# Symplectic reduction of a linear symplectic space.

I want to know what is the symplectic reduction of a symplectic linear space.

Suppose $$(V,\omega)$$ is vector space with a nondegenerate bilinear form $$\omega$$. We can assume it is $$(\mathbb R^{2n},\omega)$$ such that $$$$\omega(u,v)=(Ju)^tv$$$$ where $$$$J=\begin{pmatrix} 0&I\\-I&0 \end{pmatrix}$$$$

The symplectic group is $$$$Sp(2n)=\{A\in\mathbb R^{2n\times 2n}\colon A^tJA=J \}$$$$

The moment map for the $$Sp(2n)$$ action on $$(\mathbb R^{2n},\omega)$$ is $$$$\mu\colon \mathbb R^{2n}\to \mathfrak{sp}(2n),\quad v\mapsto \frac{-1}{2}Jvv^t$$$$ with $$\mathfrak{sp}(2n)\cong\mathfrak{sp}(2n)^*$$ by the inner product $$\langle A,B\rangle=\textbf{tr}AB^t$$.

Let $$H\le Sp(2n)$$ be a Lie subgroup, and let $$$$\phi\colon \mathbb R^{2n}\to\mathfrak h$$$$ be the composition of $$\mu$$ and $$\text{pr}\colon \mathfrak{sp}(2n)\to\mathfrak h$$, the orthogonal projection.

Then $$(\mathbb R^{2n},\omega,H,\phi)$$ is a Hamiltonian $$H$$-space.

My question is: do we have some theorems about the symplectic reduction $$\mathbb R^{2n}/H$$?