# Weights of restriction to a Levi subgroup

Let us consider the group $$G = \mathrm{GSp_{2g}/\mathbf{Q}}$$ with respect to the symplectic pairing given by the matrix $$\begin{pmatrix} & s \\ -s & \end{pmatrix}$$, where $$s$$ is the $$g\times g$$ anti-diagonal matrix with non-zero entries equal to $$1$$. Let $$P \subset G$$ be the Siegel parabolic given by matrices whose left lower corner is $$0$$ and let $$P = M \ltimes U$$ be its Levi decomposition, so $$M \cong \mathrm{GL}_g$$ via $$\mathrm{GL}_g \rightarrow G$$ given by $$a \mapsto \begin{pmatrix} a & 0 \\ 0 & ^ta^{-1} \end{pmatrix}$$.

Let us write the weights of the torus $$T_G$$ via $$\mathbf{Z}^{g+1} \cong X^\ast(T_G)$$ given by $$(k_1,\ldots, k_{g+1}) \mapsto (\mathrm{diag}(t_1,\ldots,t_g,\nu t_1^{-1}, \ldots, \nu t_g^{-1}) \mapsto t_1^{k_1} \cdots t_g^{k_g})$$. Let us fix the Borel $$B_G = \left\{\begin{pmatrix} a & \ast \\ 0 & b \end{pmatrix} \in G \right\}$$ where $$a$$ is upper triangular and $$b$$ is lower triangular (I am not completely sure this a natural choice, but it seems to me the one that is compatible with the usual Borel of upper triangular matrices in $$M$$).

If $$V$$ is a representation of $$G$$ we obtain, by restriction, a representation $$\tilde V$$ of $$M$$. Of course $$\tilde V$$ does not need to be irreducible even if $$V$$ is.

If $$k = (k_1,\ldots,k_g)$$ is a dominant weight of $$M$$ (with the usual description of the weight space for $$\mathrm{GL}_g$$) we have the standard irreducible representation $$\rho_k$$ of $$M$$ of highest weight $$k$$. Let us consider $$V_{k,0}$$, the irreducible representation of $$G$$ of highest weight $$k_1,\ldots,k_g,0$$ and its restriction $$\tilde V_{k,0}$$ to $$M$$. I am interested in the weights appearing in $$\tilde V_{k,0}$$. More precisely my question is the following.

Question: does the original representation $$\rho_k$$ appear in $$\tilde V_{k,0}$$? Or does the representation $$\rho_{k'}$$ appear in $$\tilde V_{k,0}$$, where $$k'=(-k_g,\ldots,-k_1)$$?

I have checked with sage and it seems to be true (I am not completely sure that I have understood how to do this with sage...). It is probably related with Frobenius reciprocity, but I am unable to prove it (the adjunct ion is the wrong direction).

Thank you!