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

TL;DR

This paper establishes an equivalence between the groupoid of adjoint nilpotent $(G,)$-displays and the groupoid of $(G,)$-windows, generalizing the concept of windows in the context of algebraic groups over finite fields.

Contribution

It introduces and proves the equivalence between $(G,)$-displays and $(G,)$-windows, extending the framework of windows to new algebraic structures.

Findings

Proves the equivalence of $(G,\mu)$-displays and $(G,\mu)$-windows.

Generalizes the notion of windows in the theory of algebraic groups.

Provides a new perspective on 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.