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Surjectivity of the Taylor map for complex nilpotent Lie groups. (English) Zbl 1158.22007

Authors’ summary: A Hermitian form q on the dual space, \(\mathfrak g^*\), of the Lie algebra, \(\mathfrak g\), of a simply connected complex Lie group, \(G\), determines a sub-Laplacian, \(\Delta \), on \(G\). Assuming Hörmander’s condition for hypoellipticity, there is a smooth heat kernel measure, \(\rho _{t}\), on \(G\) associated to \(e^{t\Delta /4}\). In a companion paper [the authors, Holomorphic functions and subelliptic heat kernels over Lie groups (to appear in J. European Math. Soc.)], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions in \(L^{2}(G, \rho_{t})\) onto \(J_{t}^{0}\) (a subspace of) the dual of the universal enveloping algebra of \(\mathfrak g\). 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 \(J_{t}^{0}\) 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 \(L^{2}(G, \rho _{t})\), for appropriate \(t\), when \(G\) is the complex Heisenberg group.

MSC:

22E30 Analysis on real and complex Lie groups
47B25 Linear symmetric and selfadjoint operators (unbounded)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35K05 Heat equation
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