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 ρ 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) ∈ 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, ρ t ) onto a natural Hilbert space lying in U .