Critical values of automorphic \(L\)-functions for \(\text{GL}(r)\times\text{GL}(r)\)
Let \(F\) be a totally imaginary number field and let \(G = \text{GL}_r\). Let \(\pi\) and \(\pi'\) be two irreducible cuspidal representations of \(G(\mathbb A)\). There is a well-known Rankin-Selberg integral for the L-function of \(\pi\times\pi'\), which involves an Eisenstein series attached to a maximal parabolic subgroup of \(G\). When \(\pi\) and \(\pi'\) are cohomological (i.e. contribute to the cohomology of \(G(F)\setminus G(\mathbb A)/K\)) and satisfy a condition of compatibility, the special values of the \(L\)-function can be computed by interpreting the Rankin-Selberg integral as a cup product of differential forms. This is done in the paper. The result is an expression for \(L_f(\pi\times\pi',s)\) as a product of periods for the three representations involved, divided by an Archimedean integral, times a factor from a finite extension of \(F\) determined by \(\pi\) and \(\pi'\). Rationality of cohomology was known for the cuspidal representations and is proved here for the Eisenstein cohomology. It is not proved that the Archimedean integral in the final formula does not vanish.
scientific article; zbMATH DE number 1945276