Weyl group actions on standard parabolic subgroups of classical groups [closed]

Question

$\DeclareMathOperator\U{U}\DeclareMathOperator\GL{GL}$Let $E/F$ be a quadratic extension of local fields and $G=U(V)$ a unitary group associated to hermitian space $V$ over $E/F$. We fix a minimal parabolic subgroup $P_0$ of $G$ and call $P=NM$ a standard parabolic subgroup of $G$ if it contains $P_0$. Write $M=\GL_{a_1}(E) \times \dotsb \times \GL_{a_r}(E) \times \U(W)$, where $W$ is a subspace of $V$.

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

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

Answer 1

Not quite, since the Weyl group of the quasi-split unitary group is not the symmetric group. See page 1272 of Goldberg, "R-Groups and Elliptic Representations of Unitary Groups," http://www.jointmathematicsmeetings.org/proc/1995-123-04/S0002-9939-1995-1224616-6/S0002-9939-1995-1224616-6.pdf