I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura varieties and moduli''.
My question is as follows: let $G$ be a connected reductive algebraic group over $\mathbb{R}$, $\rho:G\rightarrow GSp(V)$ a faithful algebraic representation on a finite-dimensional real vector space $V$, preserving a symplectic form $\psi$ on $V$ up to scalars. Assume that there exists a homomorphism $h:\mathbb{S}\rightarrow G$, $\mathbb{S}=Res_{\mathbb{C}/\mathbb{R}}\mathbb{G}_\mathrm{m}$, such that $\rho\circ h$ is a Hodge structure of type $\{(-1,0),(0,-1)\}$, namely a complex structure on $V$, and such that $\psi$ serves as a polarization of this complex structure. If we know further that the action of $G^{der}$ the derived group of $G$ on $V$ does not admit trivial subrepresentation of dimension $>0$, can we deduce that the connected center $C$ of $G$ acts on $V$ through a split torus over $\mathbb{R}$?
Note that if $G$ stabilizes a real subrepresentation $V'$ of $V$ on which $G^{der}$ acts trivially, then the Hodge structure on $V'$ is determined by the action of the connected center $C$ of $G$, and thus $C$ could not split over $\mathbb{R}$, otherwise $V'$ would be a sum of Tate twists. So my question is put in the inverse direction: if $G^{der}$ acts without trivial subrepresentation, does this force $C$ to be split?
An easy case to be considered is when $V$ is a simple real representation of $G$, which is given as $V=M\otimes N$ for $M$ a simple representation of $G^{der}$ and $N$ a simple representation of $C$. Here I pass from $\mathbb{S}$ to the action of Lie algebra $\mathbb{C}\rightarrow\mathfrak{g}^{der}\oplus\mathfrak{c}$, with $\mathfrak{g}$ the Lie algebra of $G$, and $\mathfrak{c}$ the Lie algebra of $C$. Since there are only two indices appearing in the Hodge type of $M\otimes N$, we deduce that one of $M$ and $N$ is of a single Hodge type, hence a Tate object, and the other is of Hodge type $\{(r-1,r),(r,r-1)\}$. And one must have $N$ to be of type $(-r,-r)$ by the assumption of $G^{der}$, which shows that $C$ acts on $V=M\otimes N$ through a split torus.
The case I fail to attack in this way is when $V$ is a subrepresentation of a representation of the form $M\otimes N$, where $M$ is a simple representation of $G^{der}$ and $N$ a simple representation of $C$. By Bourbaki Algebra Chap.8 Sect.7, we know that finite-dimensional simple modules of $\mathfrak{g}=\mathfrak{g}^{der}\oplus\mathfrak{c}$ does arise in this way. However in this case my arguments via Hodge types do not work, and it seems possible that $M\otimes N$ admits a non-trivial simple subrepresentation.
And thus I would like to know if there are other ingredients I'm missing for the split center $C$, or perhaps there are counter examples? I also expect the same situation happen for the rational case, namely $G$ comes from some Shimura subdatum in the Siegel datum $(GSp(V),H(V))$, and the claim on its connected center $C$ is modified as: $C$ splits over a totally real field.
thanks a great deal for reading my lengthy descriptions.
