Surjectivity of the Taylor map for complex nilpotent Lie groups

Abstract

AbstractA Hermitian formqon the dual space,*, of the Lie algebra,, of a simply connected complex Lie group,G, determines a sub-Laplacian, Δ, onG. Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρt, onGassociated toetΔ/4. In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions inL2(G, ρt) ontoJt0(a subspace of) the dual of the universal enveloping algebra of. Here we give a very different proof of the surjectivity of the Taylor map under the assumption thatGis nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense inJt0when the Lie algebra is graded and the Laplacian is adapted to the gradation. We also show how the Fourier–Wigner transform produces a natural family of holomorphic functions inL2(G, ρt), for appropriatet, whenGis the complex Heisenberg group.