Symmetric stable processes on amenable groups

We show that if G is a countable amenable group, then every stationary non-Gaussian symmetric alpha-stable (S alpha S) process indexed by G is ergodic if and only if it is weakly mixing, and it is ergodic if and only if its Rosinski minimal spectral representation is null. This extends previous results for Zd, and answers a question of P. Roy on discrete nilpotent groups in the range of all countable amenable groups. As a result, we construct on the Heisenberg group and on many Abelian groups, for all alpha is an element of (0, 2), stationary S alpha S processes that are weakly mixing but not strongly mixing.