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- Title
Surjectivity of the Taylor map for complex nilpotent Lie groups.
- Authors
DRIVER, BRUCE K.; GROSS, LEONARD; SALOFF-COSTE, LAURENT
- Abstract
A Hermitian form q on the dual space, - *, of the Lie algebra, - , of a simply connected complex Lie group, G , determines a sub-Laplacian, ∆, on G . Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρ - t , on G associated to e - t ∆/4 . In a companion paper [ 6 ], we proved the existence of a unitary "Taylor" map from the space of holomorphic functions in L 2 ( G , ρ - t ) onto J - t0 (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 that G is nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense in J - t0 when 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 in L 2 ( G , ρ - t ), for appropriate t , when G is the complex Heisenberg group.
- Subjects
HERMITIAN forms; LIE groups; LIE algebras; SYMMETRIC spaces; TOPOLOGICAL groups; MATHEMATICAL forms; HOLOMORPHIC functions; HYPOELLIPTIC operators; UNIVERSAL algebra
- Publication
Mathematical Proceedings of the Cambridge Philosophical Society, 2009, Vol 146, Issue 1, p177
- ISSN
0305-0041
- Publication type
Article
- DOI
10.1017/S0305004108001692