Ronald J. Stanke

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 D 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 ).

Analytic uniformly bounded representations of SU(1,n+1)

On construit une famille de representations uniformement bornees de SU(1,n+1) qui agissent toutes sur le meme espace de Hilbert L 2 (R en+1 )