## Fourier transform of symmetric Gauss measures on the Heisenberg group.(English)Zbl 0989.60006

An explicit form is derived for the Fourier transform of symmetric Gauss measures on the Heisenberg group for 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 the commutability of two symmetric Gauss measures. It turns out that the convolution of two symmetric Gauss measures is almost never a symmetric Gauss measure. This contrasts the well know case of Gauss measures on Euclidean spaces.

### MSC:

 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60E07 Infinitely divisible distributions; stable distributions
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