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

Abstract

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 $X^{\text{rig}}_{0}$ gives rise to locally analytic representations on the topological duals of the spaces $H^{0}(X^{\text{rig}}_{0},(\mathcal{M}^{s}_{0})^{\mathrm{rig}})$ of global sections of certain equivariant vector bundles $(\mathcal{M}^{s}_{0})^{\mathrm{rig}}$ over $X^{\mathrm{rig}}_{0}$. 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 $(\mathcal{M}^{s}_{m})^{\mathrm{rig}}$ over the deformation spaces $X^{\mathrm{rig}}_{m}$ with Drinfeld level-$m$-structures. Finally, we determine the space of locally finite vectors in $H^{0}(X^{\text{rig}}_{m},(\mathcal{M}^{s}_{m})^{\mathrm{rig}})$. 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.