# Weyl group actions on standard subgroups

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!