Locally analytic representations in the étale coverings of the Lubin-Tate moduli space

The Lubin-Tate moduli space X0rig is a p-adic analytic open unit polydisc which parametrizes deformations of a formal group H0 of finite height defined over an algebraically closed field of characteristic p. It is known that the natural action of the automorphism group Aut(H0) on X0rig gives rise to locally analytic representations on the topological duals of the spaces H0(X0rig , (ℳ0 )rig) of global sections of certain equivariant vector bundles (ℳ0 )rig over X0rig . In this article, we show that this result holds in greater generality. On the one hand, we work in the setting of deformations of formal modules over the valuation ring of a finite extension of ℚp. On the other hand, we also treat the case of representations arising from the vector bundles (ℳ )rig over the deformation spaces Xrig with Drinfeld level-m-structures. Finally, we determine the space of locally finite vectors in H0(Xrig , (ℳ )rig). Essentially, all locally finite vectors arise from the global sections of invertible sheaves over the projective space via pullback along the Gross-Hopkins period map.