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

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.