  Let $\mathcal K$ be an imaginary quadratic field. Let $\Pi$ and $\Pi'$ be
irreducible generic cohomological automorphic representation of
$GL(n)/{\mathcal K}$ and $GL(n-1)/{\mathcal K}$, respectively. Each of them can
be given two natural rational structures over number fields. One is defined by
the rational structure on topological cohomology, the other is given in terms
of the Whittaker model. The ratio between these rational structures is called a
{\it Whittaker period}. An argument presented by Mahnkopf and Raghuram shows
that, at least if $\Pi$ is cuspidal and the weights of $\Pi$ and $\Pi'$ are in
a standard relative position, the critical values of the Rankin-Selberg product
$L(s,\Pi \times \Pi')$ are essentially algebraic multiples of the product of
the Whittaker periods of $\Pi$ and $\Pi'$. We show that, under certain
regularity and polarization hypotheses, the Whittaker period of a cuspidal
$\Pi$ can be given a motivic interpretation, and can also be related to a
critical value of the adjoint $L$-function of related automorphic
representations of unitary groups. The resulting expressions for critical
values of the Rankin-Selberg and adjoint $L$-functions are compatible with
Deligne's conjecture.
