Once you have made the polar decomposition, it is sufficient to find the Haar measure on the compact symplectic group. This can be calculated starting from your favorite parameterization $U(\{\alpha_n\})$ of the unitary symplectic matrix $U$, via the metric tensor,

$$g_{mn}=-{\rm tr}\,U^{\dagger}(\partial U/\partial\alpha_m)U^{\dagger}(\partial U/\partial\alpha_n),$$

and then the Haar measure is $d\mu= \sqrt{{\rm det}\,g}\prod_n d\alpha_n$.

Alternatively, you can use a computer to generate random matrices with the desired measure, as explained by Franco Mezzadri.

Once you have made the polar decomposition, it is sufficient to find the Haar measure on the compact symplectic group. This can be calculated starting from your favorite parameterization $U(\{\alpha_i\})$ of the unitary symplectic matrix $U$, via the metric tensor,

$$g_{ij}=-{\rm tr}\,U^{\dagger}(\partial U/\partial\alpha_i)U^{\dagger}(\partial U/\partial\alpha_j),$$

and then the Haar measure is $d\mu= \sqrt{{\rm det}\,g}\prod_i d\alpha_i$.

Alternatively, you can use a computer to generate random matrices with the desired measure, as explained by Franco Mezzadri.

Once you have made the polar decomposition, it is sufficient to find the Haar measure on the compact symplectic group. This can be calculated starting from your favorite parameterization $U(\{\alpha_n\})$ of the unitary symplectic matrix $U$, via the metric tensor,

$$g_{mn}=-{\rm tr}\,U^{\dagger}(\partial U/\partial\alpha_m)U^{\dagger}(\partial U/\partial\alpha_n),$$

and then the Haar measure is $d\mu= \sqrt{{\rm det}\,g}\prod_n d\alpha_n$.

Alternatively, you can use a computer to generate random matrices with the desired measure, as explained by Franco Mezzadri.

Once you have made the polar decomposition, it is sufficient to find the Haar measure on the compact symplectic group. This can be calculated starting from your favorite parameterization $U(\{\alpha_n\})$ of the unitary symplectic matrix $U$, via the metric tensor,

$$g_{mn}=-{\rm tr}\,U^{\dagger}(\partial U/\partial\alpha_m)U^{\dagger}(\partial U/\partial\alpha_n),$$

and then the Haar measure is $d\mu= \sqrt{{\rm det}\,g}\prod_n d\alpha_n$.

Alternatively, you can use a computer to generate random matrices with the desired measure, as explained by Franco Mezzadri.