Gradient-type differential operators and unitary highest weight representations of SU( p, q)

Abstract

Let C p + q be equipped with a hermitian form of signature ( p, q) and let SU( p, q) denote the subgroup of the corresponding invariance group U( p, q) consisting of matrices with determinant 1. To certain highest weights λ, we associate a first-order group invariant linear differential operator D λ; whose kernel contains a unitary highest weight representation with highest weight λ. The Fock model realization of unitary highest weight representations of U( p, q) is the fundamental tool used to implement this construction. The operator D λ is shown to be equivalent to an operator D ̃ λ which acts on Hol( G K , H λ) , the space of holomorphic vector valued functions defined on G K . We identify a set Λ 1 of highest weights such that Ker( D λ) is a proper subspace of Hol( G K , H λ) and show that those λ in Λ 1 correspond to points occurring at the far right of the discrete set in the classification scheme of Enright, Howe, and Wallach. First-order differential equations arising from this proper containment are explicitly derived from the operator D̃ λ. We illustrate the fundamental nature of these first-order equations by deriving from them a system which completely determines the irreducible spaces for ladder representations of SU( p, q).