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 on the dual space, , of the Lie algebra, , of a Lie group, , determines a sub-Laplacian, , on . 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 on the dual of the universal enveloping algebra, , is non-degenerate. The subelliptic heat semigroup, , is given by convolution by a probability density . When is complex and is a holomorphic function, the collection of derivatives of at the identity in gives rise to an element, . We will show that if is complex, connected, and simply connected then the “Taylor” map, , defines a unitary map from the space of holomorphic functions in onto a natural Hilbert space lying in .

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