EXPONENTS AND SYMMETRY OF OPERATOR LEVYS PROBABILITY-MEASURES ON FINITE-DIMENSIONAL VECTOR-SPACES

We show that for a full Operator Levy's measure on a finite dimensiona l vector space there exists an exponent with suitable spectral propert ies commuting with the symmetry group of the measure. Such exponents l ead to a simple description of the symmetry group, and allow one to ob tain new (commuting or not) exponents; moreover, for them a simple rel ation exists between the symmetry group of the operator Levy's measure and the symmetry group of the mixing measure. We also show that full operator Levy's measures having ''large'' symmetry group need not be m ultivariate Levy's, correcting some earlier result.((2))