# Holomorphic functions and subelliptic heat kernels over Lie groups

### Bruce K. Driver

University of California, San Diego, United States### Leonard Gross

Cornell University, Ithaca, United States### Laurent Saloff-Coste

Cornell University, Ithaca, United States

## Abstract

A Hermitian form $q$ on the dual space, $g_{∗}$, of the Lie algebra, $g$, of a Lie group, $G$, determines a sub-Laplacian, $Δ$, 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, $e_{tΔ/4}$, 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, $u↦u^(e)$, deﬁnes a unitary map from the space of holomorphic functions in $L_{2}(G,ρ_{t})$ onto a natural Hilbert space lying in $U_{′}$.

## Cite this article

Bruce K. Driver, Leonard Gross, Laurent Saloff-Coste, Holomorphic functions and subelliptic heat kernels over Lie groups. J. Eur. Math. Soc. 11 (2009), no. 5, pp. 941–978

DOI 10.4171/JEMS/171