unique and natural unitary operator on L2(C), as is the case with the Euclidean Fourier transform FRn on L 2(Rn). Inspired by recent(More)
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(R+, x μ+ν+1 dx) with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions,(More)
We introduce orthogonal polynomials M j (x) as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters μ ∈ C and l ∈ N0. These polynomials arise as K-finite vectors in the L-model of the minimal unitary representations of indefinite orthogonal groups, and reduce to the classical Laguerre polynomials Lμj (x)(More)
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