Let me ask some basic question on Weyl group action of standard parabolic subgroups.

Let $E/F$ be a quadratic extension of local fields and $G=U(V)$ a unitary group associated to hermition space $V$ assoicated to $E/F$. Let $P=NM$ is its standard parabolic subgroup and write $M=GL_{a_1}(E) \times \cdots GL_{a_r}(E) \times U(W)$ where $W$ is a subspace of $V$.

Let $A_0$ be the maximal split torus of $G$ and $N_G(A_0), Z_G(A_0)$ its normalizer and centralizer respecively. If $P'=N'M'$ is another associated standard parabolic subgroup, (i.e, there is a Weyl group element $w \in N_G(A_0) / Z_G(A_0)$ such that $w \cdot P=P'$, then I am wodering whether $M'$ should be of the form $GL_{b_1}(E) \times \cdots GL_{b_r}(E) \times U(W)$ and $\{b_1,\cdots,b_r\}$ is a permutation of $\{a_1,\cdots,a_r\}$?

And if $\rho=\rho_1 \boxtimes \cdots \boxtimes \rho_r \boxtimes \tau$ is a representation of $M$, the $w\cdot \rho$ is also the form of $\rho_{S(1)} \boxtimes \cdots \boxtimes \rho_{S(r)} \boxtimes \tau$, where $S$ is a permutation of $\{1,2,\cdots,r\}$?

I would appreciate if anyone answers on this question.

Thank you very much!


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