Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group

Abstract

The heat kernel measure $\mu\sb{t}$ is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, ${\cal H}L\sp2(SO\sb{HS},\mu\sb{t}),$ is one of two spaces of holomorphic functions we consider. The second space, ${\cal H}L\sp2(SO(\infty)),$ consists of functions which are holomorphic on an analog of the Cameron-Martin subspace for the group. It is proved that there is an isometry from the first space to the second one. The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from ${\cal H}L\sp2(SO(\infty))$ into the Hilbert space associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups. All the results of this paper are formulated for one concrete group, the Hilbert-Schmidt complex orthogonal group, though our methods can be applied in more general situations.