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Surjectivity of the Taylor map for complex nilpotent Lie groups

Published online by Cambridge University Press:  01 January 2009

BRUCE K. DRIVER ,
LEONARD GROSS  and
LAURENT SALOFF-COSTE
BRUCE K. DRIVER
Affiliation:
Department of Mathematics, 0112University of California, San Diego, La Jolla, CA 92093-0112, U.S.A. e-mail: driver@math.ucsd.edu
LEONARD GROSS
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, U.S.A. e-mail: gross@math.cornell.edu, lsc@math.cornell.edu
LAURENT SALOFF-COSTE
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, U.S.A. e-mail: gross@math.cornell.edu, lsc@math.cornell.edu
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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, Δ, on G. Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρt, on G associated to etΔ/4. In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions in L2(G, ρt) onto Jt0 (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 Jt0 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 L2(G, ρt), for appropriate t, when G is the complex Heisenberg group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 177 - 195
Copyright
Copyright © Cambridge Philosophical Society 2008

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