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Let $Sh_{K}(G,X)$ be a Shimura variety and $Z\subset Sh_{K}(G,X)$ be a special subvariety. $Z$ is given by a Shimura sub-datum $(H,Y)$ with $H\subset G$ an algebraic subgroup which I call the $Mumford-Tate$ $group$ of the special subvariety $Z$. For $\gamma \in G(\mathbb{A}_{f})$ we have a Hecke transform $T_{\gamma}:Sh_{K}(G,X)\to Sh_{K}(G,X)$. It is well-known that $T_{\gamma}(Z)$ is again a special subvariety of $Sh_{K}(G,X)$ and hence is given by a Shimura sub-datum $(H^{\prime},Y^{\prime})$. Now my question is that can one explicitely describe the Mumford-Tate group of $T_{\gamma}(Z)$ i.e. $H^{\prime}$ in terms of the Mumford-Tate group of $Z$ i.e. $H$? For simplicity you can assume that $Sh_{K}(G,X)=A_{g}$ (moduli of principally polarized abelian varieties).

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