Hilbert spaces of tensor-valued holomorphic functions on the unit ball of ℂⁿ

Abstract

We study expansion of reproducing kernels for Hilbert spaces of holomorphic functions on the unit ball in C n with values in the antisymmetric tensor of the tangent and the cotangent spaces. As an application we find the composition series for the analytic continuation of certain families of holomorphic discrete series.

Decomposition of the tensor product of two Hilbert modules

“…where, as before, we have assumed that K t is well defined for all t ∈ R. Clearly, we have the following inclusion The proof is obtained by putting together a number of lemmas which are of independent interest. Before computing the generalized Wallach set GW(B Bm ) for the Bergman kernel of the Euclidean ball B m , we point out that the result is already included in [23,Theorem 3.7], see also [19,15]. The justification for our detailed proofs in this particular case is that it is direct and elementary in nature.…”

Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Intertwining operators and Weyl transform

“…is the invariant measure on B 5 with dm(z) the Lebesgue measure; see e.g. [20]. Now the space π −kδ 0 −3λ 5 contains all W −kδ 0 -valued holomorphic polynomials f 1 , and π −3λ 5 all scalar holomorphic polynomials f 2 , since π −kδ 0 −3λ 5 are not reduction points in the analytic continuation of the holomorphic discrete series.…”

Discrete components in restriction of unitary representations of rank one semisimple Lie groups

“…scalar holomorphic discrete series, on the unit ball G/K in C n with reproducing kernel (1 − (z, w)) −µ at the reducible point µ = 0; see e.g. [11] where a reproducing kernel and its expansion are found for the space. A full decomposition under SU(n − 1, 1) of the series and their quotient can be obtained easily.…”