Fourier Transform of Symmetric Gauss Measures on the Heisenberg Group

An explicit form is derived for the Fourier transform of symmetric Gauss measures on the Heisenberg group at the Schrödinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two symmetric Gauss measures to be a symmetric Gauss measure and for commutability of two symmetric Gauss measures. Moreover, necessary and sufficient conditions are presented for the convolution of two symmetric Gauss convolution semigroups to be a convolution semigroup. May 8, 2001