Why is the set of parabolic reductions of a G-torsor E bijective to the set of parabolic subgroups of Aut(E)?

Question

Let $G$ be a reductive group scheme over some base $X$ and $P \subseteq G$ a parabolic subgroup. To a $P$-torsor $\mathscr{E}_P$, we may associate a $G$-torsor $\mathscr{E} = G \times^P \mathscr{E}_P$, which is $G \times \mathscr{E}_P$ mod the relation $(gp, s) \sim (g, ps)$, with $G$ acting by $g \cdot (h, s) = (gh, s)$. This gives a $P$-equivariant monomorphism $\mathscr{E}_P \hookrightarrow \mathscr{E}$ sending $s$ to $(1, s)$, and this gives us an inclusion of group schemes $\mathrm{Aut}(\mathscr{E}_P) \xrightarrow{\sim} \mathrm{Stab}(\mathscr{E}_P) \subseteq \mathrm{Aut}(\mathscr{E})$.

$\mathrm{Aut}(\mathscr{E}_P)$ is an inner form of $P$ (indeed, it is given by twisting $P$ by $\mathscr{E}_P$), and likewise $\mathrm{Aut}(\mathscr{E})$ is an inner form of $G$. Upon passing to an étale cover $\widetilde{X} \rightarrow X$ and choosing a trivialization $\mathscr{E}_P$ (which automatically gives a trivialization of $\mathscr{E}$ compatible with the map $\mathscr{E}_P \rightarrow \mathscr{E}$), we obtain an isomorphism $G|_{\widetilde{X}} \rightarrow \mathrm{Aut}(E)|_{\widetilde{X}}$ sending $P$ to $\mathrm{Aut}(\mathscr{E}_P)|_{\widetilde{X}}$.

This discussion defines a functor from the groupoid of $P$-torsors on $X$ to the groupoid of pairs $(\mathscr{E}, \mathscr{P})$ where $\mathscr{E}$ is a $G$-torsor on $X$ and $\mathscr{P}$ is a parabolic subgroup of $\mathrm{Aut}(\mathscr{E})$. A morphism $(\mathscr{E}, \mathscr{P}) \rightarrow (\mathscr{E}', \mathscr{P}')$ is an isomorphism of $G$-torsors $\varphi \colon \mathscr{E} \rightarrow \mathscr{E}'$ carrying $\mathscr{P}$ to $\mathscr{P}'$ under the induced map on automorphism groups (given by conjugation by $\varphi$).

Example 10.6.2 of the book Weil's conjecture for function fields by Gaitsgory-Lurie, it is claimed that this functor is an equivalence of categories. What's the quasi-inverse functor? I can't seem to find a proof anywhere.

If it helps, feel free to assume that $X$ is a curve over a finite field $k$ and that $G = G_0 \times_{\mathrm{Spec}(k)} X$ for an adjoint split reductive group $G_0$ over $\mathrm{Spec}(k)$.

Answer 1

Thanks to Laurent Moret-Bailly for pointing out that I missed a crucial hypothesis! Now I can construct the quasi-inverse, which I'll record below in case some future person is confused by the same problem:

The hypothesis is that "$\mathscr{P} \subseteq \mathrm{Aut}(\mathscr{E})$ is an inner form of $P \subseteq G$". By this, I mean:

(*): There is an étale cover $\widetilde{X} \rightarrow X$ and a trivialization of $\widetilde{\mathscr{E}}$ such that the induced isomorphism $\widetilde{G} \rightarrow \mathrm{Aut}(\widetilde{\mathscr{E}})$ takes $\widetilde{P}$ to $\widetilde{\mathscr{P}}$.

(I'm using the notation $\widetilde{Y}$ for $Y|_{\widetilde{X}}$). Note that if we change the trivialization, $\widetilde{P}$ gets mapped to a conjugate of $\widetilde{\mathscr{P}}$.

Gaitsgory-Lurie give an equivalent formulation of this condition, which is more intrinsic to $\mathscr{P} \subseteq \mathrm{Aut}(\mathscr{E})$ and extends to general inner forms of $G$. Namely: that there is an étale cover $\widetilde{X}$ of $X$ and an isomorphism $\varphi \colon \widetilde{G} \rightarrow \mathrm{Aut}(\widetilde{\mathscr{E}})$ taking $\widetilde{P}$ to $\widetilde{\mathscr{P}}$ which "is compatible with the inner structure on $\mathrm{Aut}(\widetilde{\mathscr{E}})$". This compatibility means that if we use $\mathscr{E}$ to realize $\mathrm{Aut}(\mathscr{E})$ as an inner form of $G$, then (after passing to a further étale cover of $X$ if necessary), the resulting isomorphism $\widetilde{G} \rightarrow \mathrm{Aut}(\widetilde{\mathscr{E}})$ differs from $\Phi$ by an inner automorphism of $\widetilde{G}$.

Now, we define the quasi-inverse functor as follows:

Let $(\mathscr{E}, \mathscr{P})$ be as above, and define a subsheaf $\mathscr{E}_{\mathscr{P}}$ of $\mathscr{E}$ by $\mathscr{E}_{\mathscr{P}}(U) = \{s \in \mathscr{E}(U) \colon \forall V \rightarrow U, \forall \varphi \in \mathscr{P}(V), \varphi(s|_V) \in P(V) \cdot s|_V \}$. Note that for $s \in \mathscr{E}_\mathscr{P}(U)$, $\mathscr{P}|_U$ is the full stabilizer of $s$ mod $P$ in $\mathrm{Aut}(\mathscr{E}|_U)$ (as can be seen from (*) by passing to $\widetilde{U} \rightarrow U$ and using $s$ to trivialize $\mathrm{Aut}(\mathscr{E}|_{\widetilde{U}})$ and noting that the stabilizer of $s$ mod $P$ must be a conjugate of $\mathscr{P}$).

We claim that $\mathscr{E}_\mathscr{P}$ is a $P$-torsor, so our quasi-inverse functor is $(\mathscr{E}, \mathscr{P}) \mapsto \mathscr{E}_{\mathscr{P}}$. It's easy to see that it is a subsheaf of $\mathscr{E}$ preserved by the action of $P$, so it suffices to prove:

1. The action of $P$ on $\mathscr{E}_{\mathscr{P}}$ is transitive.
2. $\mathscr{E}_{\mathscr{P}}$ has sections over some étale cover of $X$.

To prove 1., suppose that $s, s' \in \mathscr{E}_{\mathscr{P}}(U)$ are two sections. Then there is a unique automorphism $\varphi \in \mathrm{Aut}(\mathscr{E}|_U)$ such that $s' = \varphi(s)$. It suffices to prove that $\varphi \in \mathscr{P}(U)$ (since then by definition of $\mathscr{E}_{\mathscr{P}}$, $s' = \varphi(s) \in P(U) \cdot s$). Since $\mathscr{P}$ is the stabilizer of $s$ mod $P$, $\varphi \mathscr{P} \varphi^{-1}$ is the stabilizer of $s'$ mod $P$, so we have $\varphi \mathscr{P} \varphi^{-1} = \mathscr{P}$. Since parabolic subgroups are self-normalizing, this implies that $\varphi \in \mathscr{P}$.

To prove 2., choose $\widetilde{X}$ and $s_0 \in \mathscr{E}(\widetilde{X})$ be as in (*). We claim that $s_0 \in \mathscr{E}_{\mathscr{P}}(\widetilde{X})$. Indeed, the resulting isomorphism $\widetilde{G} \rightarrow \mathrm{Aut}(\widetilde{\mathscr{E}})$ maps $g$ to the unique automorphism sending $s_0$ to $gs_0$, so the stabilizer of $s_0$ mod $\widetilde{P}$ is the image of $\widetilde{P}$ under this isomorphism, which is $\mathscr{P}$.