Let ${\mathcal{K}}$ be an imaginary quadratic field. Let ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ be irreducible generic cohomological automorphic representation of $\text{GL}(n)/{\mathcal{K}}$ and $\text{GL}(n-1)/{\mathcal{K}}$, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if ${\rm\Pi}$ is cuspidal and the weights of ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ are in a standard relative position, the critical values of the Rankin–Selberg product $L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$ are essentially algebraic multiples of the product of the Whittaker periods of ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$. We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal ${\rm\Pi}$ 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.

We consider twisted multivariable zeta series associated to polynomials of several variables. We introduce a new class of polynomials, namely HDF, that contains strictly non-degenerate and hypoelliptic polynomials. For polynomials belonging to the HDF class, we show that we can extend holomorphically our series to $\mathbb{C}^T$. Then, thanks to a new principle called ‘the Exchange Lemma’, we give very simple formulae for the values of our series at $T$-tuples of negative integers. Finally, we make the $p$-adic interpolation of those values. Thus, we have generalized the results of Cassou-Noguès (that she used to construct the $p$-adic $L$-functions of totally real fields) in two ways: we consider multivariable series and our series are associated to more general polynomials. In addition, our proof is completely different.