  We define a version of spectral invariant in the vortex Floer theory for a
$G$-Hamiltonian manifold $M$. This defines potentially new (partial) symplectic
quasi-morphism and quasi-states when $M//G$ is not semi-positive. We also
establish a relation between vortex Hamiltonian Floer homology and Woodward's
quasimap Floer homology by constructing a closed-open string map between them.
This yields applications to study non-displaceability problems of subsets in
$M//G$
