Let $\pi$ be a generic $A$-parameter, that is an isobaric automorphic representation of linear group.
Decompose $\pi= \otimes \pi_v$ as a restricted tensor product. Then by the local Langlands correpondence, we can attach the corresponding Weil-Deligne representation $\phi_v$ and we can think the local component group $S_{\phi_v}$ such that $|S_{\phi_v}|$, the number of elements of $S_{\phi_v}$ is $2^{n_{\phi_v}}$. (Here, $n_{\phi_v}$ is the number of irreducible symplectic type sub-representation of $\phi_v$.)
Then I am wondering that if $\pi_v$ is unramified, then $S_{\phi_v}$ is signleton? Because, irreducible symplectic type sub-representation of $\phi_v$ correspond to essentially square-integrable representation of linear groups, but ess. sqaure-integrable representation cannot be unramified.
Please correct me if there is some mistake in the above reasoning!
