New constructions of homogeneous operators

Abstract. New examples of homogeneous operators involving infinitely many parameters are constructed. They are realized on Hilbert spaces of holomorphic functions with reproducing kernels which are computed explicitly. All the examples are irreducible and belong to the Cowen- Douglas class. Even though the construction is completely explicit, it is based on certain facts about Hermitian holomorphic homogeneous vector bundles. These facts also make possible a description of all homogeneous Cowen- Douglas operators, in a somewhat less explicit way. An operator T on a Hilbert space is said to be homogeneous if its spectrum is contained in the closure of the unit disc D in C and if g(T) is unitarily equivalent to T for every element g of the holomorphic automorphism group G of D. There are general results about such operators, but relatively few examples are known (cf. [7, 2, 3, 1]). In this note a large family of examples is constructed and a step is made towards the description of all such operators in the Cowen-Douglas class of D (see [4]). 1. Construction of the examples Let G ̃ denote the universal covering group of G. For λ> 0, let A(λ) denote the Hilbert space of holomorphic functions on D with reproducing kernel (1 − zw̄)−2λ. The well-known discrete series D+λ of unitary representation of G ̃ acts on A (λ) by (D+λ (g)f)(z) = (∂(g−1z) ∂z)λ f(g−1(z)) with the power defined to ensure continuity on G̃ × D. Let m ≥ 1 be an integer and let λ> m2. For 0 ≤ j ≤ m we write λj = λ − m2 + j and define the operator Γj: A(λj) → Hol(D,Cm+1) by (Γjf) ` = 0 if ` < j(