$\DeclareMathOperator\U{U}\DeclareMathOperator\GL{GL}$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 $\{1,2,\dotsc,r\}$?



Weyl group actions on standard parabolic subgroups of classical groups
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>


