Locally analytic representations in the \'{e}tale coverings of the Lubin-Tate moduli space

The Lubin-Tate moduli space $X_0^{\text{rig}}$ is a $p$-adic analytic open unit polydisc which parametrizes deformations of a formal group $H_0$ of finite height defined over an algebraically closed field of characteristic $p$. It is known that the natural action of the automorphism group $\text{Aut}(H_0)$ on $X_0^{\text{rig}}$ gives rise to locally analytic representations on the topological duals of the spaces $H^0(X_0^{\text{rig}},({\mathcal{M}}_0^s{)}^{\mathrm{r}\mathrm{i}\mathrm{g}})$ of global sections of certain equivariant vector bundles $({\mathcal{M}}_0^s{)}^{\mathrm{r}\mathrm{i}\mathrm{g}}$ over $X_0^{\mathrm{r}\mathrm{i}\mathrm{g}}$. 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 ${\mathbb{Q}}_p$. On the other hand, we also treat the case of representations arising from the vector bundles $({\mathcal{M}}_m^s{)}^{\mathrm{r}\mathrm{i}\mathrm{g}}$ over the deformation spaces $X_m^{\mathrm{r}\mathrm{i}\mathrm{g}}$ with Drinfeld level-$m$-structures. Finally, we determine the space of locally finite vectors in $H^0(X_m^{\text{rig}},({\mathcal{M}}_m^s{)}^{\mathrm{r}\mathrm{i}\mathrm{g}})$. 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.