Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ attached to New- and Oldforms

Yao Cheng

Published 2021-06-23Version 1

Let $\pi$ and $\tau$ be a smooth generic representation of ${\rm SO}_{2n+1}$ and ${\rm GL}_r$ over a non-archimedean field respectively. Suppose that $\pi$ is irreducible and $\tau$ has finite length that admits a unique Whittaker model. We consider the Rankin-Selberg integrals attached to (conjecutral) new- and oldforms in $\pi$ in this paper. We show that such integrals always vanish if $\tau$ is ramified and compute the integrals when $\tau$ is unramified.