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

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 $$\mathrm{Spec}(k)$$.

• As stated, the functor is not essentially surjective: clearly, $\mathscr{P}$ should be (at least) an inner form of $P$. – Laurent Moret-Bailly May 6 at 7:34

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 $$\mathscr{P}$$.

• For what it's worth, the hypothesis that $P$ and $\mathscr{P}$ are parabolic is only used in the proof of point 1. above. In general, this conversation should carry over to the case that $H \subseteq G$ is a (connected, smooth) closed subgroup to show an equivalence between the category of $N_G(H)$-torsors and the category of pairs $(\mathscr{E}, \mathscr{H} \subseteq \mathrm{Aut}(\mathscr{E})$ where $\mathscr{H}$ is an "inner form of $H$" in the above sense. – dorebell May 6 at 16:27