Branching laws for minimal holomorphic representations

In this paper we study the branching law for the restriction from SU(n,m) to SO(n,m) of the minimal representation in the analytic continuation of the scalar holomorphic discrete series. We identify the group decomposition with the spectral decomposition of the action of the Casimir operator on the subspace of S(O(n)×O(m))-invariants. The Plancherel measure of the decomposition defines an L²-space of functions, for which certain continuous dual Hahn polynomials furnish an orthonormal basis. It turns out that the measure has point masses precisely when n−m>2. Under these conditions we construct an irreducible representation of SO(n,m), identify it with a parabolically induced representation, and construct a unitary embedding into the representation space for the minimal representation of SU(n,m).     

Branching laws for discrete Wallach points

We consider the (projective) representations of the group of holomorphic automorphisms of a symmetric tube domain V⊕iΩ that are obtained by analytic continuation of the holomorphic discrete series. For a representation corresponding to a discrete point in the Wallach set, we find the decomposition under restriction to the identity component of GL(Ω). Using Riesz distributions, an explicit intertwining operator is constructed as an analytic continuation of an integral operator. The density of the Plancherel measure involves quotients of Γ-functions and the c-function for a symmetric cone of smaller rank.     

Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials

Let be a complex bounded symmetric domain of tube type in a complex Jordan algebra V and let be its real form in a formally real Euclidean Jordan algebra J⊂V; is a bounded realization of the symmetric cone in J. We consider representations of H that are gotten by the generalized Segal-Bargmann transform from a unitary G-space of holomorphic functions on to an L²-space on . We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extension to of the spherical functions on and find their expansion in terms of the L-spherical polynomials on , which are Jack symmetric polynomials. We prove that the coefficients are orthogonal polynomials in an L²-space, the measure being the Harish-Chandra Plancherel measure multiplied by the symbol of the Berezin transform. We prove the difference equation and recurrence relation for those polynomials by considering the action of the Lie algebra and the Cayley transform on the polynomials on . Finally, we use the Laplace transform to study generalized Laguerre functions on symmetric cones.     

Basic representations for Eisenstein series from their differential equations

In this paper we provide a new approach for the derivation of parameterizations for the Eisenstein series. We demonstrate that a variety of classical formulas may be derived in an elementary way, without knowledge of the inversion formulae for the corresponding Schwarzian triangle functions. In particular, we provide a new derivation for the parametric representations of the Eisenstein series in terms of the complete elliptic integral of the first kind. The proof given here is distinguished from existing elementary proofs in that we do not employ the Jacobi-Ramanujan inversion formula relating theta functions and hypergeometric series. Our alternative approach is based on a Lie symmetry group for the differential equations satisfied by certain Eisenstein series. We employ similar arguments to obtain parameterizations from Ramanujan's alternative signatures and those associated with the inversion formula for the modular J-function. Moreover, we show that these parameterizations represent the only possible signatures under a certain assumed form for the Lie group parameters.     

The Generalized Segal–Bargmann Transform and Special Functions

Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner–Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equations and recursion relations satisfied by those special functions.