Introduction
In [Reference FarguesFar25], Fargues formulated a conjecture on a geometrisation of the local Langlands correspondence motivated by a formulation of the geometric Langlands conjecture in [Reference Frenkel, Gaitsgory and VilonenFGV02].
Let E be a p-adic number field with residue field
$\mathbb {F}_q$
. Let G be a quasi-split reductive group over E. Then we can define a moduli stack
$ {\mathrm {Bun}}_G$
of G-bundle on the Fargues–Fontaine curve, and a moduli
$ {\mathrm {Div}}_X^1$
of Cartier divisors of degree
$1$
on the Fargues–Fontaine curve. Further, we have a diagram
where
$ {\mathrm {Hecke}}^{\leq \mu }$
is a moduli stack of modifications of G-bundle on the Fargues–Fontaine curve with some condition determined by a cocharacter
$\mu $
of G, which is called a Hecke stack. For a discrete Langlands parameter
$\varphi \colon W_E \to {}^L G$
, Fargues’ conjecture predicts the existence of a sheaf
$\mathscr {F}_{\varphi }$
on
$ {\mathrm {Bun}}_{G}$
satisfying some conditions, the most intriguing one of which is the Hecke eigensheaf property
$$\begin{align*}\overrightarrow{h}_{\natural} (\overleftarrow{h}^* \mathscr{F}_{\varphi} \otimes {\mathrm{IC}}_{\mu}') = \mathscr{F}_{\varphi} \boxtimes (r_{\mu} \circ \varphi ) , \end{align*}$$
where
$r_{\mu }$
is a representation of
${}^L G$
determined by
$\mu $
, and
$ {\mathrm {IC}}_{\mu }'$
is an object of the derived category of sheaves determined by
$\mu $
via the geometric Satake correspondence. The conjecture is stated based on some conjectural objects. However, in the case
$\varphi $
is cuspidal and
$\mu $
is minuscule, we can define every object in the conjecture assuming only the local Langlands correspondence, which is constructed in many cases.
Assume that
$\varphi $
is cuspidal and
$\mu $
is minuscule. Then the support of the sheaf
$\mathscr {F}_{\varphi }$
is contained in the semi-stable locus
$ {\mathrm {Bun}}_{G}^{\mathrm {ss}}$
of
$ {\mathrm {Bun}}_{G}$
. The Hecke eigensheaf property then predicts that
$$\begin{align*}{\mathrm{supp}} \overrightarrow{h}_{\natural} (\overleftarrow{h}^* \mathscr{F}_{\varphi} \otimes {\mathrm{IC}}_{\mu}') \subset {\mathrm{Bun}}_{G}^{\mathrm{ss}} \times {\mathrm{Div}}_X^1. \end{align*}$$
This is non-trivial since the inclusion
$$\begin{align*}\overleftarrow{h}^{-1} \big( {\mathrm{Bun}}_{G}^{\mathrm{ss}} \big) \subset \overrightarrow{h}^{-1} \big( {\mathrm{Bun}}_{G}^{\mathrm{ss}} \times {\mathrm{Div}}_X^1 \big) \end{align*}$$
does not hold. The vanishing of
$\overrightarrow {h}_{\natural } (\overleftarrow {h}^* \mathscr {F}_{\varphi } \otimes {\mathrm {IC}}_{\mu }')$
outside the semi-stable locus involves geometry of a non-semi-stable locus of the Hecke stack
$ {\mathrm {Hecke}}^{\leq \mu }$
.
One aim of this paper is to give a partial result in this direction. Assume that
$\varphi $
is cuspidal, but
$\mu $
can be general in the following. Let
$B(G)$
be the set of
$\sigma $
-conjugacy classes in
$G(\breve {E})$
, where
$\breve {E}$
is the completion of the maximal unramified extension of E. Then we have a decomposition
$$\begin{align*}{\mathrm{Bun}}_{G} = \coprod_{[b] \in B(G)} {\mathrm{Bun}}_{G}^{[b]} \end{align*}$$
into strata, where the strata corresponding to basic elements of
$B(G)$
form the semi-stable locus. Let
$[b],[b'] \in B(G)$
. We define
$ {\mathrm {Hecke}}_{[b],[b']}^{\leq \mu }$
by the fibre products
We assume that
$[b]$
is not basic, and
$[b']$
is basic. Let
$ {\mathrm {Hecke}}_{[b],[b']}^{\mu }$
be an open substack of
$ {\mathrm {Hecke}}_{[b],[b']}^{\leq \mu }$
, where the modifications have type
$\mu $
. We find that a generalisation
$\mathcal {M}_{b,b'}^{\mu }$
of a diamond of a non-basic Rapoport–Zink space at infinite level covers
$ {\mathrm {Hecke}}_{[b],[b']}^{\mu }$
.
We can define a Levi subgroup
$L^b$
of G such that
$[b]$
is an image of a basic element
$[b_{00}]$
of
$B(L^b)$
. Take a proper Levi subgroup L of G containing
$L^b$
. Let
$[b_0]$
be the image of
$[b_{00}]$
in
$B(L)$
. We assume that
$[b']$
is in the image of an element
$[b^{\prime }_0] \in B(L)$
. Further, we assume that
$([b],[b'],\mu )$
satisfies a twisted analogue of Hodge–Newton reducibility. Our main theorem is the following:
Theorem. The compactly supported cohomology of
$\mathcal {M}_{b,b'}^{\mu }$
is a parabolic induction of the compactly supported cohomology of
$\mathcal {M}_{b_0,b^{\prime }_0}^{\mu }$
with some degree shift and twist.
See Theorem 4.26 for the precise statement. This theorem is a generalisation of the Harris–Viehmann conjecture on cohomology of non-basic Rapoport–Zink spaces in [Reference Rapoport and ViehmannRV14, Conjecture 8.5] (cf. [Reference HarrisHar01, Conjecture 5.2]) up to a character twist under the Hodge–Newton reducibility condition. We also show that the compactly supported cohomology of
$\mathcal {M}_{b,b'}^{\mu }$
does not contain any supercuspidal representation. These results can be viewed as generalisation of results in [Reference MantovanMan08]. Using the above theorem, we can show the following:
Theorem. The compactly supported cohomology of
$ {\mathrm {Hecke}}_{[b],[b']}^{\mu }$
with coefficient in
$\overleftarrow {h}^* \mathscr {F}_{\varphi }$
vanishes.
See Theorem 4.30 for the precise statement. This result is partial, since we are assuming Hodge–Newton reducibility. On the other hand, the assumption is automatically satisfied if
$ {\mathrm {Hecke}}_{[b],[b']}^{\leq \mu }$
is not empty in the case where
$G= {\mathrm {GL}}_2$
and
$\mu (z) = {\mathrm {diag}} (z,1)$
. As an application, we can show the following:
Theorem. Assume that
$G= {\mathrm {GL}}_2$
and
$\mu (z) = {\mathrm {diag}} (z,1)$
. Then the Hecke eigensheaf property for a cuspidal Langlands parameter holds.
During the course of this work, Hansen put a related preprint [Reference HansenHan21a] on his webpage, which shows the Harris–Viehmann conjecture for
$ {\mathrm {GL}}_n$
under the Hodge–Newton reducibility condition. We learned his result on canonical filtrations and some consequences of Scholze’s work [Reference ScholzeSch17] on cohomology of diamonds from [Reference HansenHan21a]. Note that the result of [Reference HansenHan21a] is enough for the application to Fargues’ conjecture in
$ {\mathrm {GL}}_2$
-case. Our main points are proving the Harris–Viehmann conjecture under the Hodge–Newton reducibility condition for general reductive groups and making the relation to Fargues’ conjecture clear. After this work was done, Fargues’ conjecture for cuspidal Langlands parameters in the
$ {\mathrm {GL}}_n$
-case was proved in [Reference Anschütz and Le BrasALB21] and [Reference HansenHan21b] by a different method.
In Section 1, we recall a definition of the stack of G-bundle on the Fargues–Fontaine curve, and its structure. In Section 2, we recall a definition of the Hecke stack and explain a cohomological formula. In Section 3, we construct a sheaf which satisfies properties (1), (2) and (3) of [Reference FarguesFar25, Conjecture 4.4] and explain the Hecke eigensheaf property in Fargues’ conjecture for cuspidal Langlands parameters.
In Section 4, we study a non-semi-stable locus in the Hecke stack. We find that a generalisation of a diamond of a non-basic Rapoport–Zink space at infinite level covers the non-semi-stable locus in the Hecke stack. We show that the cohomology of the generalised space can be written as a parabolic induction of the cohomology of smaller space associated with a Levi subgroup under the Hodge–Newton reducibility condition. In particular, we see that the cohomology does not contain any supercuspidal representation in each degree. As a result, we show that the cohomology of the non-semi-stable locus in the Hecke stack with a coefficient coming from a cuspidal Langlands parameter vanishes.
In Section 5, we see that we can recover Hecke eigensheaf property on some part of the semi-stable locus from non-abelian Lubin–Tate theory in the
$ {\mathrm {GL}}_n$
-case. In Section 6, we show that the Hecke eigensheaf property in the
$ {\mathrm {GL}}_2$
-case, using the results in the preceding sections.
1 Stack of G-bundles
In this section we recall various results regarding the stack of G-bundles on the curve. Let p be a prime number. Fix E a finite extension of
$\mathbb {Q}_p$
with residue field
$\mathbb {F}_q$
. We follow the definition of perfectoid algebra in [Reference FontaineFon13, 1.1] (cf. [Reference ScholzeSch12, Definition 5.1]). For an algebraic extension k of
$\mathbb {F}_q$
, let
$ {\mathrm {Perf}}_{k}$
be the category of perfectoid spaces over k equipped with v-topology (cf. [Reference ScholzeSch17, Definition 8.1(iii)]). For
$S \in {\mathrm {Perf}}_{\mathbb {F}_q}$
, we have the relative Fargues–Fontaine curve
$X_S = Y_{S}/\varphi ^{\mathbb {Z}}$
as in [Reference Fargues and ScholzeFS21, Definition II.1.15]. For an affinoid perfectoid
$ {\mathrm {Spa}} (R,R^+) \in {\mathrm {Perf}}_{\mathbb {F}_q}$
, we have also the schematic relative Fargues–Fontaine curve
$X_{ {\mathrm {Spa}}(R,R^{+})}^{ {\mathrm {sch}}}$
as defined just after [Reference Fargues and ScholzeFS21, Remark II.2.8]. The schematic version
$X_{ {\mathrm {Spa}}(R,R^{+})}^{ {\mathrm {sch}}}$
only depends on R and so we denote it by
$X_R^{ {\mathrm {sch}}}$
. We have an equivalence between categories of vector bundles on
$X_{ {\mathrm {Spa}} (R,R^+)}$
and
$X_R^{ {\mathrm {sch}}}$
by [Reference Kedlaya and LiuKL15, Theorem 8.7.7].
Let G be a connected reductive group over E. Let
$ {\mathrm {Bun}}_G$
be the fibred category in groupoids whose fibre at
$S \in {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
is the groupoid of G-bundles on
$X_S$
. Then
$ {\mathrm {Bun}}_G$
has a reasonable geometry. Let us just mention that, in particular, it is a small v-stack (cf. [Reference Fargues and ScholzeFS21, Proposition III.1.3]).
Let
$\breve {E}$
be the completion of the maximal unramified extension of E. Let
$\sigma $
be the continuous automorphism of
$\breve {E}$
lifting the q-th power Frobenius on the residue field. For
$b \in G(\breve {E})$
, we have an associated G-isocrystal
Let
$B(G)$
be the set of
$\sigma $
-conjugacy classes in
$G(\breve {E})$
. Then we have a bijection
$$\begin{align*}B(G) \longrightarrow \{ \text{the isomorphism classes of } G\text{-isocrystals over } \breve{E} \} ;\ [b] \mapsto [\mathcal{F}_b] \end{align*}$$
by [Reference Rapoport and RichartzRR96, Remarks 3.4 (i)].
Let
$S \in {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
. We have a functor
where
$\mathscr {E} (D,\varphi )$
is given by
$$\begin{align*}Y_S \times_{\varphi} D \longrightarrow Y_S /\varphi^{\mathbb{Z}} =X_S. \end{align*}$$
The composite
gives a G-bundle
$\mathscr {E}_{b,X_S}$
on
$X_S$
. We simply write
$\mathscr {E}_b$
for
$\mathscr {E}_{b,X_S}$
sometimes. If
$b'=g b \sigma (g)^{-1}$
, then we have an isomorphism
$$ \begin{align} t_g \colon \mathscr{E}_{b,X_S} \longrightarrow \mathscr{E}_{b',X_S} \end{align} $$
induced by the multiplication by g. The isomorphism class of
$\mathscr {E}_{b,X_S}$
depends only on the class of b in
$B(G)$
. Moreover by [Reference Fargues and ScholzeFS21, Theorem III.2.2], this gives a complete description of the points of
$ {\mathrm {Bun}}_G$
.
Let
$\pi _1 (G)$
be an algebraic fundamental group of G defined in [Reference BorovoiBor98, 1.4]. Let
$\overline {E}$
be a separable closure of E and let
$\Gamma = {\mathrm {Gal}} (\overline {E}/E)$
be its absolute Galois group. Let
$$\begin{align*}\kappa \colon B(G) \longrightarrow \pi_1 (G)_{\Gamma} \end{align*}$$
be the Kottwitz map in [Reference Rapoport and RichartzRR96, Theorem 1.15] (cf. [Reference KottwitzKot90, Lemma 6.1]). Then [Reference Fargues and ScholzeFS21, Theorem III.2.7]) provides a decomposition
$$\begin{align*}{\mathrm{Bun}}_{G} = \coprod_{\alpha \in \pi_1 (G)_{\Gamma}} {\mathrm{Bun}}_{G}^{\alpha} \end{align*}$$
into open and closed substacks.
Let
$\mathbb {D}$
be the split pro-algebraic torus over E such that
$X_* (\mathbb {D})=\mathbb {Q}$
. For
$b \in G(\breve {E})$
, we have an associated homomorphism
$$\begin{align*}\tilde{\nu}_b \colon \mathbb{D}_{\breve{E}} \longrightarrow G_{\breve{E}} \end{align*}$$
constructed in [Reference KottwitzKot85, 4.2]. This gives a well-defined map
$$\begin{align*}\nu \colon B(G) \longrightarrow \big( {\mathrm{Hom}} (\mathbb{D}_{\breve{E}},G_{\breve{E}})/G(\breve{E}) \big)^{\sigma} ;\ [b] \mapsto [\tilde{\nu}_b], \end{align*}$$
which is called the Newton map. We say that
$b \in G(\breve {E})$
is basic, if
$\tilde {\nu }_b$
factors through the centre of
$G_{\breve {E}}$
. We say that
$[b] \in B(G)$
is basic if it consists of basic elements in
$G(\breve {E})$
. Let
$B(G)_{ {\mathrm {basic}}}$
denote the basic elements in
$B(G)$
. We recall that the Kottwitz map induces a bijection
$$\begin{align*}\kappa \colon B(G)_{{\mathrm{basic}}} \stackrel{\sim}{\longrightarrow} \pi_1 (G)_{\Gamma}. \end{align*}$$
Assume that G is quasi-split in the sequel. We fix subgroups
$A \subset T \subset B$
of G, where A is a maximal split torus, T is a maximal torus and B is a Borel subgroup. We write
$X_* (A)^+$
for the dominant cocharacters of A. Then we have a natural isomorphism
$$\begin{align*}X_* (A)^+_{\mathbb{Q}} \stackrel{\sim}{\longrightarrow} \big( {\mathrm{Hom}} (\mathbb{D}_{\breve{E}},G_{\breve{E}})/G(\breve{E}) \big)^{\sigma}. \end{align*}$$
Let
$b \in G(\breve {E})$
. We write
$\nu _b \in X_* (A)^+_{\mathbb {Q}}$
for the representative of
$[\tilde {\nu }_b]$
. Let w be the maximal length element in the Weyl group of G with respect to T. Then the map
$$\begin{align*}\mathrm{HN} \colon B(G) \to X_* (A)^+_{\mathbb{Q}}; \ [b] \mapsto w \cdot (- \nu_b) \end{align*}$$
is called the Harder–Narasimhan map. After equipping
$X_* (A)^+_{\mathbb {Q}}$
with the natural order topology, as discussed in [Reference Rapoport and RichartzRR96, Section 2], the map
$\mathrm {HN}$
is upper semicontinuous by [Reference Fargues and ScholzeFS21, Theorem III.2.3].
We define an algebraic group
$J_b$
over E by
$$\begin{align*}J_b (R)= \{ g \in G(R \otimes_E \breve{E} ) \mid g b \sigma (g)^{-1} = b \} \end{align*}$$
for any E-algebra R. Then we have
$J_b (E)= {\mathrm {Aut}} (\mathcal {F}_b)$
. We define a v-sheaf
$\widetilde {J}_b$
on
$ {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
by
$$\begin{align*}\widetilde{J}_b (S) = {\mathrm{Aut}} (\mathscr{E}_{b,S}) \end{align*}$$
for an
$S \in {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
. We note that the isomorphism class of
$J_b$
and
$\widetilde {J}_b$
depends only on
$[b] \in B(G)$
.
For a locally profinite group H, we write
$\underline {H}$
for v-sheaf on
$ {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
associated to H. Then we have an inclusion
$$\begin{align*}\underline{J_b(E)} \subset \widetilde{J}_b. \end{align*}$$
Let
$\widetilde {J}_b^0$
be the connected component of the unit section of
$\widetilde {J}_b$
. Then we have
$$\begin{align*}\widetilde{J}_b = \widetilde{J}_b^0 \rtimes \underline{J_b(E)} \end{align*}$$
and
$\widetilde {J}_b^0$
is of dimension
$\langle 2\rho , \nu _{b} \rangle $
by [Reference Fargues and ScholzeFS21, Proposition III.5.1]. In particular
$\underline {J_b(E)} = \widetilde {J}_b$
if and only if b is basic.
Let
$ {\mathrm {Bun}}_G^{\mathrm {ss}}$
be the semi-stable locus of
$ {\mathrm {Bun}}_G$
. Then
$ {\mathrm {Bun}}_G^{\mathrm {ss}}$
is an open substack of
$ {\mathrm {Bun}}_G$
by [Reference Fargues and ScholzeFS21, Theorem III.4.5]. Let
$\alpha \in \pi _1 (G)_{\Gamma }$
. Then the upper semicontinuity of
$\mathrm {HN}$
provides a stratification
$$\begin{align*}{\mathrm{Bun}}_{G}^{\alpha} = \coprod_{\nu \in X_*(A)^+_{\mathbb{Q}}} {\mathrm{Bun}}_{G}^{\alpha,\mathrm{HN}=\nu}. \end{align*}$$
Take
$\nu \in X_*(A)^+_{\mathbb {Q}}$
and assume that
$ {\mathrm {Bun}}_{G}^{\alpha ,\mathrm {HN}=\nu }$
is not empty. Then we have a unique
$[b] \in B(G)$
such that
$\kappa ([b])=\alpha $
and
$\mathrm {HN}([b])=\nu $
. Take any representative b of
$[b]$
. Then by [Reference Fargues and ScholzeFS21, Proposition III.5.3] we have an isomorphism
$$\begin{align*}x_b \colon [{\mathrm{Spa}} (\overline{\mathbb{F}}_q )/\widetilde{J}_b ] \stackrel{\sim}{\longrightarrow} {\mathrm{Bun}}_{G}^{\alpha,\mathrm{HN}=\nu} \end{align*}$$
defined by
$\mathscr {E}_{b}$
. If b is basic, then
$ {\mathrm {Bun}}_{G}^{\alpha ,\mathrm {HN}=\nu }$
is equal to the semi-stable locus
$ {\mathrm {Bun}}_{G}^{\alpha ,\mathrm {ss}}$
of
$ {\mathrm {Bun}}_{G}^{\alpha }$
by [Reference Fargues and ScholzeFS21, Theorem III.4.5]].
The
$\widetilde {J}_b$
-torsor
$\mathscr {T}_b$
over
$ {\mathrm {Bun}}_{G}^{\alpha ,\mathrm {HN}=\nu }$
given by
$x_b$
is the torsor defined by the functor which sends
$S \in {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
to
$$\begin{align*}\Big( f \colon S \longrightarrow {\mathrm{Bun}}_{G}^{\alpha,\mathrm{HN}=\nu}, \phi \colon \mathscr{E}_{b,S} \stackrel{\sim}{\longrightarrow} \mathscr{E}_f \Big) , \end{align*}$$
where
$\mathscr {E}_f$
is the G-bundle on
$X_S$
determined by f, and
$g \in \widetilde {J}_b (S)$
acts on
$\mathscr {T}_b (S)$
(on the right) by
$$ \begin{align} (f,\phi) \mapsto (f,\phi \circ g ). \end{align} $$
Then we have
$ {\mathrm {Frob}}^* x_b =x_{\sigma (b)}$
and
$ {\mathrm {Frob}}^* \mathscr {T}_b =\mathscr {T}_{\sigma (b)}$
. Since we have
$\sigma (b)=b^{-1} b \sigma (b)$
, we have a Weil descent datum
$$ \begin{align} w_b \colon {\mathrm{Frob}}^* \mathscr{T}_b \longrightarrow \mathscr{T}_b \end{align} $$
induced by
$t_{b^{-1}} \colon \mathscr {E}_{b,S} \to \mathscr {E}_{\sigma (b),S}$
in (1.1). Explicitly at the level of S-points, (1.3) sends
$(f,\phi )$
to
$(f, \phi \circ t_{b^{-1}})$
. If
$b'=g b \sigma (g)^{-1}$
, then
$t_g^{-1}$
induces an isomorphism
$\mathscr {T}_b \to \mathscr {T}_{b'}$
, which is compatible with the Weil descent data
$w_b$
and
$w_{b'}$
. Hence the isomorphism class of
$(\mathscr {T}_b,w_b)$
depends only on
$[b] \in B(G)$
.
2 The global Hecke stack
Let
$ {\mathrm {Div}}^1_{X,\mathbb {F}_q}$
be the moduli space of degree
$1$
closed Cartier divisors defined in [Reference Fargues and ScholzeFS21, Definition II.1.19], which sends
$S \in {\mathrm {Perf}}_{\mathbb {F}_q}$
to the set of isomorphism classes of degree
$1$
closed Cartier divisors on
$X_S$
. By [Reference Fargues and ScholzeFS21, Proposition II.1.21],
$ {\mathrm {Div}}_{X,\mathbb {F}_q}^1 \to {\mathrm {Spa}} (\mathbb {F}_q)$
is representable in spatial diamonds and we have an isomorphism
$$\begin{align*}{\mathrm{Spa}} (E)^{\diamond} / \varphi_{E^{\diamond}}^{\mathbb{Z}} \stackrel{\sim}{\longrightarrow} {\mathrm{Div}}_{X,\mathbb{F}_q}^1 , \end{align*}$$
where
$\varphi _{E^{\diamond }}$
is a q-th power Frobenius action on
$E^{\diamond }$
. We put
$ {\mathrm {Div}}_{X}^1 = {\mathrm {Div}}_{X,\mathbb {F}_q}^1 \times _{\mathbb {F}_q} \overline {\mathbb {F}}_q$
.
We write
$X_{*}(T)^+$
for the set of dominant cocharacters of T. Let
$\mu \in X_*(T)^+/\Gamma $
. We define a Hecke stack
$ {\mathrm {Hecke}}^{\leq \mu }$
as the fibred category in groupoids whose fibre at an affinoid perfectoid
$ {\mathrm {Spa}} (R,R^+) \in {\mathrm {Perf}}_{\mathbb {F}_q}$
is the groupoid of quadruples
$(\mathscr {E}, \mathscr {E}', D, f)$
, where
-
•
$\mathscr {E}$
and
$\mathscr {E}'$
are G-bundles on
$X_R^{ {\mathrm {sch}}}$
, -
• D is an effective Cartier divisor of degree
$1$
on
$X_R^{ {\mathrm {sch}}}$
given by some untilt of R, -
• the isomorphism
is a modification, which is bounded by
$$\begin{align*}f \colon \mathscr{E}|_{X_R^{{\mathrm{sch}}} \setminus D} \stackrel{\sim}{\longrightarrow} \mathscr{E}'|_{X_R^{{\mathrm{sch}}} \setminus D} \end{align*}$$
$\mu $
geometric fibrewisely.
Then we have morphisms
defined by
$\overleftarrow {h} (\mathscr {E}, \mathscr {E}', D, f) =\mathscr {E}'$
and
$\overrightarrow {h} (\mathscr {E}, \mathscr {E}', D, f) =(\mathscr {E},D)$
.
In the sequel, a diamond means a diamond on
$ {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
. Let
$\ell $
be a prime number different from p. As we will need the natural functor (i.e. relative homology) constructed in [Reference Fargues and ScholzeFS21], let us briefly review it. For X a small v-stack, the derived category of solid
$\overline {\mathbb {Q}}_{\ell }$
-sheaves 
is constructed in [Reference Fargues and ScholzeFS21, Definition VII.1.17]. For what follows all tensor products are solid tensor products as constructed in [Reference Fargues and ScholzeFS21, Proposition VII.2.2]. For a map
$f \colon X \to Y$
of small v-stacks, there is a functor
constructed in [Reference Fargues and ScholzeFS21, §VII.3]. See [Reference Fargues and ScholzeFS21, Proposition VII.3.1] for basic properties of this functor. For an
$\ell $
-cohomologically smooth morphism
$f \colon X \to Y$
of diamonds, we put
For an Artin v-stack X, let 
be the subcategory defined in [Reference Fargues and ScholzeFS21, Definition VII.6.1].
Let
$\mathcal {D}_{\infty }$
be a diamond over
$\mathbb {C}_p^{\flat }$
with an action of a profinite group
$K_0$
. Let
$f_{\infty } \colon \mathcal {D}_{\infty } \to {\mathrm {Spa}} (\mathbb {C}_p^{\flat })$
be the structure morphism. Assume that the action of
$K_0$
on geometric points of
$\mathcal {D}_{\infty }$
is free and the quotient diamond
$\mathcal {D}_{\infty }/K_0$
is an
$\ell $
-cohomologically smooth diamond over
$\mathbb {C}_p^{\flat }$
. For an open subgroup K of
$K_0$
, we put
$\mathcal {D}_K =\mathcal {D}_{\infty }/K$
, and let
$f_K \colon \mathcal {D}_K \to {\mathrm {Spa}} (\mathbb {C}_p^{\flat })$
be the induced morphism. Then we put
$$\begin{align*}H_{\mathrm{c}}^i (\mathcal{D}_{\infty},\overline{\mathbb{Q}}_{\ell}) = \varinjlim_{K \subset K_0} R^i f_{K,\natural} ((f_K^!\overline{\mathbb{Q}}_{\ell})^{\vee}) \end{align*}$$
for
$i \geq 0$
. Let
$f \colon \mathcal {D} \to {\mathrm {Spa}} (\mathbb {C}_p^{\flat })$
be an
$\ell $
-cohomologically smooth morphism of diamonds. For 
and
$i \geq 0$
, we put
$$\begin{align*}H_{\mathrm{c}}^i (\mathcal{D},\mathscr{F} ) = R^i f_{\natural} (\mathcal{F} \otimes (f^!\overline{\mathbb{Q}}_{\ell})^{\vee}). \end{align*}$$
Let
$h \colon \mathcal {M} \to \mathcal {D}$
be a
$G_0$
-torsor such that
$$\begin{align*}\varinjlim_{K \subset G_0} R f_{K,\natural} ((f_K^!\overline{\mathbb{Q}}_{\ell})^{\vee}) \in D_{{\mathrm{lis}}}({\mathrm{Spa}} (\mathbb{C}_p^{\flat}),\overline{\mathbb{Q}}_{\ell}) , \end{align*}$$
where
$G_0$
is a locally profinite group, K runs along compact open subgroups of
$G_0$
and
$f_K \colon \mathcal {M} /K \to {\mathrm {Spa}} (\mathbb {C}_p^{\flat })$
. Then we can regard
$H_{\mathrm {c}}^j (\mathcal {M} ,\overline {\mathbb {Q}}_{\ell })$
as a smooth representation of
$G_0$
. Let
$\pi $
be a smooth representation of
$G_0$
over
$\overline {\mathbb {Q}}_{\ell }$
. We define
$\mathscr {F}_{\pi } \in D_{ {\mathrm {lis}}}( \mathcal {D}, \overline {\mathbb {Q}}_{\ell })$
as the pushforward of
$\mathcal {M}$
by
$\pi $
. Then we have a spectral sequence
$$ \begin{align} H_i \big( G_0, H_{\mathrm{c}}^j (\mathcal{M} ,\overline{\mathbb{Q}}_{\ell}) \otimes \pi \big) \Rightarrow H_{\mathrm{c}}^{j-i} (\mathcal{D},\mathscr{F}_{\pi} ). \end{align} $$
This follows from [Reference Fargues and ScholzeFS21, Proposition VII.3.1] as in the proof of [Reference ImaiIma19, Lemma 1.4].
3 Fargues’ conjecture
We recall the Hecke eigensheaf property in Fargues’ conjecture in the case where the Langlands parameter is cuspidal and
$\mu $
is minuscule. Up to some technicalities which were worked out in [Reference Fargues and ScholzeFS21], we refer the reader to [Reference FarguesFar25, Conjecture 4.4(4)] for the general case.
Let
$\widehat {G}$
and
${}^L G$
be the dual group and L-group of G over
$\overline {\mathbb {Q}}_{\ell }$
. Let
$\varphi \colon W_E \rightarrow {}^L G$
be a cuspidal
$\ell $
-adic L-parameter for G (cf. [Reference ImaiIma24a, Definition 1.15], [Reference FarguesFar25, Definition 4.1]). Let
$S_{\varphi }$
be the centraliser of
$\varphi $
in
$\widehat {G}$
. We fix a Whittaker datum. For
$b \in B(G)_{ {\mathrm {basic}}}$
, let
$\{ \pi _{\varphi ,b,\rho } \}_{\rho \in \widehat {S}_{\varphi }}$
be the L-packet corresponding to
$\varphi $
by the local Langlands correspondence for the extended pure inner form
$J_b$
of G (cf. [Reference KalethaKal14, Conjecture 2.4.1]). We recall that we have a decomposition
$$\begin{align*}{\mathrm{Bun}}_{G}^{\mathrm{ss}} = \coprod_{\alpha \in \pi_1 (G)_{\Gamma}} {\mathrm{Bun}}_{G}^{\alpha ,\mathrm{ss}} \end{align*}$$
into open and closed substacks. Let
$\mathscr {F}_{\varphi }$
be the object of
$D_{ {\mathrm {lis}}} ( {\mathrm {Bun}}_{G},\overline {\mathbb {Q}}_{\ell })$
with an action of
$S_{\varphi }$
determined by the following conditions:
-
• The support of
$\mathscr {F}_{\varphi }$
is contained in
$ {\mathrm {Bun}}_{G}^{\mathrm {ss}}$
. -
• Let
$\alpha \in \pi _1 (G)_{\Gamma }$
. Take a basic element
$b \in G(\breve {E})$
such that
$\alpha =\kappa ([b])$
. Let
$\rho \in \widehat {S}_{\varphi }$
. Let
$\underline {\rho }$
be the constant
$\overline {\mathbb {Q}}_{\ell }$
-sheaf with action of
$S_{\varphi }$
on
$ {\mathrm {Bun}}_{G}^{\alpha ,\mathrm {ss}}$
associated to
$\rho $
. Let
$\underline {\pi _{\varphi ,b,\rho }}$
be the object of
$D_{ {\mathrm {lis}}} ( {\mathrm {Bun}}_{G}^{\alpha ,\mathrm {ss}},\overline {\mathbb {Q}}_{\ell })$
obtained as the pushforward of the
$\underline {J_b (E)}$
-torsor
$\mathscr {T}_b$
under
$\pi _{\varphi ,b,\rho }$
. Then we have (3.1)where we view
$$ \begin{align} \mathscr{F}_{\varphi}|_{{\mathrm{Bun}}_{G}^{\alpha ,\mathrm{ss}}} = \bigoplus_{\rho \in \widehat{S}_{\varphi},\, \rho|_{Z(\widehat{G})^{\Gamma}} =\alpha} \underline{\rho} \otimes \underline{\pi_{\varphi,b,\rho}}, \end{align} $$
$\alpha $
as an element of
$X^* (Z(\widehat {G})^{\Gamma })$
under the canonical isomorphism
$\pi _1 (G)_{\Gamma } \simeq X^* (Z(\widehat {G})^{\Gamma })$
. The isomorphism class of the right-hand side of (3.1) does not depend on the choice of b, since the same is true for
$\mathscr {T}_b$
.
Then properties (1), (2) and (3) of [Reference FarguesFar25, Conjecture 4.4] are immediate.
Take a representative
$\mu ' \in X_* (T)^+$
of
$\mu $
. Let
$\Gamma '$
be the stabiliser of
$\mu '$
in
$\Gamma $
. We put
$$\begin{align*}r_{\mu} = {\mathrm{Ind}}_{\widehat{G} \rtimes \Gamma'}^{{}^L G} r_{\mu'} , \end{align*}$$
where
$r_{\mu '}$
is the highest-weight-
$\mu '$
irreducible representation of
$\widehat {G} \rtimes \Gamma '$
.
As in [Reference Fargues and ScholzeFS21, IX.2], we can construct a functor
via the geometric Satake equivalence (cf. [Reference ImaiIma24b, §10]). Let
$ {\mathrm {IC}}_{\mu }'$
be the image of
$r_{\mu }$
under the functor (3.2).
Now we can state the Hecke eigensheaf property in Fargues’ conjecture:
Conjecture 3.1. We have
$$ \begin{align*}\overrightarrow{h}_{\natural}(\overleftarrow{h}^{*} \mathscr{F}_{\varphi} {\otimes}_{\overline{\mathbb{Q}}_{\ell}} {\mathrm{IC}}_{\mu}') \cong \mathscr{F}_{\varphi} \boxtimes (r_{\mu} \circ \varphi )\end{align*} $$
as objects of 
with actions of
$S_{\varphi }$
.
4 Non-semi-stable locus
Let
$b ,b' \in G(\breve {E})$
. We have a natural morphism
$$\begin{align*}y_b \colon [{\mathrm{Div}}_{X}^1 /\widetilde{J}_b ] \simeq [{\mathrm{Spa}} ( \overline{\mathbb{F}}_q ) /\widetilde{J}_b ] \times {\mathrm{Div}}_X^1 \xrightarrow{(x_b,{\mathrm{id}} )} {\mathrm{Bun}}_{G} \times {\mathrm{Div}}_X^1. \end{align*}$$
We consider the cartesian diagram (i.e. every sub-square is cartesian)
By the construction, for a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra
$(R,R^+)$
, the groupoid
$ {\mathrm {Hecke}}_{b,b'}^{\leq \mu } (R,R^+)$
consists of quadruples
$(\mathscr {E}, \mathscr {E}', D, f)$
, where
-
•
$\mathscr {E}$
and
$\mathscr {E}'$
are G-bundles on
$X_R^{\mathrm {sch}}$
which are isomorphic to
$\mathscr {E}_b$
and
$\mathscr {E}_{b'}$
fibrewisely over
$ {\mathrm {Spa}} (R,R^+)$
. -
• D is an effective Cartier divisor of degree
$1$
on
$X_R^{ {\mathrm {sch}}}$
, -
•
$f \colon \mathscr {E}|_{X_R^{\mathrm {sch}} \setminus D} \to \mathscr {E}'|_{X_R^{\mathrm {sch}} \setminus D}$
is a modification bounded by
$\mu $
geometric fiberwisely over
$ {\mathrm {Spa}} (R,R^+)$
.
Let
$\mathcal {T}_{b,b'}^{\leq \mu }$
be the
$\widetilde {J}_b$
-torsor over
$ {\mathrm {Hecke}}_{b,b'}^{\leq \mu }$
obtained by considering an isomorphism
$\phi \colon \mathscr {E}_b \xrightarrow {\sim } \mathscr {E}$
. Let
$ {\mathrm {Gr}}_{b,b'}^{\leq \mu }$
and
$\mathcal {M}_{b,b'}^{\leq \mu }$
be the
$\widetilde {J}_{b'}$
-torsors over
$ {\mathrm {Hecke}}_{b,b'}^{\leq \mu }$
and
$\mathcal {T}_{b,b'}^{\leq \mu }$
obtained by considering an isomorphism
$\phi ' \colon \mathscr {E}_{b'} \xrightarrow {\sim } \mathscr {E}'$
, respectively. Then
$\mathcal {M}_{b,b'}^{\leq \mu }$
is a
$\widetilde {J}_{b'}$
-equivariant
$\widetilde {J}_b$
-torsor over
$ {\mathrm {Gr}}_{b,b'}^{\leq \mu }$
. We have commutative diagrams
where the sub-squares are cartesian.
By [Reference FarguesFar25, Proposition 3.20],
$\mathcal {T}_{b,b'}^{\leq \mu }$
is a diamond. Furthermore by [Reference ScholzeSch17, Lemma 10.13, Proposition 11.5],
$\mathcal {M}_{b,b'}^{\leq \mu }$
is a diamond if
$b'$
is basic.
Remark 4.1. The maps
$\mathcal {M}_{b,b'}^{\leq \mu } \to {\mathrm {Gr}}_{b,b'}^{\leq \mu }$
and
$\mathcal {M}_{b,b'}^{\leq \mu } \to \mathcal {T}_{b,b'}^{\leq \mu }$
appearing in the above diagram are generalised versions of the Hodge–Tate period map and the Gross–Hopkins period map. Indeed if
$b'=1$
and
$\mu $
is minuscule then
$\mathcal {M}_{b,b'}^{\leq \mu } \to {\mathrm {Gr}}_{b,b'}^{\leq \mu }$
is the usual Hodge–Tate period map of a Rapoport–Zink space at infinite level associated to the isocrystal b and
$\mathcal {M}_{b,b'}^{\leq \mu } \to \mathcal {T}_{b,b'}^{\leq \mu }$
is the usual Gross–Hopkins period map. On the other hand if
$b=1$
and
$\mu $
is minuscule then
$\mathcal {M}_{b,b'}^{\leq \mu } \to {\mathrm {Gr}}_{b,b'}^{\leq \mu }$
is the Gross–Hopkins map and
$\mathcal {M}_{b,b'}^{\leq \mu } \to \mathcal {T}_{b,b'}^{\leq \mu }$
is the Hodge–Tate map associated to the isocrystal
$b'$
.
For a finite-dimensional algebraic representation V of G and a rational number
$\alpha $
, we put
$$\begin{align*}{\mathrm{Fil}}_b^{\alpha} V = \bigoplus_{\alpha' \leq - \alpha} V_{\alpha'} , \end{align*}$$
where
$$\begin{align*}V = \bigoplus_{\alpha \in \mathbb{Q}} V_{\alpha} \end{align*}$$
is the slope decomposition given by
$\nu _b \in X_* (A)_{\mathbb {Q}}^+$
. This gives a filtration
$ {\mathrm {Fil}}_b$
on the forgetful fibre functor
$\omega \colon {\mathrm {Rep}} (G) \to \mathrm {Vect}_E$
(cf. [Reference Saavedra RivanoSR72, IV, 2.1]). The stabiliser of
$ {\mathrm {Fil}}_b \omega $
gives a parabolic subgroup
$P^b$
of G. Let
$L^b$
be the centraliser of
$\nu _b \in X_* (A)_{\mathbb {Q}}^+$
. Take a Levi subgroup L of G containing
$L^b$
. We put
$P=L P^b$
. Then, P is a parabolic subgroup of G and
$[b] \in B(G)$
is the image of an element
$b_{00} \in L^b (\breve {E})$
. Let
$b_0$
be the image of
$b_{00}$
in
$L (\breve {E})$
.
We take a cocharacter
$\lambda \in X_* (A)$
so that P is associated to
$\lambda $
in the sense of [Reference SpringerSpr98, 13.4.1]. Then we have a filtration
$ {\mathrm {Fil}}_{\lambda }$
on
$\omega $
associated to
$\lambda $
.
We assume that
$[b']$
is in the image of
$B(L) \to B(G)$
. Then
$ {\mathrm {Fil}}_{\lambda } \omega $
induces the filtrations
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_b$
and
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_{b'}$
as fibre functors by the construction, because
$[b], [b']$
are in the image of
$B(L) \to B(G)$
and L is the centraliser of
$\lambda $
in G.
We define a closed subspace
$\mathcal {C}_{b,b'}^{\leq \mu }$
of
$ {\mathrm {Gr}}_{b,b'}^{\leq \mu }$
as a functor that sends a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra
$(R,R^+)$
to the isomorphism classes of
$(\mathscr {E}, \mathscr {E}', D, f, \phi ')$
, where
-
•
$(\mathscr {E}, \mathscr {E}', D, f)$
is as in
$ {\mathrm {Hecke}}_{b,b'}^{\leq \mu } (R,R^+)$
, -
•
$\phi ' \colon \mathscr {E}_{b'} \xrightarrow {\sim } \mathscr {E}'$
and f are compatible with
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_b$
and
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_{b'}$
geometric fibrewisely in the sense that following holds for any geometric point
$ {\mathrm {Spa}} (F,F^+)$
of
$ {\mathrm {Spa}} (R,R^+)$
: Take an isomorphism
$\mathscr {E}_{b} \xrightarrow {\sim } \mathscr {E}$
over
$X_F^{\mathrm {sch}}$
. Let
$D_F$
be a Cartier divisor of
$X_F^{\mathrm {sch}}$
determined by D. Then the composite respects the filtrations
$$\begin{align*}\mathscr{E}_{b}|_{X_F^{\mathrm{sch}}\setminus D_F} \stackrel{\sim}{\longrightarrow} \mathscr{E}|_{X_F^{\mathrm{sch}}\setminus D_F} \stackrel{f}{\longrightarrow} \mathscr{E}'|_{X_F^{\mathrm{sch}}\setminus D_F} \xrightarrow{\phi^{\prime-1}} \mathscr{E}_{b'}|_{X_F^{\mathrm{sch}}\setminus D_F} \end{align*}$$
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_b |_{X_F^{\mathrm {sch}}\setminus D_F}$
and
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_{b'} |_{X_F^{\mathrm {sch}}\setminus D_F}$
.
Remark 4.2. The condition that
$\phi '$
and f are compatible with
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_b$
and
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_{b'}$
is independent of choice of an isomorphism
$\mathscr {E}_{b} \xrightarrow {\sim } \mathscr {E}$
, because the automorphism group
$\widetilde {J}_b$
of
$\mathscr {E}_{b}$
respects the filtration
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_b$
.
For
$\mu \in X_* (T)$
, we put
$$\begin{align*}\overline{\mu} = \frac{1}{[\Gamma :\Gamma_{\mu}]} \sum_{\tau \in \Gamma / \Gamma_{\mu}} \tau (\mu) , \end{align*}$$
where
$\Gamma _{\mu }$
is a stabiliser of
$\mu $
in
$\Gamma $
, and let
$\mu ^{\natural }$
denote the image of
$\mu $
in
$\pi _1 (G)_{\Gamma }$
.
Definition 4.3. (cf. [Reference Rapoport and ViehmannRV14, Definition 2.5]) We say that
$[b] \in B(G)$
is acceptable for
$(\mu ,[b'])$
if
$\nu _b -\nu _{b'} \leq \overline {\mu }$
. We say that
$[b] \in B(G)$
is neutral for
$(\mu ,[b'])$
if
$\kappa _G ([b]) -\kappa _G ([b']) = \mu ^{\natural }$
.
Let
$B(G,\mu ,[b'])$
be the set of acceptable neutral elements in
$B(G)$
for
$(\mu ,[b'])$
.
Remark 4.4. The set
$B(G,\mu ,[b'])$
is a twisted analogue of the set
$B(G,\mu )$
, the latter due to Kottwitz. We refer the reader to [Reference KottwitzKot97, §6.2] for this definition.
To state our main results we need the notion of Hodge–Newton reducibility.
Definition 4.5 (cf. [Reference Rapoport and ViehmannRV14, Definition 4.28])
A triple
$([b],[b'],\mu )$
such that
$[b] \in B(G,\mu ,[b'])$
and
$b'$
is basic is called Hodge–Newton reducible, if there is a standard proper Levi subgroup L of G and
$[b_0] ,[b_0'] \in B(L)$
such that
$[b]$
and
$[b']$
are the images of
$[b_0]$
and
$[b_0']$
, respectively,
$\mu $
factors through L,
$[b_0] \in B(L,\mu ,[b^{\prime }_0])$
and the action of
$\nu _{b_0}$
on
$R_{\mathrm {u}}(B)$
is non-negative.
Lemma 4.6. Let R be a DVR with maximal ideal
$\mathfrak {m}$
, and M be an R-module such that
$M \simeq \bigoplus _{1 \leq i \leq n} R/\mathfrak {m}^{k_i}$
, where
$k_1 \geq \cdots \geq k_n$
is a sequence of non-negative integers. Let N be a quotient of M generated by j elements, where
$j \leq n$
. Then we have
$l (N) \leq k_1 + \cdots + k_j$
. Further, if the equality holds, then N is a direct summand of M.
Proof. This follows from [Reference HansenHan21a, Lemma 3.2] by taking the Pontryagin dual.
The following proposition is a slight generalisation of [Reference HansenHan21a, Theorem 3.1], where the slope of a semi-stable bundle is assumed to be zero.
Proposition 4.7. Assume that
$G= {\mathrm {GL}}_n$
. Let
$(k_1 \geq \cdots \geq k_n)$
be the sequence of integers corresponding to
$\mu \in X_*(T)^+$
. Let
$(R,R^+)$
be a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra. Let
$$\begin{align*}f \colon \mathscr{E}|_{X_R^{{\mathrm{sch}}} \setminus D} \stackrel{\sim}{\longrightarrow} \mathscr{E}'|_{X_R^{{\mathrm{sch}}} \setminus D} \end{align*}$$
be a modification between G-bundles
$\mathscr {E}$
and
$\mathscr {E}'$
over
$X_R^{ {\mathrm {sch}}}$
along an effective Cartier divisor of degree
$1$
which is equal to
$\mu $
geometric fibrewisely. We view
$\mathscr {E}$
and
$\mathscr {E}'$
as vector bundles of rank n. Let
$\mathscr {E}^+$
be a saturated sub-vector bundle of
$\mathscr {E}$
such that
$$ \begin{align} \deg (\mathscr{E}_x^+ ) + \sum_{1 \leq j \leq {\mathrm{rk}} (\mathscr{E}^+)} k_{n+1-j} = {\mathrm{rk}} (\mathscr{E}^+) s \end{align} $$
for every point x of
$Spa (R,R^+)$
.
Assume that
$\mathscr {E}'$
is semi-stable of slope s geometric fibrewisely. Let
$j \colon X_R^{ {\mathrm {sch}}} \setminus D \to X_R^{ {\mathrm {sch}}}$
be the open immersion. We put
$$\begin{align*}\mathscr{E}^{\prime+} =j_* f(j^*\mathscr{E}^+ ) \cap \mathscr{E}'. \end{align*}$$
Then
$\mathscr {E}^{\prime +}$
is a semi-stable vector bundle of slope s such that
$ {\mathrm {rk}} (\mathscr {E}^{\prime +}) = {\mathrm {rk}} (\mathscr {E}^+)$
.
Proof. We follow arguments in the proof of [Reference HansenHan21a, Theorem 3.1].
Take a modification
$f_1 \colon \mathcal {O}|_{X_R^{ {\mathrm {sch}}} \setminus D} \xrightarrow {\sim } \mathcal {O} (1)|_{X_R^{ {\mathrm {sch}}} \setminus D}$
of degree
$1$
along D. For a large N, changing
$\mathscr {E}'$
, f and
$(k_1 ,\ldots ,k_n)$
by
$\mathscr {E}' (N)$
,
$$\begin{align*}({\mathrm{id}}_{\mathscr{E}'} \otimes f_1^{\otimes N}) \circ f \colon \mathscr{E} |_{X_R^{{\mathrm{sch}}} \setminus D} \stackrel{\sim}{\longrightarrow} \mathscr{E}'(N)|_{X_R^{{\mathrm{sch}}} \setminus D} \end{align*}$$
and
$(k_1 +N ,\ldots ,k_n +N )$
, respectively, we may assume that f extends to an injective morphism
$f \colon \mathscr {E} \to \mathscr {E}'$
, which induces a morphism
$f^+ \colon \mathscr {E}^+ \to \mathscr {E}^{\prime +}$
. We put
$\mathscr {E}^- =\mathscr {E} / \mathscr {E}^+$
and
$\mathscr {E}^{\prime -} =\mathscr {E}' / \mathscr {E}^{\prime +}$
. Let
$f^- \colon \mathscr {E}^- \to \mathscr {E}^{\prime -}$
be the morphism induced by f.
First, we treat the case where R is a perfectoid field. In this case,
$\mathscr {E}^{\prime +}$
and
$\mathscr {E}^{\prime -}$
are vector bundles such that
$ {\mathrm {rk}} (\mathscr {E}^{\prime +}) = {\mathrm {rk}} (\mathscr {E}^+)$
and
$ {\mathrm {rk}} (\mathscr {E}^{\prime -}) = {\mathrm {rk}} (\mathscr {E}^-)$
. Let
$Q^+$
and
$Q^-$
be the cokernel of
$f^+$
and
$f^-$
, respectively. Then we have
$$\begin{align*}l (Q^-) \leq \sum_{1 \leq i \leq {\mathrm{rk}} (\mathscr{E}^-) } k_i \end{align*}$$
by Lemma 4.6, since
$Q^-$
is generated by
$ {\mathrm {rk}} (\mathscr {E}^-)$
-elements. Hence we have
$$\begin{align*}l (Q^+) \geq \sum_{1 \leq j \leq {\mathrm{rk}} (\mathscr{E}^+) } k_{n+1-j}. \end{align*}$$
By this and (4.1), we have
$$\begin{align*}\deg (\mathscr{E}^{\prime+} )= \deg (\mathscr{E}^+) +l (Q^+) \geq {\mathrm{rk}} (\mathscr{E}^+) s. \end{align*}$$
On the other hand, we have
$\deg (\mathscr {E}^{\prime +} ) \leq {\mathrm {rk}} (\mathscr {E}^+) s$
, since
$\mathscr {E}'$
is semi-stable. Therefore,
$\mathscr {E}^{\prime +}$
is a semi-stable vector bundle of slope s.
The general case is reduced to the above case by the same argument as in [Reference HansenHan21a, §3.2].
Lemma 4.8. Let
$(R,R^+)$
be a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra. For any element
$\alpha $
of
$H_{ {\mathrm {et}}}^1 (X_R^{ {\mathrm {sch}}},\mathcal {O} )$
, there is a pro-etale extension
$(R',R^{\prime +})$
of
$(R,R^+)$
such that the image of
$\alpha $
in
$H_{ {\mathrm {et}}}^1 (X_{R'}^{ {\mathrm {sch}}},\mathcal {O} )$
is zero.
Proof. Any extension of
$\mathcal {O}$
by
$\mathcal {O}$
on
$X_R^{ {\mathrm {sch}}}$
splits after a pro-etale extension of
$(R,R^+)$
by [Reference Fargues and FontaineFF14, 6.3.1] and [Reference FarguesFar25, Theorem 2.26] (cf. [Reference Kedlaya and LiuKL15, Corollary 8.7.10]). This implies the claim, since
$H_{ {\mathrm {et}}}^1 (X_R^{ {\mathrm {sch}}},\mathcal {O} )$
parametrises the extensions of
$\mathcal {O}$
by
$\mathcal {O}$
on
$X_R^{ {\mathrm {sch}}}$
.
Assume that
$b'$
is basic. Let U be the unipotent radical of P. Note that we have a surjection
$$\begin{align*}P \longrightarrow P/U \simeq L, \end{align*}$$
where the second isomorphism is given by
$L \hookrightarrow P \to P/U$
.
Lemma 4.9. Let
$(R,R^+)$
be a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra. Let
$\mathscr {E}_P$
be a P-bundle on
$X_R^{ {\mathrm {sch}}}$
such that
$\mathscr {E}_P \times ^P L \simeq \mathscr {E}_{b^{\prime }_0}$
. Then we have an isomorphism
$\mathscr {E}_P \simeq \mathscr {E}_{b^{\prime }_0} \times ^L P $
after a pro-etale extension of
$(R,R^+)$
.
Proof. We follow arguments in the proof of [Reference FarguesFar20, Proposition 5.16]. Let P act on U by the conjugation. We put
$$\begin{align*}\mathscr{U} = \mathscr{E}_P \times^P U. \end{align*}$$
Then
$H_{ {\mathrm {et}}}^1 (X_R^{ {\mathrm {sch}}},\mathscr {U})$
parametrises the fibre of
$$\begin{align*}H_{{\mathrm{et}}}^1 (X_R^{{\mathrm{sch}}}, P) \longrightarrow H_{{\mathrm{et}}}^1 (X_R^{{\mathrm{sch}}},L) \end{align*}$$
over the image of
$\mathscr {E}_P$
. Hence, it suffices to show that
$H_{ {\mathrm {et}}}^1 (X_R^{ {\mathrm {sch}}},\mathscr {U})$
is trivial after a pro-etale extension of
$(R,R^+)$
. This follows from Lemma 4.8, since
$\mathscr {U}$
has a filtration whose graded subquotients are semi-stable vector bundles of slope zero.
Lemma 4.10. Let
$\mu _1,\mu _2 \in X_{*}(T)^{+}$
such that
$\mu _1 \leq \mu _2$
. Then
$ {\mathrm {Hecke}}^{\leq \mu _1} \subset {\mathrm {Hecke}}^{\leq \mu _2}$
is a closed substack.
Proof. By [Reference FarguesFar25, Proposition 3.20], it is enough to prove
$ {\mathrm {Gr}}_G^{\leq \mu _1} \subset {\mathrm {Gr}}_G^{\leq \mu _2}$
is closed substack. The latter follows from the semi-continuity of the map
$\lvert {\mathrm {Gr}} \rvert \to X_* (T)^+/\Gamma $
in [Reference FarguesFar25, 3.3.2] (cf. [Reference Scholze and WeinsteinSW20, Proposition 19.2.3]).
We define a substack
$ {\mathrm {Hecke}}^{\mu }$
of
$ {\mathrm {Hecke}}^{\leq \mu }$
by requiring the condition that modifications are equal to
$\mu $
geometric fibrewisely. Then
$ {\mathrm {Hecke}}^{\mu }$
is an open substack of
$ {\mathrm {Hecke}}^{\leq \mu }$
by Lemma 4.10. We use similar definitions and notations also for other spaces.
Let X be a scheme over E. Let
$\mathrm {FilVect}_{X}$
be the category of filtered vector bundles on X. We consider the functor
$$\begin{align*}\omega_{\lambda} \colon {\mathrm{Rep}}(G) \longrightarrow \mathrm{FilVect}_{X} ;\ V \mapsto (V \otimes_E \mathcal{O}_X , ({\mathrm{Fil}}_{\lambda} V) \otimes_E \mathcal{O}_X). \end{align*}$$
Let
$\mathrm {Fil}_{\lambda }\mathrm {Bun}^{G}_{X}$
be the category of functors
$\omega \colon {\mathrm {Rep}}(G) \to \mathrm {FilVect}_{X}$
which are isomorphic to
$\omega _{\lambda }$
fpqc locally on X. Let
$\mathrm {Bun}^P_{X}$
be the category of P-bundles on X.
Lemma 4.11. There is an equivalence of categories
$$\begin{align*}\mathrm{Fil}_{\lambda}\mathrm{Bun}^{G}_{X} \longrightarrow \mathrm{Bun}^P_{X} ;\ \omega \mapsto \underline{{\mathrm{Isom}}}^{\otimes}_{X} (\omega_{\lambda},\omega), \end{align*}$$
where
$\underline { {\mathrm {Isom}}}^{\otimes }_{X} (\omega _{\lambda },\omega )$
is a functor from the category of schemes over X to the category of sets which sends
$X'$
to the set of isomorphisms
$\omega _{\lambda }|_{X'} \to \omega |_{X'}$
as filtered tensor functors.
Proof. This follows from [Reference ZieglerZie15, Theorem 4.42 and Theorem 4.43].
Proposition 4.12. Assume that
$([b],[b'],\mu )$
is Hodge–Newton reducible for L. Let
$(R,R^+)$
be a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra, and
$(\mathscr {E}, \mathscr {E}', D, f) \in {\mathrm {Hecke}}_{b,b'}^{\mu } (R,R^+)$
. Then, after taking a pro-etale extension of
$(R,R^+)$
, there is a reduction
$$\begin{align*}f_P \colon \mathscr{E}_P|_{X_R^{{\mathrm{sch}}} \setminus D} \stackrel{\sim}{\longrightarrow} \mathscr{E}^{\prime}_P|_{X_R^{{\mathrm{sch}}} \setminus D} \end{align*}$$
of f to P such that
$\mathscr {E}_P \simeq \mathscr {E}_{b_0} \times ^L P$
and
$\mathscr {E}^{\prime }_P \simeq \mathscr {E}_{b_0'} \times ^L P$
.
Proof. By taking a pro-etale extension of
$(R,R^+)$
, we can take an isomorphism
$\mathscr {E}_b \simeq \mathscr {E} $
. We put
$\mathscr {E}_P =\mathscr {E}_{b_0} \times ^L P$
. Then
$\mathscr {E}_P$
and the isomorphism
$$\begin{align*}\mathscr{E}_P \times^P G \cong \mathscr{E}_{b_0} \times^L G \cong \mathscr{E}_b \stackrel{\sim}{\longrightarrow} \mathscr{E} \end{align*}$$
give a reduction of
$\mathscr {E}$
to P. We put
$\phi _P = {\mathrm {id}}_{\mathscr {E}_{b_0} \times ^L P}$
. Then
$\phi _P$
is a reduction of
$\phi $
to P.
For any irreducible
$V \in {\mathrm {Rep}}(G)$
, the vector bundle
$\mathscr {E}'(V)$
is semi-stable geometric fibrewisely. By Proposition 4.7, we have a functorial construction of a filtration of
$\mathscr {E}'(V)$
that is compatible under
$f(V)$
with the filtration of
$\mathscr {E}(V)$
coming from
$\mathscr {E}_P$
by Lemma 4.11. Since the category
$ {\mathrm {Rep}}(G)$
is semi-simple, the construction extends to all
$V \in {\mathrm {Rep}}(G)$
in a functorial way. Hence, by Lemma 4.11, we have a reduction
$$\begin{align*}f_P \colon \mathscr{E}_P|_{X_R^{{\mathrm{sch}}} \setminus D} \stackrel{\sim}{\longrightarrow} \mathscr{E}^{\prime}_P|_{X_R^{{\mathrm{sch}}} \setminus D} \end{align*}$$
of f to P for some P-bundle
$\mathscr {E}^{\prime }_P$
. By Lemma 4.9,
$\mathscr {E}^{\prime }_P$
is isomorphic to
$\mathscr {E}_{b_0'} \times ^L P$
after taking a pro-etale extension of
$(R,R^+)$
.
Let
$\widetilde {P}_{b'}$
be the stabiliser of
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_{b'}$
in
$\widetilde {J}_{b'}$
. Then
$\widetilde {P}_{b'} =\underline {P_{b'} (E)}$
for a parabolic subgroup
$P_{b'}$
of
$J_{b'}$
.
Proposition 4.13. Assume that
$([b],[b'],\mu )$
is Hodge–Newton reducible for L. Then the action of
$\widetilde {P}_{b'}$
on
$ {\mathrm {Gr}}_{b,b'}^{\mu }$
stabilises
$\mathcal {C}_{b,b'}^{\mu }$
, and we have a natural
$\widetilde {J}_{b'}$
-equivariant isomorphism
$$\begin{align*}\mathcal{C}_{b,b'}^{\mu} \times^{\widetilde{P}_{b'}} \widetilde{J}_{b'} \stackrel{\sim}{\longrightarrow} {\mathrm{Gr}}_{b,b'}^{\mu}. \end{align*}$$
Proof. The first claim follows from the definitions of
$\widetilde {P}_{b'}$
and
$ {\mathrm {Gr}}_{b,b'}^{\mu }$
. The morphism
$$\begin{align*}\mathcal{C}_{b,b'}^{\mu} \times^{\widetilde{P}_{b'}} \widetilde{J}_{b'} \longrightarrow {\mathrm{Gr}}_{b,b'}^{\mu} \end{align*}$$
induced by the action of
$\widetilde {J}_{b'}$
on
$ {\mathrm {Gr}}_{b,b'}^{\mu }$
is an epimorphism by Proposition 4.12.
We show the injectivity. Let
$g \in \widetilde {J}_{b'} (R,R^+)$
for a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra
$(R,R^+)$
. Assume that g sends a point of
$\mathcal {C}_{b,b'}^{\mu } (R,R^+)$
to a point of
$\mathcal {C}_{b,b'}^{\mu } (R,R^+)$
. Then g stabilises
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_{b'}$
outside the Cartier divisor corresponding to
$R^{\sharp }$
. This implies g stabilises
$ {\mathrm {Fil}}_{\lambda } \mathscr {E}_{b'}$
on
$X_R^{\mathrm {sch}}$
, since g stabilises
$\mathscr {E}_{b'}$
itself. Hence, we have
$g \in \widetilde {P}_{b'} (R,R^+)$
.
Let
$\mathcal {P}_{b,b'}^{\mu }$
be the inverse image of
$\mathcal {C}_{b,b'}^{\mu }$
under
$\mathcal {M}_{b,b'}^{\mu } \to {\mathrm {Gr}}_{b,b'}^{\mu }$
.
Corollary 4.14. Assume that
$([b],[b'],\mu )$
is Hodge–Newton reducible for L. Then the action of
$\widetilde {P}_{b'}$
on
$\mathcal {M}_{b,b'}^{\mu }$
stabilises
$\mathcal {P}_{b,b'}^{\mu }$
, and we have a natural
$( \widetilde {J}_b \times \widetilde {J}_{b'})$
-equivariant isomorphism
$$\begin{align*}\mathcal{P}_{b,b'}^{\mu} \times^{\widetilde{P}_{b'}} \widetilde{J}_{b'} \stackrel{\sim}{\longrightarrow} \mathcal{M}_{b,b'}^{\mu}. \end{align*}$$
Proof. This follows from Proposition 4.13.
We define a subsheaf
$\widetilde {J}_{b}^{U}$
of
$\widetilde {J}_b$
by
$$\begin{align*}\widetilde{J}_{b}^{U} (S) = \Big\{ g \in \widetilde{J}_b (S) \Bigm| g|_{{\mathrm{Fil}}_{\lambda}^j \mathscr{E}_b} \equiv {\mathrm{id}}_{{\mathrm{Fil}}_{\lambda}^j \mathscr{E}_b} \quad \mod {\mathrm{Fil}}_{\lambda}^{j+1} \mathscr{E}_b \ \text{for all } j \Big\} \end{align*}$$
for
$S \in {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
.
Let
$U_{b'}$
be the unipotent radical of
$P_{b'}$
. The inner form of L determined by
$b'$
gives a Levi subgroup
$L_{b'}$
of
$P_{b'}$
.
We use a notation that
$$\begin{align*}{\mathrm{gr}}_{\lambda}^i ={\mathrm{Fil}}_{\lambda}^i / {\mathrm{Fil}}_{\lambda}^{i+1} \end{align*}$$
for any integer i. Let
$\rho _U$
be the half-sum of the positive roots
$\alpha $
of T such that
$-\alpha $
occurs in the adjoint action of T on
$ {\mathrm {Lie}}(U)$
. We put
$N_{U,b} = \langle 2 \rho _U , \nu _b \rangle $
.
Definition 4.15. Let F be a non-archimedean field with a valuation subring
$F^+$
. Let
$f \colon D \to {\mathrm {Spa}}(F,F^+)^{\diamond }$
be an
$\ell $
-cohomologically smooth morphism of locally spatial diamonds (cf. [Reference ScholzeSch17, Definition 23.8]). We say that D is
$\ell $
-contractible of pure dimension d if
$f^! \mathbb {F}_{\ell } = \mathbb {F}_{\ell } (d)[2d]$
and the trace morphism
$Rf_! f^! \mathbb {F}_{\ell } \to \mathbb {F}_{\ell }$
is a quasi-isomorphism.
Remark 4.16. In the situation of Definition 4.15, we have
$f_{\natural }\mathbb {F}_{\ell } \cong Rf_{!}f^{!}\mathbb {F}_{\ell }$
by [Reference Fargues and ScholzeFS21, Proposition VII.5.2].
Let
$\varpi $
be a uniformiser of E. Let
$\mathbb {B}$
denote the v-sheaf on
$ {\mathrm {Perf}}_{\mathbb {F}_{q}}$
given by
$\mathbb {B}(S) = \mathcal {O}(Y_{S})$
(cf. [Reference Fargues and ScholzeFS21, Proposition II.2.1]).
Lemma 4.17. Let d and h be positive integers. Let
$f_{d,h} \colon \mathbb {B}^{\varphi ^d =\varpi ^h} \times {\mathrm {Spa}}({\breve {E}})^{\diamond } \to {\mathrm {Spa}}({\breve {E}})^{\diamond }$
be the natural morphism.
-
(1) The v-sheaf
$\mathbb {B}^{\varphi ^d =\varpi ^h} \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is an
$\ell $
-cohomologically smooth
$\ell $
-contractible locally spatial diamond of pure dimension h over
$ {\mathrm {Spa}}({\breve {E}})^{\diamond }$
. -
(2) The action of
$E^{\times }$
on
$f_{d,h,!} \mathbb {Z}_{\ell }$
is given by
$|| \cdot ||^{-d}$
. -
(3) Let F be a perfectoid field over
$\breve {E}$
and
$a \in \mathbb {B}^{\varphi ^d =\varpi ^h}(F^{\flat })$
. Let
$f_{d,h,F^{\flat }} \colon \mathbb {B}^{\varphi ^d =\varpi ^h} \times {\mathrm {Spa}}(F^{\flat }) \to {\mathrm {Spa}}(F^{\flat })$
denote the base change of
$f_{d,h}$
. Then the action of a on
$f_{d,h,F^{\flat },!} \mathbb {Z}_{\ell }$
induced by the addition on
$\mathbb {B}^{\varphi ^d =\varpi ^h}$
is trivial.
Proof. Replacing E by the unramified extension of degree d, we may assume that
$d=1$
(cf. [Reference Fargues and FontaineFF18, Remarque 4.2.2]). We proceed by induction on
$h \geq 1$
. For
$h=1$
, the diamond
$\mathbb {B}^{\varphi = \varpi } \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is isomorphic to
$ {\mathrm {Spa}} (\mathbb {F}_q [[x^{1/p^{\infty }}]]) \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
by [Reference FarguesFar25, 1.5.3]. The action of
$\varpi $
on
$ {\mathrm {Spa}} (\mathbb {F}_q [[x^{1/p^{\infty }}]]) \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is induced from the morphism
$$\begin{align*}{\mathrm{Spa}} (\mathbb{F}_q [[x^{1/q^m}]]) \to {\mathrm{Spa}} (\mathbb{F}_q [[x^{1/q^m}]]) ;\ x^{1/q^m} \mapsto x^{1/q^{m-1}} \end{align*}$$
of degree q by taking limit with respect to
$m \geq 0$
. On the other hand, the action of
$\mathcal {O}_E^{\times }$
on
$ {\mathrm {Spa}} (\mathbb {F}_q [[x^{1/p^{\infty }}]]) \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is induced from an isomorphism on
$ {\mathrm {Spa}} (\mathbb {F}_q [[x^{1/q^m}]])$
by taking limit with respect to
$m \geq 0$
. Further the addition of
$a \in {\mathrm {Spa}} (\mathbb {F}_q [[x^{1/p^{\infty }}]])(F^{\flat })$
on
$ {\mathrm {Spa}} (F^{\flat } [[x^{1/p^{\infty }}]])$
is induced from an isomorphism on
$ {\mathrm {Spa}} (\mathbb {F}_q [[x^{1/q^m}]])$
by taking limit with respect to
$m \geq 0$
. Hence the claims hold for
$h=1$
by [Reference ImaiIma19, Lemma 1.3].
Assume that the result is true for
$\mathbb {B}^{\varphi = \varpi ^{h-1}}$
. We have an exact sequence
$$ \begin{align} 0 \longrightarrow \mathbb{B}^{\varphi = \varpi^{h-1}} \times {\mathrm{Spa}}(\breve{E})^{\diamond} \longrightarrow \mathbb{B}^{\varphi = \varpi^{h}} \times {\mathrm{Spa}}(\breve{E})^{\diamond} \longrightarrow \mathbb{A}^{1,\diamond}_{\breve{E}} \longrightarrow 0 \end{align} $$
of diamonds which splits pro-etale locally on
$\mathbb {A}^{1,\diamond }_{\breve {E}}$
as in [Reference Scholze and WeinsteinSW20, Example 15.2.9 (4)]. Therefore
$\mathbb {B}^{\varphi = \varpi ^{h}} \times {\mathrm {Spa}}(\breve {E})^{\diamond }$
satisfies the claims (1) and (2), since
$\mathbb {A}^{1,\diamond }_{\breve {E}}$
is an
$\ell $
-cohomologically smooth
$\ell $
-contractible diamond of pure dimension
$1$
over
$ {\mathrm {Spa}}({\breve {E}})^{\diamond }$
and the action of
$c \in E^{\times }$
on
$\mathbb {A}^{1,\diamond }_{\breve {E}}$
is induced from the isomorphism
$\mathbb {A}^{1}_{\breve {E}} \to \mathbb {A}^{1}_{\breve {E}};\ x \mapsto cx$
.
The action of
$a \in \mathbb {B}^{\varphi = \varpi ^{h}}(F^{\flat })$
on
$f_{d,h,F^{\flat },!} \mathbb {Z}_{\ell }$
depends only on the image
$\overline {a} \in \mathbb {A}^{1,\diamond }_{\breve {E}}(F^{\flat })$
of a under (4.2) since the claim (3) is true for
$\mathbb {B}^{\varphi = \varpi ^{h-1}}$
. Hence it suffices to show that the action of
$\overline {a}$
on
$f_{\mathbb {A},!} \mathbb {Z}_{\ell }$
is trivial, where
$f_{\mathbb {A}} \colon \mathbb {A}^{1,\diamond }_{F} \to {\mathrm {Spa}} (F^{\flat })$
is the natural morphism. This follows from the fact that the addition by
$\overline {a}$
on
$\mathbb {A}^{1,\diamond }_{F}$
is induced from an automorphism on
$\mathbb {A}^{1}_{F}$
by [Reference Scholze and WeinsteinSW20, Proposition 10.2.3].
Let
$\delta _P \colon P(E) \to \overline {\mathbb {Q}}_{\ell }^{\times }$
be the modulus character of
$P(E)$
. Let
$A_b$
be the split centre of
$J_b$
. Since
$J_b$
is an inner form of
$L^b$
, we can view
$A_b$
as an algebraic subgroup of
$L^b$
. We put
$\delta _{P,A_b}=\delta _P|_{A_b(E)}$
. Let
$g \in J_b(E)$
act on
$\widetilde {J}_b^U$
by the conjugation right action
$u \mapsto g^{-1}ug$
.
Lemma 4.18. Let
$f_J \colon \widetilde {J}_b^U \times {\mathrm {Spa}}({\breve {E}})^{\diamond } \to {\mathrm {Spa}}({\breve {E}})^{\diamond }$
be the natural morphism.
-
(1) The functor
$\widetilde {J}_b^U \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is an
$\ell $
-cohomologically smooth
$\ell $
-contractible diamond of pure dimension
$N_{U,b}$
over
$ {\mathrm {Spa}}({\breve {E}})^{\diamond }$
. -
(2) Let
$\kappa \colon J_b(E) \to \overline {\mathbb {Q}}_{\ell }^{\times }$
be the character of the action of
$J_b(E)$
on
$f_{J,!} \overline {\mathbb {Q}}_{\ell }$
induced by the conjugation right action of
$J_b(E)$
on
$\widetilde {J}_b^U$
. Then we have
$\kappa |_{A_b(E)}=\delta _{P,A_b}^{-1}$
. -
(3) Let F be a perfectoid field over
$\breve {E}$
. Then the action of
$\widetilde {J}_b^U(F^{\flat })$
on
$f_{J,!} \overline {\mathbb {Q}}_{\ell }$
induced by the addition on
$\widetilde {J}_b^U$
is trivial.
Proof. For
$i \geq 0$
, we define an algebraic subgroup
$U_i$
of P by
$$\begin{align*}U_i (R) = \Big\{ g \in P(R) \Bigm| g|_{{\mathrm{Fil}}_{\lambda}^j V_R} \equiv {\mathrm{id}}_{{\mathrm{Fil}}_{\lambda}^j V_R} \quad \mod {\mathrm{Fil}}_{\lambda}^{j+i+1} V_R \ \text{for all } j \text{ and } V \in {\mathrm{Rep}} (G) \Big\} \end{align*}$$
for any E-algebra R, where
$V_R =V \otimes _E R$
. Then
$U_0=U$
, and
$U_i$
are normal in P for all i. Similarly, we define a subsheaf
$\widetilde {J}^U_{b,i}$
of
$\widetilde {J}_b$
for
$i \geq 0$
by
$$\begin{align*}\widetilde{J}^U_{b,i} (S)= \Big\{ g \in \widetilde{J}_b (S) \Bigm| g|_{{\mathrm{Fil}}_{\lambda}^j \mathscr{E}_b} \equiv {\mathrm{id}}_{{\mathrm{Fil}}_{\lambda}^j \mathscr{E}_b} \quad \mod {\mathrm{Fil}}_{\lambda}^{j+i+1} \mathscr{E}_b \ \text{for all } j \Big\} \end{align*}$$
for
$S \in {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
. Then
$\widetilde {J}^U_{b,0} = \widetilde {J}^U_b$
. Let
$\varphi $
act on
$G_{\breve {E}}$
and its subgroup
$U_{i,\breve {E}}$
by
$g \mapsto b_0 \sigma (g) b_0^{-1}$
. Let S be a perfectoid space over
$ {\mathrm {Spa}}({\breve {E}})^{\diamond }$
. By the internal definition of a G-torsor on the Fargues–Fontaine curve, we see that
$\widetilde {J}^U_{b,i}(S)$
is equal to the sections of
$$\begin{align*}Y_S \times_{\varphi} U_{i,\breve{E}} \longrightarrow X_S. \end{align*}$$
Hence,
$(\widetilde {J}^U_{b,i}/\widetilde {J}^U_{b,i+1})(S)$
is equal to the sections of
$$\begin{align*}Y_S \times_{\varphi} (U_{i,\breve{E}}/U_{i+1,\breve{E}}) \longrightarrow X_S. \end{align*}$$
Let L act on
$U_i$
by the conjugation. Let
$ {\mathrm {Lie}} (G)$
be the adjoint representation of G. Then the action of L on
$ {\mathrm {Lie}} (G)$
induces an action of L on
$ {\mathrm {Lie}} (U_i / U_{i+1})$
. We have an isomorphism
$$\begin{align*}U_i / U_{i+1} \simeq {\mathrm{Lie}} ( U_i / U_{i+1} ) \end{align*}$$
as representations of L, since
$U_i / U_{i+1}$
is isomorphic to
$\mathbb {G}_a^{d_i}$
for some
$d_i$
as linear algebraic groups. We have the equality
$$\begin{align*}{\mathrm{Lie}} (U_i) = {\mathrm{Fil}}_{\lambda}^i {\mathrm{Lie}} (G) \end{align*}$$
by the definition of the both sides. Hence we have an isomorphism
$$\begin{align*}{\mathrm{Lie}} ( U_i / U_{i+1} ) \simeq {\mathrm{gr}}_{\lambda}^i {\mathrm{Lie}} (G) \end{align*}$$
as representations of L. As a result we have an isomorphism
$$ \begin{align} U_i / U_{i+1} \simeq {\mathrm{gr}}_{\lambda}^i {\mathrm{Lie}} (G) \end{align} $$
as representations of L. The element
$b_0 \in L$
gives an L-bundle
$\mathscr {E}_{b_0,S} \colon {\mathrm {Rep}} (L) \to {\mathrm {Bun}}_{X_S}$
. Then we have
$$\begin{align*}Y_S \times_{\varphi} (U_{i,\breve{E}}/U_{i+1,\breve{E}}) \simeq \mathscr{E}_{b_0,S} ({\mathrm{gr}}_{\lambda}^i {\mathrm{Lie}} (G) ) \end{align*}$$
by (4.3). Hence,
$(\widetilde {J}^U_{b,i}/\widetilde {J}^U_{b,i+1})(S)$
is equal to the sections of
$$\begin{align*}\mathscr{E}_{b_0,S} ({\mathrm{gr}}_{\lambda}^i {\mathrm{Lie}} (G) ) \longrightarrow X_S. \end{align*}$$
Then
$\mathbb {D}$
acts on
$ {\mathrm {gr}}_{\lambda }^i {\mathrm {Lie}} (G)$
via
$\nu _b$
and the conjugation. This action gives a slope decomposition
$$\begin{align*}{\mathrm{gr}}_{\lambda}^i {\mathrm{Lie}} (G) = \bigoplus_{1 \leq j \leq m_i} V_{-\alpha_{i,j}} \end{align*}$$
where
$\alpha _{i,j}$
are positive rational numbers, since L contains the centraliser
$L^b$
of
$\nu _b$
. Then we have an isomorphism
$$ \begin{align} \mathscr{E}_{b_0} ({\mathrm{gr}}_{\lambda}^i {\mathrm{Lie}} (G) ) \simeq \bigoplus_{1 \leq j \leq m_i} \mathcal{O}(\alpha_{i,j}). \end{align} $$
Hence
$(\widetilde {J}^U_{b,i}/\widetilde {J}^U_{b,i+1}) \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is an
$\ell $
-cohomologically smooth
$\ell $
-contractible diamond by (4.4) and Lemma 4.17.
We show that
$\widetilde {J}^U_{b,i} \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is an
$\ell $
-cohomologically smooth
$\ell $
-contractible diamond by a decreasing induction on i. The claim is trivial for enough large i, since
$\widetilde {J}^U_{b,i} \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is one point for such i. We see that
$U_{i,\breve {E}}$
is isomorphic to
$U_{i+1,\breve {E}} \times (U_{i,\breve {E}}/U_{i+1,\breve {E}})$
as schemes over
$U_{i,\breve {E}}/U_{i+1,\breve {E}}$
with actions of
$\varphi $
by [SGA70, XXVI Proposition 2.1] and its proof. Hence,
$\widetilde {J}^U_{b,i} \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is isomorphic to
$\widetilde {J}^U_{b,i+1} \times (\widetilde {J}^U_{b,i}/\widetilde {J}^U_{b,i+1}) \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
as diamonds over
$(\widetilde {J}^U_{b,i}/\widetilde {J}^U_{b,i+1}) \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
. Therefore, we see that
$\widetilde {J}^U_{b,i} \times {\mathrm {Spa}}({\breve {E}})^{\diamond } \to (\widetilde {J}^U_{b,i}/\widetilde {J}^U_{b,i+1}) \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is an
$\ell $
-cohomologically smooth morphism with
$\ell $
-contractible geometric fibre, since
$\widetilde {J}^U_{b,i+1} \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is an
$\ell $
-cohomologically smooth
$\ell $
-contractible diamond by our induction hypothesis. Then we see that
$\widetilde {J}^U_{b,i} \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is an
$\ell $
-cohomologically smooth
$\ell $
-contractible diamond, since we know that
$(\widetilde {J}^U_{b,i}/\widetilde {J}^U_{b,i+1}) \times {\mathrm {Spa}}({\breve {E}})^{\diamond }$
is an
$\ell $
-cohomologically smooth
$\ell $
-contractible diamond. The claim on the dimension follows from the above arguments. The claim (2) follows from the arguments above, Lemma 4.17 (2) and a calculation of
$\delta _P$
(cf. [Reference RenardRen10, V.5.4]). The claim (3) follows from Lemma 4.17 (3) by induction on i for
$\widetilde {J}^U_{b,i}$
in the same way as the proof of Lemma 4.17 (3).
Remark 4.19. Some integral version of
$\widetilde {J}_b$
is studied in [Reference Caraiani and ScholzeCS17, Proposition 4.2.11]. The character
$\kappa $
in Lemma 4.18 (2) is explicitly determined in [Reference Hamann and ImaiHI25, Corollary 4.6].
Let
$X_* (T)^{L+}$
be the set of L-dominant cocharacters in
$X_* (T)$
. We put
$$\begin{align*}I_{b_0,b^{\prime}_0,\mu,L} = \Big\{ [\mu'] \in X_* (T)^{L+}/\Gamma \Bigm| \mu' \text{ is } G\text{-conjugate to } \mu \text{ and } [b_0] \in B(L,\mu',[b^{\prime}_0]) \Big\}. \end{align*}$$
We claim the set
$I_{b_0,b_0',\mu ,L}$
consists of a single element. To prove this we begin with a preliminary lemma.
Lemma 4.20. Given two cocharacters
$\mu ,\mu ' \in X_{*}(T)$
which are G-conjugate, then there exists an element w of the absolute Weyl group of T in G such that
$w \cdot \mu = \mu '$
.
Proof. Let
$L_{\mu }$
be the centraliser of the cocharacter
$\mathbb {G}_{m} \xrightarrow {\mu } T \rightarrow G$
and define similarly
$L_{\mu '}$
. Then, since
$\mu ' = g\mu g^{-1}$
for some
$g \in G(\overline {E})$
, it follows that
$L_{\mu '} = gL_{\mu }g^{-1}$
. Since
$gTg^{-1} \subseteq L_{\mu '}$
is a maximal torus, there exists
$l \in L_{\mu '}$
such that
$gTg^{-1} = lTl^{-1}$
. This means that
$l^{-1}g$
normalises T and gives an element w in the absolute Weyl group of T in G. Then we have
$w \cdot \mu = \mu '$
.
Lemma 4.21.
$I_{b_0,b^{\prime }_0,\mu ,L}$
consists of a single element.
Proof. By the definition of Hodge–Newton reducibility, we have
$[\mu ] \in I_{b_0,b^{\prime }_0,\mu ,L}$
. Let
$[\mu '] \in I_{b_0,b^{\prime }_0,\mu ,L}$
be another element. Let
$\Delta (G,T)$
be the set of simple roots of G with respect to T, where the positivity of roots is given by B. Since
$\mu $
is G-dominant,
$\mu '$
is G-conjugate to
$\mu $
and
$\mu \neq \mu '$
, we have that
$\mu '$
is not G-dominant and
$$ \begin{align} \mu -\mu' = \sum_{\alpha \in \Delta (G,T)} n_{\alpha} \alpha^{\vee} , \end{align} $$
where
$n_{\alpha } \geq 0$
by Lemma 4.20, [Reference HumphreysHum78, 10.3 Lemma B] and [Reference BourbakiBou81, VI §1 Proposition 18]. Since
$\mu '$
is not G-dominant, but L-dominant, there is
$\alpha _0 \in \Delta (G,T) \setminus \Delta (L,T)$
such that
$\langle \mu ' , \alpha _0 \rangle <0$
. Then we have
$$ \begin{align} \langle \mu - \mu' , \alpha_0 \rangle>0. \end{align} $$
Substituting (4.5) to (4.6), we have
$$\begin{align*}\sum_{\alpha \in \Delta (G,T)} n_{\alpha} \langle \alpha^{\vee} , \alpha_0 \rangle>0. \end{align*}$$
This implies
$n_{\alpha _0}>0$
, since we have
$\langle \alpha ^{\vee } , \alpha _0 \rangle \leq 0$
for
$\alpha \neq \alpha _0$
by [Reference HumphreysHum78, 10.1 Lemma]. Recall that
$$ \begin{align} \pi_{1}(L) = X_{*}(T) \big/ \textstyle \sum_{\alpha \in \Delta(L,T)} \mathbb{Z}\alpha^{\vee}, \end{align} $$
by the proof of [Reference BorovoiBor98, Proposition 1.10] (cf. [Reference Rapoport and RichartzRR96, §1.13]). Let
$\overline {\mu }^{\natural }$
and
$\overline {\mu '}^{\natural }$
be the images in
$\pi _1 (L)_{\mathbb {Q}}^{\Gamma }$
of
$\overline {\mu }$
and
$\overline {\mu '}$
in
$X_*(T)_{\mathbb {Q}}^{\Gamma }$
.
We show that
$\overline {\mu }^{\natural } \neq \overline {\mu '}^{\natural }$
. We write
$$\begin{align*}\overline{\mu} -\overline{\mu'} = \sum_{\alpha \in \Delta (G,T)} m_{\alpha} \alpha^{\vee} , \end{align*}$$
where
$m_{\alpha } \in \mathbb {Q}$
. Then the equation
$$\begin{align*}\overline{\mu} -\overline{\mu'} = [\Gamma: \Gamma_{\mu} \cap \Gamma_{\mu'}]^{-1} \left( (\mu-\mu') + \sum_{1 \not= \tau \in \Gamma/(\Gamma_{\mu} \cap \Gamma_{\mu'})} \tau(\mu-\mu') \right) \end{align*}$$
implies
$m_{\alpha _0}> 0$
, since
$n_{\alpha _0}> 0$
and
$n_{\alpha } \geq 0$
for all
$\alpha \in \Delta (G,T)$
. Thus when passing to
$\pi _1(L)^{\Gamma }$
the term
$\alpha _{0}^{\vee }$
is not killed according to (4.7) and so
$\overline {\mu }^{\natural } \neq \overline {\mu '}^{\natural }$
as claimed. This implies
$$\begin{align*}\mu^{\natural} \neq \mu^{\prime\natural} \in \pi_1 (L)_{\Gamma}, \end{align*}$$
since
$\overline {\mu }^{\natural }$
and
$\overline {\mu '}^{\natural }$
are images of
$\mu ^{\natural }$
and
$\mu ^{\prime \natural }$
under the map
$$\begin{align*}\pi_1 (L)_{\Gamma} \to \pi_1 (L)_{\mathbb{Q}}^{\Gamma}; \ [g] \mapsto \frac{1}{[\Gamma : \Gamma_{g}]} \sum_{\tau \in \Gamma/\Gamma_g} \tau (g) , \end{align*}$$
where
$g \in \pi _1 (L)$
and
$\Gamma _g$
is the stabiliser of g in
$\Gamma $
. This contradicts that
$[\mu '] \in I_{b_0,b^{\prime }_0,\mu ,L}$
, because we have
$$\begin{align*}\mu^{\prime\natural}=\kappa_L ([b_0]) -\kappa_L ([b^{\prime}_0]) =\mu^{\natural} \in \pi_1 (L)_{\Gamma} \end{align*}$$
by
$[b_0] \in B(L,\mu ',[b^{\prime }_0])$
and
$[b_0] \in B(L,\mu ,[b^{\prime }_0])$
.
Definition 4.22. Let R be a DVR with uniformiser
$\pi $
, and quotient field F. Let
$k_1 \geq \cdots \geq k_n$
be a sequence of integers. We say that the type of
$g \in {\mathrm {GL}}_n (F)$
is
$(k_1, \ldots , k_n)$
if we have
$$\begin{align*}g \in {\mathrm{GL}}_n (R) \begin{pmatrix} \pi^{k_1} & & \\ & \ddots & \\ & & \pi^{k_n} \end{pmatrix} {\mathrm{GL}}_n (R). \end{align*}$$
Lemma 4.23. Let R be a DVR with uniformiser
$\pi $
, and quotient field F. We consider the subgroups
$$\begin{align*}L= \begin{pmatrix} {\mathrm{GL}}_{n_1} & & \\ & \ddots & \\ & & {\mathrm{GL}}_{n_m} \end{pmatrix} \subset P= \begin{pmatrix} {\mathrm{GL}}_{n_1} & & 0 \\ & \ddots & \\ * & & {\mathrm{GL}}_{n_m} \end{pmatrix} \subset {\mathrm{GL}}_n \end{align*}$$
of
$ {\mathrm {GL}}_n $
. Let
$g \in P(F)$
, and
$g_L$
be the image of g in the Levi quotient. We regard
$g_L$
as an element of
$L(F)$
. We put
$N_l =n_1 + \cdots +n_l$
for
$0 \leq l \leq m$
.
Let
$k_1 \geq \cdots \geq k_n$
be a sequence of integers. Assume that the type of
$$\begin{align*}(g_{ij})_{N_l +1 \leq i,j \leq n} \in {\mathrm{GL}}_{n-N_l} (F) \end{align*}$$
is
$(k_{N_l +1}, \ldots ,k_n)$
for
$0 \leq l \leq m-1$
. Then we have
$g_L^{-1} g \in P(R)$
.
Proof. By multiplying a power of
$\pi $
to g, we may assume that
$k_n \geq 0$
. By the assumption, we see that the type of
$$\begin{align*}(g_{ij})_{N_l +1 \leq i,j \leq N_{l+1}} \in {\mathrm{GL}}_{n_{l+1}} (F) \end{align*}$$
is
$(k_{N_l +1}, \ldots ,k_{N_{l+1}})$
for
$0 \leq l \leq m-1$
using Lemma 4.6. Hence, we may assume that
$g_L = {\mathrm {diag}} (\pi ^{k_1} , \ldots , \pi ^{k_n})$
.
Let v be a normalised valuation of F. Then, it suffices to show that
$v(g_{ij}) \geq k_i$
for all
$1 \leq j < i \leq n$
. Assume it does not hold, and take the biggest
$i_0$
such that there is
$j_0 <i_0$
satisfying
$v(g_{i_0 j_0}) < k_{i_0}$
. Then the type of
$$\begin{align*}(g_{ij})_{i_0 +1 \leq i,j \leq n} \in {\mathrm{GL}}_{n-i_0} (F) \end{align*}$$
is
$(k_{i_0 +1}, \ldots ,k_n)$
. Using this and Lemma 4.6, we can show that the type of
$$\begin{align*}(g_{ij})_{1 \leq i,j \leq i_0} \in {\mathrm{GL}}_{i_0} (F) \end{align*}$$
is
$(k_1, \ldots ,k_{i_0})$
. This implies that
$v(g_{ij}) \geq k_{i_0}$
for all
$1 \leq i,j \leq i_0$
. This contradicts the choice of
$i_0$
.
In the sequel, we simply write
$(R^{\sharp },f)$
for
$$\begin{align*}(\mathscr{E}_b, \mathscr{E}_{b'}, R^{\sharp}, f, {\mathrm{id}}_{\mathscr{E}_b}, {\mathrm{id}}_{\mathscr{E}_{b'}} ) \in \mathcal{M}_{b,b'}^{\mu} (R,R^+). \end{align*}$$
Every point of
$\mathcal {M}_{b,b'}^{\mu } (R,R^+)$
is represented by a datum of the above form, since we have an isomorphism of data
$$\begin{align*}(\mathscr{E}, \mathscr{E}', R^{\sharp}, f, \phi, \phi') \simeq (\mathscr{E}_b, \mathscr{E}_{b'}, R^{\sharp}, \phi^{\prime-1} \circ f \circ \phi, {\mathrm{id}}_{\mathscr{E}_b}, {\mathrm{id}}_{\mathscr{E}_{b'}} ) \end{align*}$$
for
$$\begin{align*}(\mathscr{E}, \mathscr{E}', R^{\sharp}, f, \phi, \phi') \in \mathcal{M}_{b,b'}^{\mu} (R,R^+). \end{align*}$$
We write
$D_{R^{\sharp }}$
for the degree-
$1$
Cartier divisor given by
$R^{\sharp }$
.
We define a morphism
$$\begin{align*}\Phi \colon \mathcal{M}_{b_0,b^{\prime}_0}^{\mu} \times \widetilde{J}_{b}^{U} \longrightarrow \mathcal{P}_{b,b'}^{\mu} \end{align*}$$
by sending
$$\begin{align*}\big( (R^{\sharp}, f_L ), g \big) \in \Big( \mathcal{M}_{b_0,b^{\prime}_0}^{\mu} \times \widetilde{J}_{b}^{U} \Big) (R,R^+) \end{align*}$$
to
$$\begin{align*}\big( R^{\sharp}, (f_L \times^L P) \circ g \big) \in \mathcal{P}_{b,b'}^{\mu} (R,R^+) \end{align*}$$
for a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra
$(R,R^+)$
.
Proposition 4.24. The morphism
$$\begin{align*}\Phi \colon \mathcal{M}_{b_0,b^{\prime}_0}^{\mu} \times \widetilde{J}_{b}^{U} \longrightarrow \mathcal{P}_{b,b'}^{\mu} \end{align*}$$
is an isomorphism.
Proof. Let
$(R,R^+)$
be a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra, and
$$\begin{align*}\big( (R^{\sharp}, f_L ), g \big) \in \Big( \mathcal{M}_{b_0,b^{\prime}_0}^{\mu} \times \widetilde{J}_{b}^{U} \Big) (R,R^+). \end{align*}$$
Then we have
$\Phi \big ( (R^{\sharp }, f_L ), g \big ) \times ^P L =(R^{\sharp }, f_L )$
. Further,
$(R^{\sharp }, f_L )$
and
$\Phi \big ( (R^{\sharp }, f_L ), g \big )$
recover g. Hence, we have the injectivity of
$\Phi $
.
Let
$$\begin{align*}(R^{\sharp}, f ) \in \mathcal{P}_{b,b'}^{\mu} (R,R^+). \end{align*}$$
By the definition of
$\mathcal {P}_{b,b'}^{\mu }$
, we have a reduction
$$\begin{align*}f_P \colon (\mathscr{E}_{b_0} \times^L P) |_{X_R^{{\mathrm{sch}}} \setminus D_{R^{\sharp}}} \stackrel{\sim}{\longrightarrow} (\mathscr{E}_{b^{\prime}_0} \times^L P) |_{X_R^{{\mathrm{sch}}} \setminus D_{R^{\sharp}}} \end{align*}$$
of f to P. We put
$f_L = f_P \times ^P L$
.
We show that
$$ \begin{align} (f_L \times^L P)^{-1} \circ f_P \in \widetilde{J}_b^U (R,R^+). \end{align} $$
For this, it suffices to show (4.8) after taking realisations for all
$V \in {\mathrm {Rep}}(G)$
. Hence, we may assume that
$G= {\mathrm {GL}}_n$
.
We view
$ {\mathrm {GL}}_n$
-bundles as vector bundles. We take the diagonal torus and the upper-half Borel subgroup as T and B. Then we have
$$\begin{align*}L= \begin{pmatrix} {\mathrm{GL}}_{n_1} & & \\ & \ddots & \\ & & {\mathrm{GL}}_{n_m} \end{pmatrix} \subset P= \begin{pmatrix} {\mathrm{GL}}_{n_1} & & 0 \\ & \ddots & \\ * & & {\mathrm{GL}}_{n_m} \end{pmatrix} \subset {\mathrm{GL}}_n. \end{align*}$$
We write
$$\begin{align*}b_0 =(b_1 , \ldots , b_m), \ b^{\prime}_0 =(b^{\prime}_1 , \ldots , b^{\prime}_m) \in {\mathrm{GL}}_{n_1} (\breve{E}) \times \cdots {\mathrm{GL}}_{n_m} (\breve{E}). \end{align*}$$
Then we have a decomposition
$$\begin{align*}\mathscr{E}_b =\bigoplus_{1 \leq i \leq m} \mathscr{E}_{b_i} , \quad \mathscr{E}_{b'} =\bigoplus_{1 \leq i \leq m} \mathscr{E}_{b^{\prime}_i} \end{align*}$$
as vector bundles. We put
$$\begin{align*}{\mathrm{Fil}}^j \mathscr{E}_b = \bigoplus_{j \leq i \leq m} \mathscr{E}_{b_i}, \quad {\mathrm{Fil}}^j \mathscr{E}_{b'} = \bigoplus_{j \leq i \leq m} \mathscr{E}_{b^{\prime}_i} \end{align*}$$
for
$1 \leq j \leq m+1$
. Then
$f \colon \mathscr {E}_b|_{X_R^{ {\mathrm {sch}}} \setminus D_{R^{\sharp }}} \to \mathscr {E}_{b'}|_{X_R^{ {\mathrm {sch}}} \setminus D_{R^{\sharp }}}$
respects these filtrations. We can write
$$\begin{align*}f=\bigoplus_{1 \leq i \leq j \leq m} f_{ij} \colon \mathscr{E}_b|_{X_R^{{\mathrm{sch}}} \setminus D_{R^{\sharp}}} \longrightarrow \mathscr{E}_{b'}|_{X_R^{{\mathrm{sch}}} \setminus D_{R^{\sharp}}}, \end{align*}$$
where
$f_{ij} \colon \mathscr {E}_{b_i}|_{X_R^{ {\mathrm {sch}}} \setminus D_{R^{\sharp }}} \to \mathscr {E}_{b_j'}|_{X_R^{ {\mathrm {sch}}} \setminus D_{R^{\sharp }}}$
. Then the morphism
$$\begin{align*}f_{jj}^{-1} \circ f_{ij} \colon \mathscr{E}_{b_i}|_{X_R^{{\mathrm{sch}}} \setminus D_{R^{\sharp}}} \longrightarrow \mathscr{E}_{b_j}|_{X_R^{{\mathrm{sch}}} \setminus D_{R^{\sharp}}} \end{align*}$$
extends to a morphism
$\mathscr {E}_{b_i} \to \mathscr {E}_{b_j}$
by Lemma 4.23. Hence we have (4.8) (cf. the proof of [Reference HansenHan21a, Theorem 4.1]).
It remains to show that
$(R^{\sharp },f_L) \in \mathcal {M}_{b_0,b^{\prime }_0}^{\mu } (R,R^+)$
. It suffices to show that the type of the modification
$f_L$
is equal to
$\mu $
geometric fibrewisely. Let
$\mu '$
be the type of
$f_L$
at a geometric point of
$ {\mathrm {Spa}} (R,R^+)$
. The type of
$f_L \times ^L G$
is equal to
$\mu $
by (4.8). Hence, we have
$\mu ' =\mu $
by Lemma 4.21.
For a diamond
$\mathcal {D}$
over
$ {\mathrm {Spa}} (\breve {E})^{\diamond }$
, let
$\mathcal {D}_{\mathbb {C}_p^{\flat }}$
denote
$\mathcal {D} \times _{ {\mathrm {Spa}} (\breve {E})^{\diamond }} {\mathrm {Spa}} (\mathbb {C}_p^{\flat })$
. Let
$\kappa \colon J_b(E) \to \overline {\mathbb {Q}}_{\ell }^{\times }$
be the character in Lemma 4.18.
Lemma 4.25.
-
(1) We have
(4.9)where
$$ \begin{align} \varinjlim_{K' \subset J_{b'}(E)} R f_{K',\natural} ((f_{K'}^!\overline{\mathbb{Q}}_{\ell})^{\vee}) \in D_{{\mathrm{lis}}}({\mathrm{Spa}} (\mathbb{C}_p^{\flat}),\overline{\mathbb{Q}}_{\ell}) , \end{align} $$
$K'$
runs along compact open subgroups of
$J_{b'}(E)$
and
$f_{K'} \colon \mathcal {M}_{b ,b', \mathbb {C}_p^{\flat }}^{\mu } /K' \to {\mathrm {Spa}} (\mathbb {C}_p^{\flat })$
. Further we can regard this as an object of the derived category of smooth representations of
$J_b(E) \times J_{b'}(E)$
.
-
(2) We have an isomorphism
as smooth representations of
$$\begin{align*}H_{\mathrm{c}}^i \big( \mathcal{M}_{b_0 ,b_0', \mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_{\ell} \big) \otimes \kappa \stackrel{\sim}{\longrightarrow} H_{\mathrm{c}}^{i + 2N_{U,b} } \big( \mathcal{P}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_{\ell} \big) \end{align*}$$
$J_b(E) \times L_{b'}(E)$
.
Proof. We can define
$R\Gamma _{\mathrm {c}} (\mathcal {M}_{b ,K,b', \mathbb {C}_p^{\flat }}^{\mu })$
and
$R\Gamma _{\mathrm {c}} (\mathcal {M}_{b ,b',K', \mathbb {C}_p^{\flat }}^{\mu })$
in the same way as [Reference ImaiIma19, §3] replacing
$ {\mathrm {IC}}_{\mu }'$
by
$j_{\mu ,\natural }j_{\mu }^* {\mathrm {IC}}_{\mu }'$
, where
$j_{\mu } \colon {\mathrm {Hecke}}^{\mu } \to {\mathrm {Hecke}}^{\leq \mu }$
. Then (4.9) coincides with
$$\begin{align*}\varinjlim_{K' \subset J_{b'}(E)} R\Gamma_{\mathrm{c}} (\mathcal{M}_{b ,b',K', \mathbb{C}_p^{\flat}}^{\mu}) \end{align*}$$
up to shift and Tate twist. By the proof of [Reference Fargues and ScholzeFS21, Proposition IX.2.1] and [Reference Fargues and ScholzeFS21, Corollary VI.6.6], we can show that
$$\begin{align*}\varinjlim_{K \subset J_{b}(E)} R\Gamma_{\mathrm{c}} (\mathcal{M}_{b ,K,b', \mathbb{C}_p^{\flat}}^{\mu}), \ \varinjlim_{K' \subset J_{b'}(E)} R\Gamma_{\mathrm{c}} (\mathcal{M}_{b ,b',K', \mathbb{C}_p^{\flat}}^{\mu}) \in D_{{\mathrm{lis}}}({\mathrm{Spa}} (\mathbb{C}_p^{\flat}),\overline{\mathbb{Q}}_{\ell}). \end{align*}$$
In the same way as [Reference ImaiIma19, Proposition 3.16], we can show that
$$\begin{align*}\varinjlim_{K \subset J_{b}(E)} R\Gamma_{\mathrm{c}} (\mathcal{M}_{b ,K,b', \mathbb{C}_p^{\flat}}^{\mu}) \cong \varinjlim_{K' \subset J_{b'}(E)} R\Gamma_{\mathrm{c}} (\mathcal{M}_{b ,b',K', \mathbb{C}_p^{\flat}}^{\mu}). \end{align*}$$
Hence the claims in (1) follow.
By (1), we can regard
$H_{\mathrm {c}}^i \big ( \mathcal {M}_{b_0 ,b_0', \mathbb {C}_p^{\flat }}^{\mu } , \overline {\mathbb {Q}}_{\ell } \big )$
as a smooth representation of
$J_{b_0}(E) \times J_{b_0'}(E)$
. Then claim (2) follows from Lemma 4.18 and Proposition 4.24.
Theorem 4.26. Assume that
$([b],[b'],\mu )$
is Hodge–Newton reducible for L. Then we have an isomorphism
$$\begin{align*}H_{\mathrm{c}}^{i+2N_{U,b}} \big( \mathcal{M}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_{\ell} \big) \simeq {\mathrm{Ind}}_{P_{b'}(E)}^{J_{b'}(E)} H_{\mathrm{c}}^{i} \big( \mathcal{M}_{b_0 ,b_0',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_{\ell} \big) \otimes \kappa \end{align*}$$
as smooth
$J_b(E) \times J_{b'}(E)$
-representations.
Lemma 4.27. Let
$(R,R^+)$
be a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra. Let
$$\begin{align*}(\mathscr{E} ,\mathscr{E}' ,R^{\sharp} ,f, \phi, \phi') \in \mathcal{M}_{b,b'}^{\mu}(R,R^+). \end{align*}$$
For any
$g \in \underline {U_{b'} (E)}(R,R^+)$
, there exists
$h \in \widetilde {J}_b^U(R,R^{+})$
such that
$g \circ f' =f' \circ h$
, where we put
$$\begin{align*}f'=\phi^{\prime-1} \circ f \circ \phi \colon \mathscr{E}_b|_{X_R^{\mathrm{sch}}\setminus D_{R^{\sharp}}} \to \mathscr{E}_{b'}|_{X_R^{\mathrm{sch}}\setminus D_{R^{\sharp}}}. \end{align*}$$
Proof. Let
$j \colon X_R^{\mathrm {sch}}\setminus D_{R^{\sharp }} \to X_R^{\mathrm {sch}}$
be the open immersion. Let
$V \in {\mathrm {Rep}} (G)$
. We have an embedding
$$\begin{align*}\mathscr{E}_b (V) \hookrightarrow j_* j^* \mathscr{E}_b (V) \stackrel{\sim}{\longrightarrow} j_* j^* \mathscr{E}_{b'} (V), \end{align*}$$
where the second isomorphism is induced by
$f'$
. We have an action of g on
$j_* j^* \mathscr {E}_{b'} (V)$
. It suffices to show that g stabilises
$ {\mathrm {Fil}}_{\lambda }^{i} \mathscr {E}_b (V)$
and induces the identity on
$ {\mathrm {gr}}_{\lambda }^{i} \mathscr {E}_b (V)$
for all i.
We show this claim by a decreasing induction on i. For enough large i, we have
$ {\mathrm {Fil}}_{\lambda }^{i} \mathscr {E}_b (V) =0$
and the claim is trivial for such i. Assume that the claim is true for
$i +1$
. We have the natural embedding
$$\begin{align*}{\mathrm{gr}}_{\lambda}^i \mathscr{E}_b (V) \hookrightarrow j_* j^* {\mathrm{gr}}_{\lambda}^i \mathscr{E}_b (V) \stackrel{\sim}{\longrightarrow} j_* j^* {\mathrm{gr}}_{\lambda}^i \mathscr{E}_{b'} (V) \end{align*}$$
where the second isomorphism is induced by
$f'$
. We have a commutative diagram
where the bottom morphism is induced by the natural inclusion
$$\begin{align*}g {\mathrm{Fil}}_{\lambda}^i \mathscr{E}_b (V) \subset g \big( j_* j^* {\mathrm{Fil}}_{\lambda}^i \mathscr{E}_{b'} (V) \big) = j_* j^* {\mathrm{Fil}}_{\lambda}^i \mathscr{E}_{b'} (V). \end{align*}$$
By this diagram, we see that
$g {\mathrm {Fil}}_{\lambda }^i \mathscr {E}_b (V) = {\mathrm {Fil}}_{\lambda }^i \mathscr {E}_b (V)$
, since
$ {\mathrm {gr}}_{\lambda }^i g$
is the identity on
$ {\mathrm {gr}}_{\lambda }^i \mathscr {E}_{b'} (V)$
. Hence, g stabilises
$ {\mathrm {Fil}}_{\lambda }^i \mathscr {E}_b (V)$
. Further, g induces the identity on
$ {\mathrm {gr}}_{\lambda }^i \mathscr {E}_b (V)$
again by the above diagram, since
$ {\mathrm {gr}}_{\lambda }^i g$
is the identity.
Lemma 4.28. The action of
$U_{b'} (E)$
on
$H_{\mathrm {c}}^i \big ( \mathcal {P}_{b,b',\mathbb {C}_p^{\flat }}^{\mu } , \overline {\mathbb {Q}}_{\ell } \big )$
is trivial.
Proof. Let
$p_{\mathcal {M}} \colon \mathcal {P}_{b,b'}^{\mu } \cong \mathcal {M}_{b_0,b^{\prime }_0}^{\mu } \times \widetilde {J}_{b}^{U} \to \mathcal {M}_{b_0,b^{\prime }_0}^{\mu }$
be the projection, where the first isomorphism is given by Proposition 4.24. It suffices to show that the action of
$U_{b'} (E)$
on
$p_{\mathcal {M},!} \overline {\mathbb {Q}}_{\ell }$
is trivial. It suffices to show this after the pullback to each geometric point of
$\mathcal {M}_{b_0,b^{\prime }_0}^{\mu }$
. It follows from Lemma 4.18 (3) and Lemma 4.27.
Proposition 4.29. Let
$\pi $
be a smooth representation of
$J_{b'}(E)$
. Assume that
$([b],[b'],\mu )$
is Hodge–Newton reducible for L and that the Jacquet module of
$\pi $
with respect to
$P_{b'}$
vanishes. Then we have
$$\begin{align*}{\mathrm{Hom}}_{J_{b'}(E)} \Big( \pi, H_{\mathrm{c}}^i \big( \mathcal{M}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_{\ell} \big) \Big) =0. \end{align*}$$
We define
$ t_{b,b'} \colon \mathcal {T}_{b,b',\mathbb {C}_p^{\flat }}^{\mu } \to [ {\mathrm {Spa}} (\overline {\mathbb {F}}_q)/J_{b'}(E) ]$
as the composites
$$ \begin{align*} \mathcal{T}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} \longrightarrow \mathcal{T}_{b,b'}^{\mu} \longrightarrow {\mathrm{Hecke}}_{b,b'}^{\mu} \longrightarrow [{\mathrm{Spa}} (\overline{\mathbb{F}}_q)/J_{b'}(E) ]. \end{align*} $$
We put
$\overleftarrow {t}_{b,b'} = x_{b'} \circ t_{b,b'}$
.
Theorem 4.30. Assume that b is not basic and
$([b],[b'],\mu )$
is Hodge–Newton reducible for L. Then we have
$$\begin{align*}H_{\mathrm{c}}^i \big( \mathcal{T}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , {\overleftarrow{t}}_{b,b'}^* \mathscr{F}_{\varphi} \big) =0. \end{align*}$$
Proof. We have
$$\begin{align*}{\overleftarrow{t}}_{b,b'}^* \mathscr{F}_{\varphi} = t_{b,b'}^* x_{b'}^* \mathscr{F}_{\varphi} = t_{b,b'}^* \left( \bigoplus_{\rho \in \widehat{S}_{\varphi},\, \rho|_{Z(\widehat{G})^{\Gamma}} =\kappa (b')} \underline{\rho} \otimes \underline{\pi_{\varphi,b',\rho}} \right) \end{align*}$$
by (3.1). We take
$\rho \in \widehat {S}_{\varphi }$
such that
$\rho |_{Z(\widehat {G})^{\Gamma }} =\kappa (b')$
. Then it suffices to show that
$$\begin{align*}H_{\mathrm{c}}^i \Big( \mathcal{T}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , t_{b,b'}^* \underline{\pi_{\varphi,b',\rho}} \Big) =0. \end{align*}$$
The pullback of
$\underline {\pi _{\varphi ,b',\rho }}$
to
$\mathcal {M}_{b,b'}^{\mu }$
is a constant sheaf, since the map
$\mathcal {M}_{b,b'}^{\mu } \to [ {\mathrm {Spa}} (\overline {\mathbb {F}}_q)/\underline {J_{b'}(E)}]$
factorises via
$ {\mathrm {Spa}}(\overline {\mathbb {F}}_q)$
. Hence, there is a Hochschild–Serre spectral sequence
$$\begin{align*}H_i \Big(J_{b'}(E), H^j_{\mathrm{c}} \big(\mathcal{M}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_\ell\big) \otimes \pi_{\varphi,b',\rho} \Big) \Rightarrow H^{j-i}_{\mathrm{c}} \Big( \mathcal{T}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , t_{b,b'}^* \underline{\pi_{\varphi,b',\rho}} \Big) \end{align*}$$
by (2.1) and Lemma 4.25. We show that
$$\begin{align*}H_i \big(J_{b'}(E), H^j_{\mathrm{c}} \big(\mathcal{M}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_\ell\big) \otimes \pi_{\varphi,b',\rho}\big) =0 \end{align*}$$
for all i and j. Take a projective resolution
$$\begin{align*}\cdots \longrightarrow V_1 \longrightarrow V_0 \longrightarrow H_{\mathrm{c}}^j \big( \mathcal{P}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_{\ell} \big) \end{align*}$$
as smooth
$L_{b'}(E)$
-representations. By Lemma 4.25 and Theorem 4.26 we have
$$\begin{align*}H^j_{\mathrm{c}} \big(\mathcal{M}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_\ell\big) \simeq {\mathrm{Ind}}_{P_{b'}(E)}^{J_{b'}(E)} H_{\mathrm{c}}^j \big( \mathcal{P}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_{\ell} \big) \end{align*}$$
as smooth
$J_{b'}(E)$
-representations. Moreover, the induction on the right-hand-side is parabolic by Lemma 4.28. Parabolic induction preserves projective objects, since it has a Jacquet functor as the right adjoint functor by Bernstein’s second adjoint theorem (cf. [Reference BushnellBus01, Theorem 3]) and the Jacquet functor is exact. Note also that parabolic induction is exact. Thus we obtain the projective resolution
$$\begin{align*}\cdots \longrightarrow {\mathrm{Ind}}_{P_{b'}(E)}^{J_{b'}(E)} V_1 \longrightarrow {\mathrm{Ind}}_{P_{b'}(E)}^{J_{b'}(E)} V_0 \longrightarrow H^j_{\mathrm{c}} \big(\mathcal{M}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_\ell\big) \end{align*}$$
as smooth
$J_{b'}(E)$
-representations. Finally the right adjoint of
$-\otimes \pi _{\varphi ,b',\rho }$
in the category of smooth
$J_{b'}(E)$
-representations is
$-\otimes \pi _{\varphi ,b',\rho }^*$
, where
$\pi _{\varphi ,b',\rho }^*$
is the smooth dual of
$\pi _{\varphi ,b',\rho }$
. Both functors are exact and so in particular
$-\otimes \pi _{\varphi ,b',\rho }$
preserves exact sequences and projective objects. Thus we obtain the projective resolution
$$\begin{align*}\cdots \longrightarrow {\mathrm{Ind}}_{P_{b'}(E)}^{J_{b'}(E)} V_1 \otimes \pi_{\varphi,b',\rho} \longrightarrow {\mathrm{Ind}}_{P_{b'}(E)}^{J_{b'}(E)} V_0 \otimes \pi_{\varphi,b',\rho} \longrightarrow H^j_{\mathrm{c}} \big(\mathcal{M}_{b,b',\mathbb{C}_p^{\flat}}^{\mu} , \overline{\mathbb{Q}}_\ell\big) \otimes \pi_{\varphi,b',\rho}. \end{align*}$$
Note that
$P_{b'}$
is a proper parabolic subgroup of
$J_{b'}$
, since b is not basic. For
$i \geq 0$
, we have
$$\begin{align*}\Big( \pi_{\varphi,b',\rho} \otimes {\mathrm{Ind}}_{P_{b'}(E)}^{J_{b'}(E)} V_i \Big)_{J_{b'}(E)} =0 , \end{align*}$$
since
$\pi _{\varphi ,b',\rho }$
is cuspidal. Hence we have the claim.
5 Non-abelian Lubin–Tate theory
Assume that
$G= {\mathrm {GL}}_n$
and
$\mu (z)= {\mathrm {diag}} (z,1, \ldots , 1)$
. In this case,
$S_{\varphi } =\mathbb {G}_m$
and
$ {\mathrm {Hecke}}^{\leq \mu }= {\mathrm {Hecke}}^{\mu }$
. We simply write
$\pi _{\varphi ,b}$
for
$\pi _{\varphi ,b,1}$
for any
$[b] \in B( {\mathrm {GL}}_n)_{ {\mathrm {basic}}}$
. We put
$$\begin{align*}b_1 = \begin{pmatrix} 0&0&\cdots &0& \varpi \\ 1&0&\cdots &0&0\\ 0&1&\cdots &0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots &1&0 \end{pmatrix} \in {\mathrm{GL}}_n (E). \end{align*}$$
Then we have a bijection
$$\begin{align*}\mathbb{Z} \stackrel{\sim}{\longrightarrow} B({\mathrm{GL}}_n)_{{\mathrm{basic}}} ;\ N \mapsto b_1^N. \end{align*}$$
The following proposition is a consequence of non-abelian Lubin–Tate theory.
Proposition 5.1. We put
$b=b_1^N$
for an integer N. Assume that
$N \equiv 0, 1\ \mod n$
. Then we have
$$ \begin{align} y_b^* \overrightarrow{h}_{\natural} (\overleftarrow{h}^{*} \mathscr{F}_{\varphi} \otimes {\mathrm{IC}}_{\mu}') = y_b^* (\mathscr{F}_{\varphi} \boxtimes \varphi ). \end{align} $$
Proof. We show the claim in the case where
$N \equiv 1\ \mod n$
using arguments in [MFO16, Chapter 23]. See arguments in [Reference FarguesFar25, 8.1] for the case where
$N \equiv 0\ \mod n$
. Suppose that
$N = mn+1$
for some
$m \in \mathbb {Z}$
. The following lemma provides an explicit description of the stack
$ {\mathrm {Hecke}}^{\leq \mu }_{b}$
.
Lemma 5.2. Let
$ {\mathrm {Spa}} (F,F^{+})$
be a geometric point in
$ {\mathrm {Perf}}_{\overline {\mathbb {F}}_q}$
. Let
$\mathscr {E}$
be a vector bundle of rank n on
$ X_{F}^{\mathrm {sch}}$
having a degree-one modification fibrewise by
$\mathscr {E}_b$
$$\begin{align*}0 \to \mathscr{E}_b \to \mathscr{E} \to \mathscr{F} \to 0, \end{align*}$$
where
$\mathscr {F}$
is a torsion coherent sheaf of length
$1$
. Then
$\mathscr {E}$
is isomorphic to
$\mathcal {O}(-m)^n$
.
Proof. This follows from [Reference Fargues and FontaineFF14, Theorem 2.94] by dualising the modification and twisting by
$\mathcal {O}(-m)$
.
We put
$b'=b_1^{nm}$
. Then, we have isomorphisms
$$ \begin{align*} {\mathrm{Hecke}}_{b,b'}^{\leq \mu} \simeq {\mathrm{Hecke}}_b^{\leq \mu} \end{align*} $$
by Lemma 5.2.
Lemma 5.3. Let
$\mathcal {M}_{\mathrm {LT}}^{\infty }$
be the Lubin–Tate space over
$\breve {E}$
at infinite level. Then we have an isomorphism
$\mathcal {M}_{b,b'}^{\leq \mu } \simeq \mathcal {M}_{\mathrm {LT}}^{\infty ,\diamond }$
, that is compatible with actions of
$ {\mathrm {GL}}_n(E) \times J_b (E)$
and Weil descent data.
Proof. For a perfectoid affinoid
$\overline {\mathbb {F}}_q$
-algebra
$(R,R^+)$
, the set
$\mathcal {M}_{b,b'}^{\leq \mu } (R,R^+)$
consists of 6-tuples
$(\mathscr {E}, \mathscr {E}', R^{\sharp }, f,\phi ,\phi ')$
, where
-
•
$(\mathscr {E},\mathscr {E}', D_{R^{\sharp }},f) \in {\mathrm {Hecke}}^{\leq \mu }_{b(0)}$
-
•
$\phi \colon \mathscr {E}_b \xrightarrow {\sim } \mathscr {E}$
and
$\phi ' \colon \mathscr {E}_{b'} \xrightarrow {\sim } \mathscr {E}'$
are isomorphisms.
Hence, the claim follows from [Reference Scholze and WeinsteinSW13, Proposition 6.3.9] by dualising the modification and twisting by
$\mathcal {O}(-m)$
.
Let
$$ \begin{align} p_b \colon {\mathrm{Spa}} (\mathbb{C}_p^{\flat}) \longrightarrow {\mathrm{Spa}} ( \breve{E} )^{\diamond} \longrightarrow [{\mathrm{Div}}_{X}^1 /\widetilde{J}_b ] \end{align} $$
be the natural projection. The equality (5.1) is equivalent to the equality
$$ \begin{align} p_b^* y_b^* \overrightarrow{h}_{\natural} (\overleftarrow{h}^{*} \mathscr{F}_{\varphi} \otimes {\mathrm{IC}}_{\mu}') = p_b^* y_b^* (\mathscr{F}_{\varphi} \boxtimes \varphi ) \end{align} $$
with action of
$J_b(E) \times W_E$
. Then the right-hand side of (5.3) is
$\pi _{\varphi ,b} \otimes \varphi $
as a representation of
$J_b(E) \times W_E$
. Hence it suffices to show that the cohomology of the left-hand side of (5.3) vanishes outside degree zero, and is equal to
$\pi _{\varphi ,b} \otimes \varphi $
in degree zero as representations of
$J_b(E) \times W_E$
.
The i-th cohomology of the left-hand side of (5.3) is equal to
$$\begin{align*}H_{\mathrm{c}}^{i+n-1} \big( \mathcal{T}_{b,b',\mathbb{C}_p^{\flat}}^{\leq \mu} , \overleftarrow{t}_{b,b'}^* \mathscr{F}_{\varphi} \big) \Big( \frac{n-1}{2} \Big). \end{align*}$$
We have
$$\begin{align*}\overleftarrow{t}_{b,b'}^* \mathscr{F}_{\varphi} = t_{b,b'}^* \underline{\pi_{\varphi,1}} \end{align*}$$
by (3.1), since
$\pi _{\varphi ,b'}=\pi _{\varphi ,1}$
in our case. We have a Hochschild–Serre spectral sequence
$$\begin{align*}H_i \Big( {\mathrm{GL}}_n (E), H^j_{\mathrm{c}} \big(\mathcal{M}_{b,b',\mathbb{C}_p^{\flat}}^{\leq \mu} , \overline{\mathbb{Q}}_\ell\big) \otimes \pi_{\varphi,1} \Big) \Rightarrow H^{j-i}_{\mathrm{c}} \Big( \mathcal{T}_{b,b',\mathbb{C}_p^{\flat}}^{\leq \mu} , t_{b,b'}^* \underline{\pi_{\varphi,1}} \Big) \end{align*}$$
by (2.1) and Lemma 4.25. We put
$$\begin{align*}{\mathrm{GL}}_n(E)^{0} =\{ g \in {\mathrm{GL}}_n(E) \mid \det (g) \in \mathcal{O}_E^{\times} \}. \end{align*}$$
Then we have
for a connected component
$\mathcal {M}_{\mathrm {LT}}^{\infty ,(0)}$
of
$\mathcal {M}_{\mathrm {LT}}^{\infty }$
(cf. [Reference FarguesFar04, 4.4.2]). By Lemma 5.3, we have
Therefore one has
$$\begin{align*}H_i \Big( {\mathrm{GL}}_n (E), H^j_{\mathrm{c}} \big(\mathcal{M}_{b,b',\mathbb{C}_p^{\flat}}^{\leq \mu} , \overline{\mathbb{Q}}_\ell\big) \otimes \pi_{\varphi,1} \Big) = H_i \Big( {\mathrm{GL}}_n (E)^{0}, H^j_{\mathrm{c}} \big(\mathcal{M}_{\mathrm{LT},\mathbb{C}_p^{\flat}}^{\infty,(0),\diamond}, \overline{\mathbb{Q}}_\ell\big) \otimes \pi_{\varphi,1}|_{ {\mathrm{GL}}_n (E)^{0}} \Big) \end{align*}$$
by Shapiro’s Lemma. Now
$ \pi _{\varphi ,1}|_{ {\mathrm {GL}}_n (E)^{0}}$
is a compact representation and thus it is a projective object in the category of smooth
$ {\mathrm {GL}}_n (E)^{0}$
-representations. Hence no higher homology groups appear and so
$$\begin{align*}\Big( H^j_{\mathrm{c}} \big(\mathcal{M}_{\mathrm{LT},\mathbb{C}_p^{\flat}}^{\infty,\diamond}, \overline{\mathbb{Q}}_\ell\big) \otimes \pi_{\varphi,1} \Big)_{{\mathrm{GL}}_n (E)} = H^{j}_{\mathrm{c}} \Big( \mathcal{T}_{b,b',\mathbb{C}_p^{\flat}}^{\leq \mu} , t_{b,b'}^* \underline{\pi_{\varphi,1}} \Big). \end{align*}$$
Hence, the claim follows from the non-abelian Lubin–Tate theory.
6 Hecke eigensheaf property
Assume that
$G= {\mathrm {GL}}_2$
and
$\mu (z)= {\mathrm {diag}} (z,1)$
in this section.
Lemma 6.1. Let
$ {\mathrm {Spa}} (F,F^+)$
be a geometric point in
$ {\mathrm {Perf}}_{\mathbb {F}_q}$
. Let
$$\begin{align*}0 \longrightarrow \mathscr{E} \longrightarrow \mathscr{E}' \longrightarrow \mathscr{F} \longrightarrow 0 \end{align*}$$
be an exact sequence of coherent sheaf over
$X_F^{\mathrm {sch}}$
, where
$\mathscr {E}$
and
$\mathscr {E}'$
are vector bundles of rank
$2$
and
$\mathscr {F}$
is a torsion coherent sheaf of length
$1$
. Assume that
$\mathscr {E}$
is not semi-stable and
$\mathscr {E}'$
is semi-stable. Then
$\mathscr {E} \simeq \mathcal {O} (m) \oplus \mathcal {O} (m-1)$
and
$\mathscr {E}' \simeq \mathcal {O} (m) \oplus \mathcal {O} (m)$
for some integer m.
Proof. The vector bundle
$\mathscr {E}'$
is isomorphic to
$\mathcal {O} (m+\frac {1}{2})$
or
$\mathcal {O} (m) \oplus \mathcal {O} (m)$
for some integer m, since it is semi-stable.
If
$\mathscr {E}'$
is isomorphic to
$\mathcal {O} (m+\frac {1}{2})$
, then
$\mathscr {E}$
is isomorphic to
$\mathcal {O} (m) \oplus \mathcal {O} (m)$
by [Reference Fargues and FontaineFF14, Theorem 2.9]. This contradicts the condition that
$\mathscr {E}$
is not semi-stable.
Assume
$\mathscr {E}'$
is isomorphic to
$\mathcal {O} (m) \oplus \mathcal {O} (m)$
. Then
$\mathscr {E}$
is isomorphic to
$\mathcal {O} (m_1) \oplus \mathcal {O} (m_2)$
with
$m_1, m_2 \leq m$
or
$\mathcal {O} (n+\frac {1}{2})$
with
$n \leq m-1$
by [Reference Fargues and FontaineFF14, 6.3.1]. By considering
$\deg (\mathscr {E}) +1=\deg (\mathscr {E}')$
, the possible cases are
$\mathcal {O} (m) \oplus \mathcal {O} (m-1)$
or
$\mathcal {O} (m-\frac {1}{2})$
. However, the latter case does not happen, since
$\mathscr {E}$
is not semi-stable.
Proposition 6.2. Then we have
$$\begin{align*}{\mathrm{supp}} \overrightarrow{h}_{\natural}(\overleftarrow{h}^{*}\mathscr{F}_{\varphi} \otimes {\mathrm{IC}}_{\mu}') \subset {\mathrm{Bun}}_{G}^{\mathrm{ss}} \times {\mathrm{Div}}_X^1. \end{align*}$$
Proof. Take a non-basic element
$[b] \in B(G)$
. Then it suffices to show that
$p_b^* y_b^* \overrightarrow {h}_{\natural } \overleftarrow {h}^* \mathscr {F}_{\varphi } =0$
, where
$p_b$
is defined at (5.2). We consider the following cartesian diagram:
Let
$\overleftarrow {h}_{b} \colon \mathcal {T}_{b,\mathbb {C}_p^{\flat }}^{\leq \mu } \to {\mathrm {Bun}}_{G}$
be the morphism which appears in the above diagram. Then it suffices to see that
$$\begin{align*}H_{\mathrm{c}}^i \big( \mathcal{T}_{b,\mathbb{C}_p^{\flat}}^{\leq \mu}, {\overleftarrow{h}}_{b}^* \mathscr{F}_{\varphi} \big) =0. \end{align*}$$
On the other hand, we have
$$\begin{align*}H_{\mathrm{c}}^i \big( \mathcal{T}_{b,\mathbb{C}_p^{\flat}}^{\leq \mu}, {\overleftarrow{h}}_{b}^* \mathscr{F}_{\varphi} \big) = H_{\mathrm{c}}^i \big( \mathcal{T}_{b,\mathbb{C}_p^{\flat}}^{\leq \mu,\mathrm{ss}}, {\overleftarrow{h}}_{b,\mathrm{ss}}^* j_{\mathrm{ss}}^* \mathscr{F}_{\varphi} \big) \end{align*}$$
by
$\mathscr {F}_{\varphi } = j_{\mathrm {ss},\natural } j_{\mathrm {ss}}^* \mathscr {F}_{\varphi }$
. We have a decomposition
$$\begin{align*}\mathcal{T}_{b,\mathbb{C}_p^{\flat}}^{\leq \mu, \mathrm{ss}} = \coprod_{N \in 2 \mathbb{Z}} \mathcal{T}_{b,b_1^N,\mathbb{C}_p^{\flat}}^{\leq \mu} \end{align*}$$
by Lemma 6.1. Hence, we have
$$\begin{align*}H_{\mathrm{c}}^i \big( \mathcal{T}_{b,\mathbb{C}_p^{\flat}}^{\leq \mu, \mathrm{ss}}, {\overleftarrow{h}}_{b,\mathrm{ss}}^* j_{\mathrm{ss}}^* \mathscr{F}_{\varphi} \big) =0 \end{align*}$$
by Theorem 4.30.
Theorem 6.3. Then we have
$$\begin{align*}\overrightarrow{h}_{\natural}(\overleftarrow{h}^{*}\mathscr{F}_{\varphi} \otimes {\mathrm{IC}}_{\mu}') \cong \mathscr{F}_{\varphi} \boxtimes \varphi. \end{align*}$$
Acknowledgements
The authors would like to thank Laurent Fargues and Peter Scholze for answering our questions on their forthcoming works. They also want to thank Paul Ziegler for answering a question regarding his work. They are grateful to David Hansen for his helpful comments on a previous version of this paper. They also want to thank Teruhisa Koshikawa for his comments on this paper. Finally, they thank a referee for helpful comments and suggestions.
Competing interests
The authors have no competing interests to declare.
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There is no data associated with this research.
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