μ-filtrations

Say $G / \mathbb{Q}$ is a connected reductive group, and let $\lbrace \mu \rbrace$ be a conjugacy class of geometric cocharacters $\mathbb{G}_ {m,\bar{\mathbb{Q}}} \to G_{\bar{\mathbb{Q}}}$, with reflex field $E$. Say there is a faithful representation $G \hookrightarrow \GL(V)$ and this is cut out by tensors $s_\alpha \in V^\otimes$. Given $X / E$ a scheme, a $G$-torsor on $X$ can then by identified with the data of

• a vector bundle $M$ on $X$,
• sections $s_{\alpha,M} \in \Gamma(X, M^\otimes)$,
• such that étale-locally $(M, (s_{\alpha,M}))$ looks like $(V, (s_\alpha))$.

The question we want to address is the following: what does it mean for a filtration on $M$ to be a $\lbrace \mu \rbrace$-filtration? One annoying problem is that even if the conjugacy class $\lbrace \mu \rbrace$ might be defined over $E$, there might not be a physical cocharacter $\mu$ over $E$ in that conjugacy class.

Remark 1. This doesn’t happen when $G$ is quasi-split. There are probably examples where this happens for a Shimura datum, but I don’t really know too many examples of Shimura varieties.

Thanks to Richard Taylor, Xinwen Zhu, and Pol van Hoften for teaching me this material.

Cocharacters of a connected reductive group scheme#

Let us review some facts about connected reductive group schemes.

Definition 2. A connected reductive group scheme $G \to S$ is a smooth (relatively) affine group scheme, such that geometric fibers are connected reductive groups.

This naive definition somehow behaves quite nicely. For instance, it locally has maximal tori.

Definition 3. A torus $T \to S$ is a smooth affine group scheme that is étale-locally (or fpqc-locally) on $S$ isomorphic to $\mathbb{G}_ {m, S}^n \to S$. A maximal torus of a smooth affine group $G \to S$ is a closed subgroup $T \subseteq G$ that is a torus, such that on each geometric fiber, $T_\bar{s} \subseteq G_\bar{s}$ is a maximal torus.

Theorem 4 (SGA3, XII.1.7). Let $G \to S$ be a connected reductive group. Then $G$ admits a maximal torus étale-locally on the base $S$. Moreover, any two maximal tori are étale-locally conjugate to each other.

Theorem 5 (Con14, 3.2.8). Let $G \to S$ be a connected reductive group with a maximal torus $T$. Then the Weyl group $$ W_G(T) = N_G(T) / Z_G(T) $$ is finite étale over $S$.

Using these facts, we can prove representability of the space of cocharacters up to conjugacy. Consider the presheaf on $S$-schemes defined by $$ S^\prime \mapsto \lbrace \text{homomorphisms } \mathbb{G}_ {m, S^\prime} \to G_{S^\prime} \rbrace / G(S^\prime)\text{-conjugacy}, $$ and denote by $\mathscr{C}(G)$ its étale sheafification (on the big site of $S$).

Theorem 6. The above étale sheaf $\mathscr{C}(G)$ is locally constant. In particular, it is representable by a scheme that is étale over $S$.

Here is a very general fact about smooth affine group schemes.

Theorem 7 (SGA3, XI.4.2). Let $G \to S$ be a smooth affine group scheme. Then the functor $\Hom(\mathbb{G}_ m, G)$ is represented by a smooth separated scheme over $S$.

Returning to the case of a connected reductive group scheme $G \to S$, we obtain a smooth surjective map $$ \Hom(\mathbb{G}_ m, G) \to \mathscr{C}(G) $$ over $S$, given by identifying conjugate cocharacters.

Torsors for a group scheme#

To discuss filtrations, we must first fix a faithful representation $$ \rho \colon G \hookrightarrow \GL(M) $$ that is a closed embedding, where $M$ is a locally free sheaf on $S$. In many cases, this $G$ will be realized as the stabilizer of a collection of sections $$ s_\alpha \in \Gamma(S, M^\otimes), \quad M^\otimes = \bigoplus_{i,j \ge 0} M^{\otimes i} \otimes (M^\vee)^{\otimes j}, $$ but we don’t need to assume the existence of such tensor sections $s_\alpha$.

For any $S$-scheme $X$ and a left $G$-torsor $P \to X$, the Borel construction gives a locally free sheaf $$ N = G \backslash ((P \to S)^\ast M) $$ on $X$, where the $G$-action is diagonal for $P$ and $M$. Let’s be more precise. We know that étale-locally in $X$, the torsor $P \to X$ has a trivialization, where transition maps are given by right-multiplication by points of $G$. For each trivialization of $P \to X$, say over an étale neighborhood $Y \to X$, we glue in a copy of $(Y \to S)^\ast M$. For each right-multiplication by $G(Y^\prime)$, we let its inverse act on $(Y^\prime \to S)^\ast M$ and use it as transition maps. Then we can étale-descend the vector bundle from $Y$ to $X$. TODO: direct construction using $G$-equivariant sections?

Proposition 8. If $G \hookrightarrow \GL(M)$ is cut out by tensors $s_\alpha$, then there is a natural equivalence of categories between

• $G$-torsors on $X$,
• locally free sheaves $N$ on $X$ together with sections $s_{\alpha,N} \in \Gamma(X, N^\otimes)$, satisfying the property that étale-locally on $X$ there exists an isomorphism $(N, (s_{\alpha,N})) \cong (X \to S)^\ast (M, (s_\alpha))$.

Remark 9. This “there étale-locally exists an isomorphism” condition is really an essential surjectivity condition. If we define a functor $$ F = \operatorname{Isom}((N, (s_{\alpha,N})), (X \to S)^\ast(M, (s_\alpha))) $$ then this will have a natural $G$-action and will satisfy the property that $G \times F \to F \times F$ is an isomorphism; in other words, it is always going to be a pseudo-$G$-torsor. So as soon as $F$ surjects onto $X$ in the fpqc topology, it is going to be a $G$-torsor, hence by smoothness of $G$, it étale-locally has a section.

Even when there is no such $s_\alpha$, we can still do something similar. Given a $G$-torsor $P \to X$ and a corresponding locally free sheaf $N$ on $X$ coming from the Borel construction, we may regard $P$ as a closed subfunctor $$ P \subseteq \operatorname{Isom}(N, (X \to S)^\ast M). $$ In particular, for $Y$ an $X$-scheme, a map $Y \to P$ over $X$ (equivalently, a section of $P_Y \to Y$) induces an isomorphism $$ (Y \to X)^\ast N \cong (Y \to S)^\ast M $$ on $Y$.

μ-filtrations#

Now that we understand $G$-torsors, let’s move on to filtrations. Again, we are in the setting where we fix a faithful representation $G \hookrightarrow \GL(M)$, and a $G$-torsor $P \to X$ corresponds to a vector bundle $N$ on $X$. In addition, let’s say that we are given a map $$ \lbrack \mu \rbrack \colon X \to \mathscr{C}(G) $$ over $S$, which the correct way of considering a cocharacter $\mathbb{G}_ {m, X} \to G_X$ up to conjugation. Given a finite decreasing filtration $\mathrm{Fil}^\bullet N$ on the associated locally free sheaf $N$, we want to say when it is a $[\mu]$-filtration.

Definition 10. For a cocharacter $\mu \colon \mathbb{G}_ {m,Y} \to G_Y$ over an $S$-scheme $Y$, define the finite decreasing filtration $\mathrm{Fil}_ \mu^\bullet(M_Y)$ on $M_Y$ by $$ \mathrm{Fil}_ \mu^n (M_Y) = \bigoplus_{i \ge n} (M_Y)^{\mu(t)=t^i}. $$

Using this filtration, we can describe the associated parabolic as $$ \begin{align*} P_{Y,\mu} &= \lbrace g \in G : \lim_{t \to 0} \mu(t) g \mu(t)^{-1} \text{ exists} \rbrace \br &= \lbrace g \in G : g \text{ preserves } \mathrm{Fil}_ \mu^\bullet(M_Y) \rbrace \subseteq G_Y. \end{align*} $$

But to get this filtration, we need an actual cocharacter, rather than a conjugacy class of a cocharacter. Since the natural map $\Hom(\mathbb{G}_ m, G) \to \mathscr{C}(G)$ is smooth surjective, we can find an étale cover $\tilde{X} \to X$ such that the map $[\mu] \colon X \to \mathscr{C}(G)$ lifts to $$ \tilde{\mu} \colon \tilde{X} \to \Hom(\mathbb{G}_ m, G). $$

Definition 11. Let $P \to X$ be a $G$-torsor and let $N$ on $X$ be the locally free sheaf obtained from the Borel construction with respect to the faithful representation $G \hookrightarrow \GL(M)$. A filtration $\mathrm{Fil}^\bullet N$ on the locally free sheaf $N$ is said to be a $[\mu]$-filtration when the following property holds:

• if $\tilde{\mu} \colon \mathbb{G}_ {m,\tilde{X}} \to G_{\tilde{X}}$ is a lift of $[\mu]$, then étale-locally on $\tilde{X}$, there is an section of $P_{\tilde{X}} \to \tilde{X}$ that induces an isomorphism $$ (\tilde{X} \to X)^\ast (N, \mathrm{Fil}^\bullet N) \cong (M_{\tilde{X}}, \mathrm{Fil}_ \tilde{\mu}^\bullet M_{\tilde{X}}). $$

Our first claim is that this condition does not depend on the choice of $\tilde{X}$ and the lift $\tilde{\mu}$. Why is that? If we have two different choices, first we can find a common refinement and assume that we have the same $\tilde{X}$. Let $\tilde{\mu}$ and $\tilde{\mu}^\prime$ be the two cocharacters. We can further refine $\tilde{X}$ so that $\tilde{\mu}$ and $\tilde{\mu}^\prime$ are rationally conjugate to each other. Here, note that when we replace $\tilde{X}$ by an étale cover, the condition doesn’t change because it is an étale-local condition anyways. Once $\tilde{\mu}$ and $\tilde{\mu}^\prime$ are conjugate, say by $g \in G(\tilde{X})$, we obtain an isomorphism $$ g \colon (M_{\tilde{X}}, \mathrm{Fil}_ {\tilde{\mu}}^\bullet M_{\tilde{X}}) \cong (M_{\tilde{X}}, \mathrm{Fil}_ {\tilde{\mu}^\prime}^\bullet M_{\tilde{X}}). $$ This shows that the conditions for $\tilde{\mu}$ and $\tilde{\mu}^\prime$ are equivalent.

Remark 12. This condition pretty closely related to the notion of a $G$-split filtration introduced in Kisin’s Integral models for Shimura varieties of abelian type. Given $P \to X$ a $G$-torsor, we can form its automorphism group $$ H = \Aut_X(P) \hookrightarrow \GL_X(N) $$ which is a pure inner form of $G$, hence also connected reductive group scheme over $X$. Then a finite decreasing filtration $\mathrm{Fil}^\bullet N$ is $H$-split in the sense of Kisin if and only if it is a $[\mu]$-filtration for some (uniquely determined) $[\mu] \colon X \to \mathscr{C}(G)$.

The space of μ-filtrations#

We now prove our main theorem. We start with the following data:

• a connected reductive group scheme $G \to S$,
• a faithful representation $G \hookrightarrow \GL(M)$,
• an $S$-scheme $X$ together with a $G$-torsor $P \to X$, and
• a map $[\mu] \colon X \to \mathscr{C}(G)$ over $S$.

Consider the locally free sheaf $N$ on $X$ associated to $P \to X$ and the representation $G \hookrightarrow \GL(M)$. We look at the space of all finite decreasing filtrations $\mathrm{Fil}^\bullet N$ on $N$, where the successive quotients are locally free with rank given by $\mu$. Let us denote this flag variety by $$ \operatorname{Flag}_ {[\mu]}(N) \to X, $$ where $[\mu]$ is there to only determine the ranks of the graded associated. This is evidently a smooth projective scheme over $X$.

Theorem 13. The moduli space of $[\mu]$-filtrations, defined as the functor on $X$-schemes $Y$ given by $$ \operatorname{Fil}_ {[\mu]}(N) \colon Y \mapsto \lbrace [\mu]\text{-filtrations on } N_Y \rbrace, $$ is represented by a closed subscheme of $\operatorname{Flag}_ {[\mu]}(N)$ that is smooth over $X$.

The first step is to reduce to the case when $[\mu]$ lifts to $\mu \colon X \to \Hom(\mathbb{G}_ m, G)$. Note that $\operatorname{Fil}_ {[\mu]}(N)$ is an étale sheaf by construction, because we are cutting it out as a sub-presheaf of a sheaf by an étale-local condition.

Lemma 14. If the statement is true for an étale cover $\tilde{X} \to X$, then the statement is true for $X$ as well.

So now we may pass to $X = \tilde{X}$ where $[\mu]$ lifts to an actual cocharacter $\mu \colon \mathbb{G}_ {m,X} \to G_X$. We may also pass to where the $G$-torsor $P \to X$ is the trivial $G$-torsor $G_X \to X$, so that $N = M_X$. At this point, we can define a filtration $\mathrm{Fil}_ \mu^\bullet M_X$ corresponding to $\mu$, and a filtration $\mathrm{Fil}^\bullet M_X$ is a $[\mu]$-filtration if and only if there are étale-local sections of $G_X \to X$ inducing an isomorphism $$ (M_X, \mathrm{Fil}_ \mu^\bullet M_X) \cong (M_X, \mathrm{Fil}^\bullet M_X). $$

By recall that $P_\mu$ is precisely the stabilizer of the filtration $\mathrm{Fil}_ \mu^\bullet M$, and so this functor is the quotient $G_X / P_\mu$.

Theorem 15 (Con14, 4.1.7). The subgroup $P_\mu \subseteq G_X$ is a parabolic subgroup, i.e., a smooth closed subgroup such that each geometric fiber is parabolic.

Theorem 16 (Con14, 2.3.6). If $P \subseteq G$ is a parabolic subgroup, then the quotient $G/P$ is representable by a scheme, smooth and Zariski-locally projective.

This shows that $\operatorname{Fil}_ {[\mu]}(M_X) = G_X / P_\mu$ is representable by a proper smooth scheme over $X$. To show that the map $\operatorname{Fil}_ {[\mu]}(M_X) \to \operatorname{Flag}_ {[\mu]}(M_X)$ is a closed embedding, we can use the fact that a proper monomorphism is a closed embedding (see Stacks 04XV for instance). This finishes the proof of the theorem.

μ-filtrations without filtrations#

The space of $[\mu]$-filtrations actually can be defined without choosing a faithful representation $G \hookrightarrow \GL(M)$. In fact, in the proof above, we showed that this space is étale-locally $G_X / P_\mu$, and this does not depend on the choice of the representation $M$. Of course, if we don’t have a representation, we don’t get an actual filtration.

As before, let

• $G \to S$ be a connected reductive group scheme,
• $P \to X$ be a $G$-torsor over an $S$-scheme $X$,
• $[\mu] \colon X \to \mathscr{C}(G)$ be a map over $S$.

Associated to $P$ is a pure inner form $$ H = \Aut_X(P) $$ of $G_X$, which is again a connected reductive group scheme over $X$. Note that we canonically have $$ \mathscr{C}(H) \cong \mathscr{C}(G) \times_S X, $$ because $H$ is a pure inner form of $G$. So we may regard $[\mu]$ also as a section $X \to \mathscr{C}(H)$.

Definition 17. A $[\mu]$-filtration on $P$ is a parabolic subgroup $Q \subseteq H$ such that there exists an étale cover $\tilde{X} \to X$ and a lift $\tilde{\mu} \colon \tilde{X} \to \Hom(\mathbb{G}_ m, H)$ of $[\mu]$ whose associated parabolic is $$ P_{\tilde{\mu}} = Q_\tilde{X} \subseteq H_\tilde{X}. $$

The claim is that this agrees with the previous definition.

Theorem 18. Assume $G$ has a faithful representation $G \hookrightarrow \GL(M)$. If $N$ on $X$ is the locally free sheaf associated to the $G$-torsor $P \to X$, then there is a canonical bijection between $[\mu]$-filtrations on $P$ and $[\mu]$-filtrations on $N$.

The proof also shows that the conjugation action of $H$ on the space of $[\mu]$-filtrations on $P$ is transitive.

Theorem 19 (SGA3, XXVI.3.3). Let $G \to S$ be a connected reductive group scheme. Then the space of parabolic subgroups of $G$ is represented by a smooth projective scheme over $S$. Moreover, it maps to a finite étale scheme over $S$ where the conjugation action of $G$ on each fiber is transitive.

Corollary 20. The space of $[\mu]$-filtrations on $P$ is represented by a smooth projective scheme over $X$.

References#

• [SGA3] Michel Demazure and Alexander Grothendieck, Schemas en Groupes.
• [Con14] Brian Conrad, Reductive group schemes.