The unramified computation of Rankin-Selberg integrals for SO_2l × GL_n

Let SO_2l be the special orthogonal group, either split or quasi-split over a number field, and 1 < l < n. We compute the local integral, where data are unramified, derived from the global Rankin-Selberg construction for SO_2l × GL_n. In the general case, the local integral is difficult to compute directly, so instead it is transformed to an integral related to a construction for SO_2n+1×GL_n, which carries a Bessel model on SO_2n+1. For the quasisplit case, when l = n - 1 we are able to compute the local integral, by a modification of our recently introduced approach using invariant theory. This leads to another proof of our result for 1 < l < n, as well as a new proof of a known result regarding the unramified Bessel function.