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Another math.stackexchange question (here: $\mathbb{G}_m$-torsors and line bundles) goes over a way to construct a line bundle $L$ from a $\mathbb{G}_m$-torsor $T \to B$, by using a decomposition $\mathcal{O}_T = \bigoplus_{n \in \mathbb{Z}} P_n$. $P_1$ is naturally isomorphic (as an $\mathcal{O}_B$-module) to the line bundle $L$ whose everywhere nonzero sections form $T$, while $P_n$ is isomorphic to $L^{\otimes n}$.

I'm interested in extending this to sub-torsors of $T$. Let $G$ be a subgroup of $\mathbb{G}_m$ (with a fixed inclusion), $T \to B$ a $\mathbb{G}_m$-torsor as above, and $S \to B$ a $G$-torsor that lives inside $T$. In a fairly natural way, the decomposition $\mathcal{O}_T = \bigoplus_{n \in \mathbb{Z}} P_n$ can be viewed as turning $\mathcal{O}_T$ into a graded ring, graded by the characters of $\mathbb{G}_m$. I'd like to use a similar construction to break up $\mathcal{O}_S = \bigoplus_{\chi \colon G \to \mathbb{G}_m} P_\chi$, in order to get a line bundle $L^\chi$ for each character of $G$. The character given by the fixed inclusion should give $L$ itself back, and its $n$th power should give $L^{\otimes n}$.

My motivation for this question involves $p$-adic modular forms, and the geometric construction of them as given by Pilloni and others. In this light, I think of sections of $L^{\otimes n}$ as corresponding to functions $f \colon (x \in B, \omega \in L|_x) \mapsto f(x,\omega) \in R$ which are homogeneous of degree $n$, meaning $f(x,\lambda \omega) = \lambda^nf(x,\omega)$. I hope that I can naturally think of sections of $L^\chi$ as corresponding to functions $f \colon (x \in B, \omega \in L|_x) \mapsto f(x,\omega) \in R$ which are homogeneous with character $\chi$, meaning $f(x,\lambda \omega) = \chi(\lambda)f(x,\omega)$. Any insight into how these interact with the above is welcome as well.

EDIT: As an addition, I have a feeling this decomposition I'm looking for might come from the coaction $\mu \colon \mathcal{O}_S \to \mathcal{O}_G \otimes_{\mathcal{O}_B} \mathcal{O}_S$ (as in the linked post from the first paragraph) and a grading by characters on $\mathcal{O}_G$. It seems to work out for $(\mathbb{G}_m)^r$, but I can't prove anything for algebraic groups $G$ where I don't really know what $\mathcal{O}_G$ looks like.

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    $\begingroup$ It seems you got this right. So, what exactly is your question? :) $\endgroup$ Sep 13, 2019 at 7:09
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    $\begingroup$ If I understand your question, you have the right of things, and the coaction is indeed the right place to look. This sort of procedure can be carried out for any diagonalizable group scheme $G$, where the representing Hopf algebra $\mathcal{O}_{G}$ can be recovered as the group algebra on the group of $G$-characters. If I have time, I will try to write a more detailed/coherent post later. $\endgroup$ Sep 13, 2019 at 10:08
  • $\begingroup$ @AlexWertheim, what characterizes diagonalizable group schemes? Are Abelian groups schemes all diagonalizable? Subgroups of $\mathbb{G}_m^r$? Your description of $\mathcal{O}_G$ is exactly what I want. $\endgroup$
    – Jon Aycock
    Sep 13, 2019 at 20:04

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