Semiprojectivity and very stability in moduli of symplectic and orthogonal parabolic Higgs bundles

Let $X$ be a compact Riemann surface of genus $g\ge 2$, and let $D\subset X$ be a fixed finite subset. We prove the semiprojectivity of the moduli space of semistable symplectic or orthogonal parabolic Higgs bundles over $X$. We show that a stable symplectic parabolic bundle $E$ on $X$ is strongly very stable, meaning $E$ does not have any nonzero strongly parabolic nilpotent Higgs field, if and only if the symplectic parabolic Hitchin morphism induced on the affine space $$H^0(X,{\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathrm{S}\mathrm{p}}(E)\otimes K(D))$$ is a proper morphism, where ${\mathrm{S}\mathrm{P}\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathrm{S}\mathrm{p}}(E)$ denotes the set of symplectic strongly parabolic endomorphisms of $E$. We remark that the same criterion for very stability applies to the orthogonal case.