Differential recursion relations for Laguerre functions on symmetric cones

Let Ω be a symmetric cone and V the corresponding simple Euclidean Jordan algebra. In our previous papers (some with G. Zhang) 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 L2 (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 L2 (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+. © 2005 Elsevier SAS. All rights reserved