Tohoku Math. J. (2) 62 (3), 427-474, (2010) DOI: 10.2748/tmj/1287148621
Bruce K. Driver, Leonard Gross, Laurent Saloff-Coste
KEYWORDS: Subelliptic, heat kernel, complex groups, universal enveloping algebra, Taylor map, 32W30, 35H20, 32C15, 43A15
Given a non-negative Hermitian form on the dual of the Lie algebra of a complex Lie group, one can associate to it a (possibly degenerate) Laplacian on the Lie group. Under Hörmander's condition on the Laplacian there exists a smooth time-dependent measure, convolution by which gives the semigroup generated by the Laplacian. Fixing a positive time, we may form the Hilbert space of holomorphic functions on the group which are square integrable with respect to this “heat kernel” measure. At the same time, under Hörmander's condition, the given Hermitian form extends to a time dependent norm on the dual of the universal enveloping algebra.
In previous work we have shown that, for each positive time, the Taylor map, which sends a holomorphic function to its set of Taylor coefficients at the identity element, is a unitary map from the previous Hilbert space of square integrable holomorphic functions onto a Hilbert space contained in the dual of the universal enveloping algebra.
The present paper is concerned with the behavior of these two families of Hilbert spaces when the Lie group is replaced by a product of complex Lie groups or by a quotient by a not necessarily normal subgroup. We obtain thereby the first example of unitarity of the Taylor map for a complex manifold which is not a Lie group. In addition, we determine the behavior of these spaces as the given Hermitian form varies.