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

TL;DR

This paper extends the understanding of locally analytic representations in the context of Lubin-Tate moduli spaces, covering more general deformation settings and analyzing the structure of locally finite vectors.

Contribution

It generalizes previous results to deformations over finite extensions of \\mathbb{Q}_p and includes vector bundles with Drinfeld level structures, also characterizing locally finite vectors.

Findings

Locally finite vectors originate from global sections of invertible sheaves.

Results apply to deformations over finite extensions of \\mathbb{Q}_p.

The structure of locally analytic representations is clarified in broader settings.

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.