On the Gauss map of equivariant immersions in hyperbolic space

Given an oriented immersed hypersurface in hyperbolic space (Formula presented.), its Gauss map is defined with values in the space of oriented geodesics of (Formula presented.), which is endowed with a natural para-Kähler structure. In this paper, we address the question of whether an immersion (Formula presented.) of the universal cover of an (Formula presented.) -manifold (Formula presented.), equivariant for some group representation of (Formula presented.) in (Formula presented.), is the Gauss map of an equivariant immersion in (Formula presented.). We fully answer this question for immersions with principal curvatures in (Formula presented.) : while the only local obstructions are the conditions that (Formula presented.) is Lagrangian and Riemannian, the global obstruction is more subtle, and we provide two characterizations, the first in terms of the Maslov class, and the second (for (Formula presented.) compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms.