We construct an L 2 -model of "very small" irreducible unitary representations of simple Lie groups G which, up to finite covering, occur as conformal groups Co(V ) of simple Jordan algebras V . If V is split and G is not of type A n , then the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction allows the case where G does not admit minimal representations. In particular, applying to Jordan algebras of split rank one we obtain the entire complementary series representations of SO(n, 1) 0 . A distinguished feature of these representations in all cases is that they attain the minimum of the Gelfand-Kirillov dimensions among irreducible unitary representations. Our construction provides a unified way to realize the irreducible unitary representations of the Lie groups in question as Schrödinger models in L 2 -spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits. In this realization the Lie algebra representations are given explicitly by differential operators of order at most two, and the key new ingredient is a systematic use of specific second-order differential operators (Bessel operators) which are naturally defined in terms of the Jordan structure.
For any Hermitian Lie group G of tube type we construct a Fock model of its minimal representation. The Fock space is defined on the minimal nilpotent K C -orbit X in p C and the L 2 -inner product involves a K-Bessel function as density. Here K ⊆ G is a maximal compact subgroup and g C = k C + p C is a complexified Cartan decomposition. In this realization the space of k-finite vectors consists of holomorphic polynomials on X. The reproducing kernel of the Fock space is calculated explicitly in terms of an I-Bessel function. We further find an explicit formula of a generalized Segal-Bargmann transform which intertwines the Schrödinger and Fock model. Its kernel involves the same I-Bessel function. Using the Segal-Bargmann transform we also determine the integral kernel of the unitary inversion operator in the Schrödinger model which is given by a J-Bessel function.
For a symmetric pair (G, H) of reductive groups we construct a family of intertwining operators between spherical principal series representations of G and H that are induced from parabolic subgroups satisfying certain compatibility conditions. The operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation. We further discuss uniqueness of intertwining operators, and for the rank one cases
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.
We study boundary value problems for some differential operators on Euclidean space and the Heisenberg group which are invariant under the conformal group of a Euclidean subspace resp. Heisenberg subgroup. These operators are shown to be self-adjoint in certain Sobolev type spaces and the related boundary value problems are proven to have unique solutions in these spaces. We further find the corresponding Poisson transforms explicitly in terms of their integral kernels and show that they are isometric between Sobolev spaces and extend to bounded operators between certain L p -spaces. The conformal invariance of the differential operators allows us to apply unitary representation theory of reductive Lie groups, in particular recently developed methods for restriction problems.
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