  Langlands' beyond endoscopy proposal for establishing functoriality motivates
interesting and concrete problems in the representation theory of algebraic
groups. We study these problems in a setting related to the Langlands
$L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal automorphic
representation of $\mathrm{GL}_n(\mathbb{A}_F)$ where $F$ is a global field.
