Let $V$ be a finite dimensional $\mathbf{C}$-vector space with a symplectic (non-degenerate anti-symmetric bilinear) form $\omega: V\times V \to \mathbf{C}$ and a symplectic $\mathrm{SL}_2$-representation on $V$, i.e. a homomorphism of algebraic groups $\rho: \mathrm{SL}_2(\mathbf{C}) \to \mathrm{Sp}(V, \omega)$.
Let $V = V^{-n} \supseteq V^{-n+1} \supseteq \cdots \supseteq V^{n-1}\supseteq V^n = 0$ be a decreasing filtration such that $(V^i)^{\perp} = V^{-i}$ for each $i$, where the orthogonal complement is taken with respect to $\omega$. Let $\{e, h, f\}$ be the standard basis of $\mathfrak{sl}_2$ so that $[e,f] = h$, $[h, e] = 2e$, $[h, f] = -2f$. Suppose that $h\, V^i \subseteq V^i$ and $e\, V^i\subseteq V^{i+1}$ for each $i$.
Question 1: If we denote by $s = \begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}$ the simple reflection, do we have a Hodge-like decomposition with involution $s$ in the sense that the quotient map $V^i\to V^i/V^{i+1}$ induces an isomorphism $s(V^{-i-1})\cap V^i\to V^i/V^{i+1}$ for each $i$?
For each $\lambda \in \mathbf{N}$, denote by $V(\lambda)$ the isotypic component of the representation $\rho$ of highest weight $\lambda$.
Question 2: Is the filtration $V^{\bullet}$ compatible with the isotypic decomposition in the sense that $V^i = \bigoplus_{\lambda\in \mathbf{N}} V^i\cap V(\lambda)$ for each $i$?
The first question is true if $\rho$ is irreducible, by consideration of weight spaces.
The same questions can also be asked for $\omega$ non-degenerate symmetric bilinear form and $\rho:\mathrm{SL}_2(\mathbf{C})\to \mathrm{O}(V, \omega)$.
Thanks.
