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

摘要

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+iSym(n,R). We then use the Laplace transform to carry the Lie algebra action over to L2(Ω,dμν). The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl(2,R).