On the space of SO(d,C)-invariant holomorphic functions which are square-integrable with respect to a Gaussian measure

Abstract

The Segal-Bargmann space HL2(Cd,Ut ) is the space of all holomorphic functions on Cd which are square-integrable with respect to a Guassian measure. Here, we study the space HL2(Cd,Ut )SO(d,C) consisting of all functions f in HL2(Cd,Ut ) which are invariant under the action of the special complex orthogonal group SO(d, C). It is a closed subspace of HL2(Cd,Ut ) , Hence, it is a Hilbert space. In this work, we investigate some of its properties such as an orthonormal basis, reproducing kernel and pointwise bound.