Holomorphic functions and subelliptic heat kernels over Lie groups

Abstract

A Hermitian form $q$ on the dual space, ${\mathfrak{g}}^{\ast }$, of the Lie algebra, $\mathfrak{g}$, of a Lie group, $G$, determines a sub-Laplacian, $\Delta$, 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, ${\mathcal{U}}^{\prime }$, is non-degenerate. The subelliptic heat semigroup, $e^{t\Delta /4}$, is given by convolution by a $C^{\infty }$ probability density ${\rho }_t$. When $G$ is complex and $u:G\to \mathbb{C}$ is a holomorphic function, the collection of derivatives of $u$ at the identity in $G$ gives rise to an element, $\hat{u}(e)\in {\mathcal{U}}^{\prime }$. We will show that if $G$ is complex, connected, and simply connected then the “Taylor” map, $u\mapsto \hat{u}(e)$, defines a unitary map from the space of holomorphic functions in $L^2(G,{\rho }_t)$ onto a natural Hilbert space lying in ${\mathcal{U}}^{\prime }$.