On Classical groups detected by the tensor third representation

Motivated by the Langlands' beyond endoscopy proposal for establishing functoriality, we study the representation $\otimes^3$ in a setting related to the Langlands $L$-functions $L(s,π,\,\otimes^3),$ where $π$ is a cuspidal automorphic representation of $G$ where $G$ is either $\mathrm{SO}(2n+1)$, $\mathrm{Sp}(2n)$ and $\mathrm{SO}(2n)$. In particular, under what conditions on partitions $λ$, we examine whether or not $\otimes^3$ detects the subgroups $\mathbb{S}_{[λ]}(G)$ for $G$ with type $B_n$ and $D_{2n}$ or $\mathbb{S}_{\langleλ\rangle}(G)$ for $G$ with type $C_n$. Here $\mathbb{S}_{[λ]}$ and $\mathbb{S}_{\langleλ\rangle}$ are the usual Schur functors associated to the partition $λ$.

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