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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!

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