  Every permutation invariant Borel subset of the space of countable structures
is definable in $\La_{\omega_1\omega}$ by a theorem of Lopez-Escobar. We prove
variants of this theorem relative to fixed relations and fixed non-permutation
invariant quantifiers. Moreover we show that for every closed subgroup $G$ of
the symmetric group $S_{\infty}$, there is a closed binary quantifier $Q$ such
that the $G$-invariant subsets of the space of countable structures are exactly
the $\La_{\omega_1\omega}(Q)$-definable sets.
