EVENT DETAILS AND ABSTRACT

Title: Holomorphic functions and subelliptic heat kernels over Lie groups
Speaker: Professor Bruce Driver
Speaker Info: University of California, San Diego
Brief Description:
Special Note:
Abstract:

Abstract: (Based on joint work with Laurent Saloff-Coste and Leonard Gross) A Hermitian form $q$ on the dual space, $\mathfrak{g}^{\ast}$, of a Lie algebra, $\mathfrak{g},$ of a Lie group, $G,$ determines a Laplacian, $\Delta,$ on $G.$ Assuming H\"{o}rmander's condition for hypoellipticity, the subelliptic heat semigroup, $e^{t\Delta/4},$ is given by convolution by a $C^{\infty}$ probability density $\rho_{t}.$ Analogous to earlier work in the strongly elliptic case, we are able to show that if $G$ is complex, connected, and simply connected then the Taylor expansion defines a unitary map from the space of holomorphic functions in $L^{2}\left( G,\rho_{t}\right) $ onto (a subspace of) the dual of the universal enveloping algebra in the norm induced by $q.$ This work is related to an extension of the bosonic Fock space to the noncommutative Lie group setting.

(Despite this technical abstract, the majority of this talk will be a \textbf{gentle} introduction to the subject!)