Holomorphic functions and subelliptic heat kernels over Lie groups

A Hermitian form q on the dual space, g , of the Lie algebra, g; of a Lie group, G; determines a sub-Laplacian, ; on G: It will be shown that Hormander's condition for hypoellipticity of the sub-Laplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universal enveloping algebra, U0, is nondegenerate. The subelliptic heat semigroup, et =4; is given by convolution by a C1 probability density t: When G is complex and u : G ! C is a holomorphic function, the collection of derivatives of u at the identity in G gives rise to an element, ^u(e) 2