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 Ω of positive definite real matrices. The highest weight representations of the group Sp ( n , R ) play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω + i Sym ( n , R ) . We then use the Laplace transform to carry the Lie algebra action over to L 2 ( Ω , d μ ν ) . The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl ( 2 , R ) .