Archimedean zeta integrals on ${\rm SO}_{2n} \times {\rm GL}_n$ and ${\rm Sp}_{2n} \times {\rm GL}_n$

Source abstract

We evaluate the archimedean Rankin-Selberg zeta integrals associated with spherical principal series representations of ${\rm SO}_{2n}(\mathbf R) \times {\rm GL}_n(\mathbf R)$ and ${\rm Sp}_{2n}(\mathbf R) \times {\rm GL}_n(\mathbf R)$, where ${\rm SO}_{2n}$ is split and $n\ge2$. We obtain explicit formulas expressing these integrals as quotients of products of gamma functions. Our approach uses Mellin-Barnes representations of spherical Whittaker functions, together with new integral identities for their Mellin transforms and transformations of Barnes integrals. In the even orthogonal case, we establish a recurrence in the rank and evaluate the integral by induction. The symplectic case then follows from an explicit identity relating the two families of zeta integrals.