Differential Recursion Relations for Laguerre Functions on Symmetric Cones

Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In [2, 5, 6, 8] we considered the family of generalized Laguerre functions on Ω that generalize the classical Laguerre functions on R. This family forms an orthogonal basis for the subspace of L-invariant functions in L(Ω, dμν), where dμν is a certain measure on the cone and where L is the group of linear transformations on V that leave the cone Ω invariant and fix the identity in Ω. The space L(Ω, dμν) supports a highest weight representation of the group G of holomorphic diffeomorphisms that act on the tube domain T (Ω) = Ω + iV. In this article we give an explicit formula for the action of the Lie algebra of G and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on R. Introduction It is a general understanding that special functions are closely related to representation theory of special Lie groups. Special functions are realized as coefficient functions of the representation and the action of the Lie algebra is used to derive differential equations and recursion relations satisfied by those functions. Standard references to this philosophy are the works of Vilenkin and Klimyk [24, 25]. We also refer the interested reader to the text [10], the recent text [1], and the work of T. Koornwinder. The present article reflects these general philosophies. In particular, we conclude our work on the connection between generalized Laguerre functions, highest weight representations and Jordan algebras, [2, 5, 6, 7, 8]. The classical Laguerre functions ln form an orthogonal basis for the Hilbert space L(R, xdx), λ > 0. As far as we have been able to trace, the first generalizations of the Laguerre functions and polynomials is from 1935 in the work of F. Tricomi [23]. Later, in 1955, C. S. Herz [12] considered generalized Laguerre functions in the context of Bessel functions on the space of complex m ×m-matrices. The Laguerre polynomials are defined on the cone of positive definite complex matrices in terms of the generalized 2000 Mathematics Subject Classification. Primary: 33C45; Secondary: 43A85 .