# Pushforward of functions on a frame bundle

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.

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)!