# Holomorphic functions and subelliptic heat kernels over Lie groups

### Laurent Saloff-Coste

Cornell University, Ithaca, United States### Bruce K. Driver

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

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, ∆, 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*'.

## Cite this article

Laurent Saloff-Coste, Bruce K. Driver, Leonard Gross, 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