A rationality result for the exterior and the symmetric square $L$-function

Let $G={\rm GL}_{2n}$ over a totally real number field $F$ and $n\geq 2$. Let $\Pi$ be a cuspidal automorphic representation of $G(\mathbb A)$, which is cohomological and a functorial lift from SO$(2n+1)$. The latter condition can be equivalently reformulated that the exterior square $L$-function of $\Pi$ has a pole at $s=1$. In this paper, we prove a rationality result for the residue of the exterior square $L$-function at $s=1$ and also for the holomorphic value of the symmetric square $L$-function at $s=1$ attached to $\Pi$. On the way, we also show a rationality result for the residue of the Rankin--Selberg $L$-function at $s=1$, which is very much in the spirit of our recent joint paper with Harris and Lapid, as well as of one of the main results in a recent article of Balasubramanyam--Raghuram.