There exist exponents of an operator stable measure which commute with every operator in the measure's symmetry group. These exponents together with a new norm lead to some simplifications in the representation of the Levy measure.
"The Symmetry Group and Exponents of Operator Stable Probability Measures." Ann. Probab. 14 (3) 1014 - 1023, July, 1986. https://doi.org/10.1214/aop/1176992455