Holomorphic functions and subelliptic heat kernels over Lie groups

Abstract

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 Hörmander’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, U', is non-degenerate. The subelliptic heat semigroup, _e__t_∆/4, is given by convolution by a _C_∞ probability density ρt. 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, û(e) ∈ U'. We will show that if G is complex, connected, and simply connected then the “Taylor” map, u ↦ û(e), defines a unitary map from the space of holomorphic functions in _L_2(G, ρt) onto a natural Hilbert space lying in U'.