Pushforward of functions on a frame bundle

Question

Apologies in advance for the long setup and question.

Let $L \to X$ be a line bundle. We may take its frame bundle $p \colon Fr(L) \to X$, a $\mathbb{G}_m$-torsor. We have

$$ p_*\mathcal{O}_{Fr(L)} = \bigoplus_{k \in \mathbb{Z}} {L}^{\otimes (-k)}. $$

To see this, we may take a trivializing neighborhood $U$ for $L$, and note that the fiber $L|_U$ looks like $U \times K$, where $K$ is a 1-dimensional vector space. So the functions $\mathcal{O}_{U \times K} \cong \mathcal{O}_U[x]$, where $x \in K^\vee$ is a nonzero functional. Choosing these $x$'s in families amounts to choosing a section of $L^\vee =: L^{\otimes(-1)}$. Requiring that the frames be nowhere vanishing sections of $L$ is like inverting $x$, since we don't care if they're defined where $x=0$; in families it's like adjoining $(L^\vee)^\vee = L$. So we get the above.

This $p_*\mathcal{O}_{Fr(L)}$ is graded by the characters of $\mathbb{G}_m$, with the $L^{\otimes(-k)}$ piece corresponding to the character $\lambda \mapsto \lambda^{-k}$ (or maybe its inverse).

Now let $F$ be a degree $d$ (totally real) number field, and $L_F \to X$ be a rank $d$ vector bundle with real multiplication by $\mathcal{O}_F$; the fibers look like 1-dimensional $F$-vector spaces. Take its $\mathcal{O}_F$-frame bundle $p \colon Fr(L_F) \to X$, which is a Res$_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$-torsor. Points of $Fr(L_F)$ are nonzero elements of the fiber. Taking a trivializing neighborhood as above gives us $p_*\mathcal{O}_{L_F}(U) \cong U[x_1,\dots,x_d]$ locally, and globally $p_*\mathcal{O}_{L_F} = \bigoplus_{k \geq 0} L_F^{\otimes_\mathbb{Z} (-k)}$. Requiring that sections of the frame bundle don't vanish is making sure not all of the coordinates of the section are $0$. So I want functions that aren't necessarily defined at $(0,\dots,0)$ -- but by the Koecher principle, any function defined on the $d$-dimensional fiber minus the origin extends to the origin!

This leads me to believe that we invert nothing (in contrast to the above situation where we invert $x$, and it gives us the positive tensor powers of $L$), giving $$ p_*\mathcal{O}_{Fr(L_F)} = p_*\mathcal{O}_{L_F} = \bigoplus_{k \geq 0} L_F^{\otimes_\mathbb{Z} (-k)}. $$

Is this true? It seems to disagree with my intuition that this should be graded by characters of the group Res$_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$, unless some of the graded pieces (i.e., the ones where one of the $k_\sigma$'s is negative) can be trivial. There is also the fact that we take the tensor powers of $L_F$ by tensoring over $\mathbb{Z}$, rather than $\mathcal{O}_F$, which is not obvious to me.

The above is my real question, but I'm also interested in the case when $V \to X$ is just a rank $g$ vector bundle, and $p \colon Fr(V) \to X$ is its usual frame bundle, a GL$_g$-torsor. For this, is there a good description of $p_*\mathcal{O}_{Fr(V)}$ in terms of $V$? I have separate motivation for this question, but it may also provide intuition for my current main one.

Answer 1

I think I've figured out an answer that makes sense and deals with my problems mentioned in the question.

Since $Fr(L_F)$ is a $Res_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$-torsor, its fibers should look (non-canonically) like $Res_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$, whose structure sheaf is $\mathcal{O}_{Res_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m} \cong \mathcal{O}_F[t,t^{-1}]$. So over a trivializing neighborhood $U$ we should have a (non-canonical) isomorphism with $\mathcal{O}_U \otimes_\mathbb{Z} \mathcal{O}_F[t,t^{-1}]$. The only way I can see to make this happen is to have $$ p_*\mathcal{O}_{Fr(L_F)} = \bigoplus_{k \in \mathbb{Z}} L_F^{\otimes_{\mathcal{O}_F} (-k)}. $$ Here each piece has the right size, since $t^{-k}\mathcal{O}_F \leftrightarrow L_F^{\otimes_{\mathcal{O}_F} (-k)}$ and the fibers of both look like $\mathcal{O}_F$ tensored with the residue field. When we base change to $\mathbb{R}$ we can see the grading by non-parallel weight characters, since $\mathbb{R} \otimes_\mathbb{Z} \mathcal{O}_F \cong \mathbb{R}^d$ ($F$ was totally real of degree $[F:\mathbb{Q}] = d$). Thus after base changing to $\mathbb{R}$, the structure sheaf looks (again, non-canonically) like $\prod_{\sigma \colon F \to \mathbb{R}} \mathbb{R}[t_\sigma,t_\sigma^{-1}]$. The character $\lambda \mapsto \prod_{\sigma \colon F \to \mathbb{R}} \lambda^{k_\sigma}$ corresponds to the piece $t^{-k_{\sigma_1}}\mathbb{R} \times \dots \times t^{-k_{\sigma_d}}\mathbb{R}$.

The reasons I am convinced by this answer are because the fibers looks like $Res_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$ and because we get the grading by characters. I'd love to see if anyone has a reference that confirms this answer (or an issue that would point to it being false)!