ncts

Let $\pi$ and $\tau$ be a smooth generic representation of (split) ${\rm SO}_{2n+1}$ and ${\rm GL}_r$ over a non-archimedean field respectively with $1\le r\le n$ . Suppose that $\pi$ is irreducible and $\tau$ has finite length that admits a unique Whittaker model. Then Gross and Tsai have proposed a conjectural new and old forms theory for $\pi$ , which puts results of Casselman and Roberts-Schmidt into a single framework. On the other hand, Soudry has developed a local theory for Rankin-Selberg convolutions for generic representations of ${\rm SO}_{2n+1}\times{\rm GL}_r$ , which represent the standard tensor product $L$ -factors. In this talk, we consider the Rankin-Selberg integrals for $\pi\times\tau$ attached to (conjecutral) new and old forms in $\pi$ . We show that such integrals always vanish if $\tau$ is ramified and compute the integrals when $\tau$ is unramified.