Laguerre functions on symmetric cones and recursion relations in the real case

In this article we derive differential recursion relations for the Laguerre functions on the cone ΩΩ of positive definite real matrices. The highest weight representations of the group Sp(n,R)Sp(n,R) play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω+iSym(n,R)Ω+iSym(n,R). We then use the Laplace transform to carry the Lie algebra action over to L2(Ω,dμν)L2(Ω,dμν). The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl(2,R).sl(2,R).