$(G,\mu )$-Windows and Deformations of $(G,\mu )$-Displays

Abstract

Let $k_0$ be a finite field of characteristic $p$, let $G$ be a smooth affine group scheme over ${\mathbb{Z}}_p$, and let $\mu$ be a cocharacter of $G_{W(k_0)}$ such that the set of $\mu$-weights of $\text{Lie}G$ is a subset of $\{-1,0,1,2,\dots \}$. We prove that the groupoid of adjoint nilpotent $(G,\mu )$-displays is equivalent to the groupoid of $(G,\mu )$-windows, which are the generalizations of windows.