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.

中文翻译：

---

Lubin-Tate 模空间的 étale 覆盖中的局部解析表示

Lubin-Tate 模空间 $X_{0}^{\text{rig}}$ 是一个 $p$-adic 解析开单位多圆盘，它参数化定义在一个有限高度的形式群 $H_{0}$ 的变形特征 $p$ 的代数闭域。众所周知，自同构群 $\text{Aut}(H_{0})$ 对 $X^{\text{rig}}_{0}$ 的自然作用产生了拓扑对偶上的局部解析表示的空间 $H^{0}(X^{\text{rig}}_{0},(\mathcal{M}^{s}_{0})^{\mathrm{rig}})$某些等变向量丛 $(\mathcal{M}^{s}_{0})^{\mathrm{rig}}$ 在 $X^{\mathrm{rig}}_{0}$ 上的全局部分。在本文中，我们表明该结果具有更大的普遍性。一方面，我们在 $\mathbb{Q}_{p}$ 的有限扩展的估值环上设置形式模块的变形。另一方面，我们还处理由向量丛 $(\mathcal{M}^{s}_{m})^{\mathrm{rig}}$ 在变形空间 $X^{\mathrm{rig} }_{m}$ 与 Drinfeld 级别-$m$-结构。最后，我们确定$H^{0}(X^{\text{rig}}_{m},(\mathcal{M}^{s}_{m})^{\ mathrm{rig}})$。本质上，所有局部有限向量都来自投影空间上可逆滑轮的全局部分，通过沿 Gross-Hopkins 周期图的回拉产生。