Holomorphic functions and subelliptic heat kernels over Lie groups
type: Item
instance of: scholarly article
title: Holomorphic functions and subelliptic heat kernels over Lie groups
zbMATH Open document ID: 1193.32023
DOI: 10.4171/JEMS/171
author: Bruce K. Driver
author: Leonard Gross
author: Laurent Saloff-Coste
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2009-09-29T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: Summary: A Hermitian form \(q\) on the dual space \(g^*\) of the Lie algebra \(g\) of a Lie group \(G\) determines a sub-Laplacian \(\Delta \) on \(G\). It is 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 \({\mathcal U}'\) 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 \(\hat u(e) \in{\mathcal U}'\). We show that, if \(G\) is complex, connected, and simply connected, then the ``Taylor'' map \(u \rightarrowtail \hat u(e)\) defines a unitary map from the space of holomorphic functions in \(L^{2}(G, \rho _{t})\) onto a natural Hilbert space lying in \({\mathcal U}'\).
Mathematics Subject Classification ID: 32W30
Mathematics Subject Classification ID: 35H20
Mathematics Subject Classification ID: 32C15
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 5608950
zbMATH Keywords: subelliptic
zbMATH Keywords: heat kernel
zbMATH Keywords: complex groups
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: Taylor map
Wikidata QID: Q115212478
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5608950