  Let F be a totally real number field, n a prime integer, and G a unitary
group of rank n defined over F that is compact at every infinite place. We
prove an asymptotic formula for the number of cuspidal automorphic
representations of G whose factors at finitely many places are prescribed up to
inertia. The results and the methods used are a generalization to this setting
of those used by Weinstein for GL_2.
