those which are contained in a closed Weyl chamber attached to
$w_{0}Bw_{0}^{-1}$ for some $w_{0}\in W_{0}$, the finite Weyl group of
$G_{\breve{F}}$ with respect to $S_{\breve{F}}$.

It will turn out that ${\mathcal{A}}_{{\mathcal{G}},\bar{k},\mu}$ is
functorial in $({\mathcal{G}},\mu)$, as soon as we develop a theory of local
models ${\mathcal{M}}_{{\mathcal{G}},\mu}$, see Definition 4.11, and calculate
their special fibers, confer Theorem 6.16. The admissible locus also admits
the following representation-theoretic interpretation in terms of
representations of the Langlands dual group $\widehat{G}$ (here, taken over
any algebraically closed field) with dual torus $\widehat{T}$:

######  Lemma 3.12.

Let $\widehat{\lambda}^{I}\in X^{\ast}(\widehat{T}^{I})\cong X_{\ast}(T)_{I}$
be running through the set of restrictions of all weights $\widehat{\lambda}$
for $\widehat{T}$ occurring in a finite dimensional algebraic representation
of $\widehat{G}$ with fixed highest weight $\widehat{\mu}=\mu$. Then
${\mathcal{A}}_{{\mathcal{G}},\mu}={{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\\{\widehat{\lambda}^{I}(\pi)\\}}$.

###### Proof.

Being a $\widehat{G}$-representation, $V$ contains all the weights
$\widehat{\lambda}$ conjugate to $\widehat{\mu}$ under the absolute Weyl group
with the same non-zero multiplicity. Under $X^{\ast}(\widehat{T})\cong
X_{\ast}(T)$, these correspond to the conjugates of $\mu$ compatibly with the
projection to $X^{\ast}(\widehat{T}^{I})\cong X_{\ast}(T)_{I}$. Hence, the
lemma follows from the definition of the admissible locus. ∎

######  Example 3.13.

The basic example of the admissible locus occurs for
$G={\operatorname{GL}}_{2}$, $\mu=(1,0)$ and ${\mathcal{G}}={\mathcal{I}}$ an
Iwahori. In this case, ${\mathcal{A}}_{{\mathcal{G}},\mu}$ is the union of two
copies of ${\mathbb{P}}^{1,\operatorname{perf}}_{k}$ intersecting
transversally at a point. More generally, one can enumerate the
Iwahori–Schubert orbits of the translated to the neutral component admissible
locus ${\mathcal{A}}_{{\mathcal{I}},\mu}$ in terms of alcoves in the standard
apartment ${\mathscr{A}}(G,S,F)$. For pictures in the case of unitary groups
of split rank $2$, the reader is referred to the introduction of [PR09]. For
further examples, see the survey [PRS13].

###  3.3. Canonical deperfections

Now, we wish to introduce equivariant deperfections of the Schubert perfect
schemes ${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}$ following Proposition 3.3
and discuss their geometry, at least for certain $W$. We are especially
interested in admissible loci ${\mathcal{A}}_{{\mathcal{G}},\mu}$ for $\mu$
minuscule.

First, recall that the congruence quotient $L^{\leq n}_{k}{\mathcal{G}}$ of
$L^{+}_{k}{\mathcal{G}}$ has a deperfection $\mathrm{Gr}_{n}{\mathcal{G}}$,
given by $(n+1)$-truncated Witt vectors and which is called the Greenberg
realization. We denote by $L^{>n}_{k}{\mathcal{G}}$ the kernel of
$L^{+}_{k}{\mathcal{G}}\to L^{\leq n}_{k}{\mathcal{G}}$.

######  Definition 3.14.

Let $n$ be the smallest nonnegative integer such that
$L^{>n}_{\bar{k}}{\mathcal{G}}$ acts trivially on
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},W}$ and call it the associated
depth. The canonical deperfection151515In equicharacteristic, one recovers the
weak normalization of classical Schubert schemes, which turn out to be the
classical ones under 1.9 and also
$p\nmid\lvert\pi_{1}(G_{\operatorname{der}})\rvert$, see [FHLR22, Section 3.1]
and [HLR18]. ${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}^{\operatorname{can}}$
of the perfect Schubert scheme ${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}$ is
the finite type $k_{W}$-scheme with $\mathrm{Gr}_{n}{\mathcal{G}}$-action
determined by Proposition 3.3.

Assume the $L^{+}_{\bar{k}}{\mathcal{G}}$-action on
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},W}$ factors through
$L^{0}_{\bar{k}}{\mathcal{G}}={\mathcal{G}}_{\bar{k}}^{\mathrm{perf}}$. For
$V\leq W$, we get a deperfection

${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},V}^{\operatorname{can}}\to{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},W}^{\operatorname{can}}$
(3.21)

of the closed immersion of perfect Schubert schemes, because the image is a
finite type deperfection with smaller function fields, as it carries a
${\mathcal{G}}_{\bar{k}}$-action.

However, it is not clear that the finite type morphism is a closed immersion.
To know more about the geometry of
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}^{\operatorname{can}}$, we exploit
the picture in equicharacteristic.

Assume $G$ is adjoint, and also 1.9 for $G$ over $\breve{F}$, that is, if
$p=2$, then $G$ has no odd unitary factors over $\breve{F}$. Then, for every
parahoric $\breve{O}$-group ${\mathcal{G}}$ attached to a facet in
${\mathscr{A}}(G,S,\breve{F})$, we find smooth, affine, fiberwise connected
$\breve{O}[[{t}]]$-lifts $\underline{{\mathcal{G}}}$ in the sense of [FHLR22,
Proposition 2.8]. Note that the $\bar{k}[[{t}]]$-reductions
${\mathcal{G}}^{\prime}$ are parahoric models of some adjoint connected
reductive $\bar{k}(({t}))$-group $G^{\prime}$ attached to a facet in some
appartment
${\mathscr{A}}(G^{\prime},S^{\prime},\bar{k}(({t})))\cong{\mathscr{A}}(G,S,F)$,
see [FHLR22, Lemma 2.7].

In particular, these come with isomorphisms

${\mathcal{G}}\otimes_{\breve{O}}\bar{k}\cong{\mathcal{G}}^{\prime}\otimes_{\bar{k}[[{t}]]}\bar{k},$
(3.22)

that are functorial as we vary ${\mathcal{G}}$ among parahoric models attached
to a facet in ${\mathscr{A}}(G,S,\breve{F})$, and which we now exploit to
compare their Schubert schemes. Let us note that the loop groups
$L^{+}_{\bar{k}}{\mathcal{G}}$ and its equicharacteristic cousin
$L^{+}_{\bar{k}}({\mathcal{G}}^{\prime})$ admit natural surjections on
${\mathcal{G}}_{\bar{k}}^{\operatorname{perf}}\cong{\mathcal{G}}^{\prime,\mathrm{perf}}_{\bar{k}}.$
Below, we use subscripts $({\operatorname{-}})^{\prime}$ to denote the
perfections of the equicharacteristic loop groups and Schubert varieties for
${\mathcal{G}}^{\prime}$.

######  Lemma 3.15.

Under the above constraints, there are unique equivariant isomorphisms

${{\mathcal{F}}\\!\ell}^{\operatorname{can}}_{{\mathcal{G}},\bar{k},W}\cong{{\mathcal{F}}\\!\ell}^{\operatorname{can}}_{{\mathcal{G}}^{\prime},\bar{k},W^{\prime}}$
(3.23)

for all connected ${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},W}$ of depth
$0$, that is, whose $L^{+}_{\bar{k}}{\mathcal{G}}$-action factors through
${\mathcal{G}}_{\bar{k}}^{\operatorname{perf}}$.

###### Proof.

As ${\mathcal{G}}_{\bar{k}}\cong{\mathcal{G}}_{\bar{k}}^{\prime}$, it suffices
by Proposition 3.3 to produce equivariant isomorphism
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},W}\cong{{\mathcal{F}}\\!\ell}_{{\mathcal{G}}^{\prime},\bar{k},W^{\prime}}$
of the perfect Schubert schemes. During the proof, we fix an auxiliary Iwahori
${\mathcal{I}}$ dilated from ${\mathcal{G}}$ and consider the corresponding
Iwahori–Schubert perfect scheme
${{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),\bar{k},W}$.

First assume that $W=\\{w\\}$. The perfect variety
${{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),\bar{k},w}$ can be
resolved via a Demazure variety
${\mathcal{D}}_{{\mathcal{I}},\bar{k},\dot{w}}$. If $s$ is the first letter of
the word $\dot{w}$ and $\dot{v}$ is the word obtained from deleting the first
letter, we get

${\mathcal{D}}_{{\mathcal{I}},\bar{k},\dot{w}}={{\mathcal{F}}\\!\ell}_{{\mathcal{I}},\bar{k},s}\tilde{\times}{\mathcal{D}}_{{\mathcal{I}},\bar{k},\dot{v}}$
(3.24)

where $L^{+}_{\bar{k}}{\mathcal{I}}\subset L^{+}_{\bar{k}}{\mathcal{P}}$ is
the minimal parahoric corresponding to $s$. We claim that the action of
$L^{+}_{\bar{k}}{\mathcal{I}}$ on
${\mathcal{D}}_{{\mathcal{I}},\bar{k},\dot{v}}$ is trivial when restricted to
the normal subgroup $L^{\geq 1}_{\bar{k}}{\mathcal{P}}$. Otherwise, let
$\alpha$ be the negative simple affine root corresponding to $s$ and observe
that $L^{+}_{\bar{k}}{\mathcal{U}}_{\alpha+1}$ acts non-trivially on
${\mathcal{D}}_{{\mathcal{I}},\bar{k},\dot{v}}$. But conjugating by $s$ yields
that $L^{+}_{\bar{k}}{\mathcal{U}}_{-\alpha+1}\subset L^{\geq
1}_{\bar{k}}{\mathcal{I}}$ does not act trivially on
${\mathcal{D}}_{{\mathcal{I}},\bar{k},\dot{w}}$.

Arguing inductively on $\dot{w}$, and exploting the above claim, we reach at
an ${\mathcal{I}}_{\bar{k}}^{\operatorname{perf}}$-equivariant identification
of the Demazure perfect varieties

${\mathcal{D}}_{{\mathcal{I}},\bar{k},\dot{w}}\cong{\mathcal{D}}_{{\mathcal{I}}^{\prime},\bar{k},\dot{w}^{\prime}}$
(3.25)

bounded by $\dot{w}$ resp. $\dot{w}^{\prime}$ and attached to ${\mathcal{I}}$,
respectively the $\bar{k}(({t}))$-reduction ${\mathcal{I}}^{\prime}$ of the
Iwahori $\breve{O}[[{t}]]$-lift. In the case $l(\dot{w})=1$, then we get the
unique equivariant identification of one-dimensional Iwahori–Schubert perfect
varieties, which are perfected projective lines.

As the Picard group of the Demazure varieties have already been determined,
see Theorem 3.8, respectively [HZ20, Section 3.2] for the equicharacteristic
case, the previous isomorphism descends uniquely by Proposition 3.2 to an
${\mathcal{I}}_{\bar{k}}^{\operatorname{perf}}$-equivariant identification

${{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),\bar{k},w}\cong{{\mathcal{F}}\\!\ell}_{({\mathcal{I}}^{\prime},{\mathcal{G}}^{\prime}),\bar{k},w^{\prime}}$
(3.26)

of the perfect Schubert varieties, see Remark 3.9. If the left side is stable
under ${\mathcal{G}}_{\bar{k}}$, then we need to show the map is not only
${\mathcal{I}}_{\bar{k}}^{\operatorname{perf}}$-equivariant, but furthermore
${\mathcal{G}}_{\bar{k}}^{\operatorname{perf}}$-equivariant.

Let $\bar{Q}\subset{\mathcal{G}}_{\bar{k}}$ denote the image of
${\mathcal{I}}_{\bar{k}}$. By assumption, the
${\mathcal{I}}_{\bar{k}}^{\operatorname{perf}}$-action on both sides factors
through the perfection of $\bar{Q}$. Using the convolution product

${\mathcal{G}}_{V}^{\operatorname{perf}}\times^{\bar{Q}^{\operatorname{perf}}}{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},w},$
(3.27)

we get a perfect $\bar{k}$-variety mapping
${\mathcal{I}}_{\bar{k}}^{\operatorname{perf}}$-equivariantly to
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},w}$ and that can be identified
with its equicharacteristic analogue in a
${\mathcal{G}}_{\bar{k}}^{\operatorname{perf}}$-equivariant fashion. Since
${\mathcal{G}}_{\bar{k}}^{\operatorname{perf}}/{\bar{Q}^{\operatorname{perf}}}\subset{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},\bar{k}}$
is a Schubert perfect variety at Iwahori level, we know its Picard group by
Theorem 3.8 and Remark 3.9. Applying again Proposition 3.2, we not only
recover the original isomorphism, by Proposition 3.10, but also conclude it is
${\mathcal{G}}_{\bar{k}}^{\operatorname{perf}}$-equivariant and the unique
such map.

For a general $W$ as in the statement, we now can glue the above isomorphism
to non-irreducible Schubert perfect schemes

${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},W}\cong{{\mathcal{F}}\\!\ell}_{{\mathcal{G}}^{\prime},\bar{k},W^{\prime}}$
(3.28)

appealing again to Proposition 3.10. ∎

From now on $G$ is no longer assumed to be adjoint. We approach the canonical
admissible locus ${\mathcal{A}}_{{\mathcal{G}},\mu}^{\operatorname{can}}$ for
minuscule $\mu$, that is, the canonical perfection of the admissible locus,
with our comparison result, describing its singularities (thereby confirming
[Zhu17a, Conjecture III] for Schubert varieties in the admissible locus) and
computing its coherent cohomology.

######  Theorem 3.16.

Let $\mu$ be minuscule and assume 1.9. Then,
${\mathcal{A}}_{{\mathcal{G}},\mu}^{\operatorname{can}}$ is Cohen–Macaulay and
Frobenius split compatibly with its ${\mathcal{G}}_{\bar{k}}$-stable reduced
$\bar{k}$-subschemes.

Moreover, for every ample line bundle ${\mathcal{L}}$ on
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}$ that descends to
${\mathcal{A}}_{{\mathcal{G}},\bar{k},\mu}^{\operatorname{can}}$, there is an
equality

$\dim_{\bar{k}}\operatorname{H}^{0}({\mathcal{A}}_{{\mathcal{G}},\bar{k},\mu}^{\operatorname{can}},{\mathcal{L}})=\dim_{C}\operatorname{H}^{0}({\mathcal{F}}_{G,C,\mu},{\mathcal{O}}(c_{\mathcal{L}})).$
(3.29)

Here, ${\mathcal{F}}_{G,\mu}=G_{E}/P_{\mu}^{-}$ is the classical flag variety
attached to $\mu$, the central charge
$c_{\mathcal{L}}\in{\mathbb{Z}}^{\operatorname{rk}\Psi_{G}-\operatorname{rk}\Phi_{G}}$
is given by Kac–Moody coefficients, see [PR08, Section 10] and [BS17, Section
10], and the line bundle ${\mathcal{O}}(c_{\mathcal{L}})$ is the corresponding
power of the ample generator of $\operatorname{Pic}({\mathcal{F}}_{G,C,\mu})$.

###### Proof.

We want to apply Lemma 3.15, in order to reduce the statements to
equicharacteristic, where we refer to [FHLR22, Theorem 3.1, Theorem 4.1].

In order to do this, we first notice that there is an equivariant isomorphism
${\mathcal{A}}_{{\mathcal{G}},\bar{k},\mu}^{\mathrm{can}}\cong{\mathcal{A}}_{{\mathcal{G}}_{\operatorname{ad}},\bar{k},\mu_{\operatorname{ad}}}^{\mathrm{can}}$
via the natural map. Here, $\mu_{{\operatorname{ad}}}$ denotes the composition
of $\mu$ with $G_{C}\to{G_{\operatorname{ad}}}_{C}$. Indeed, this can be
checked on perfections and then at the level of geometric points, where it
follows from the assertion that
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}\to{{\mathcal{F}}\\!\ell}_{{\mathcal{G}}_{\mathrm{ad}},\bar{k}}$
is an open and closed immersion.

We still have to show that
${\mathcal{A}}_{{\mathcal{G}},\bar{k},\mu}^{\mathrm{can}}$ has minimal depth,
that is, $L^{+}_{\bar{k}}{\mathcal{G}}$ acts via ${\mathcal{G}}_{\bar{k}}$.
Since $L^{\geq 1}_{\bar{k}}{\mathcal{G}}$ is a normal subgroup, it suffices to
check that it fixes each of the sections ${\lambda}$ defining the admissible
locus. By the combinatorial dictionary, see our proof of Proposition 3.10, it
suffices to show that $\lvert a({\lambda}_{I})\rvert\leq 1$, that is, the
translation ${\lambda}_{I}$ moves every affine root to a parallel one at
distance at most one. By definition, one has

$a({\lambda}_{I})=[K:\breve{F}]^{-1}\sum_{{\sigma}\in{\operatorname{Gal}}(K/\breve{F})}{\sigma}\tilde{a}({\lambda}),$
(3.30)

where $K$ is a finite Galois extension of $\breve{F}$ splitting
$G_{\breve{F}}$, and $\tilde{a}$ is an absolute root restricting to $a$, so
its absolute value is at most $1$, since ${\lambda}$ is minuscule. ∎

##  4\. Affine Grassmannians and local models

In this section, we start by gathering several basic facts on the
$B_{\mathrm{dR}}^{+}$-affine Grassmannian over $\operatorname{{Spd}}C$. Most
of them were established in [SW20, Lecture XIX] and [FS21, Chapters VI.2,
VI.5], but our approach is sufficiently different and relevant to later
sections that it merits some elaboration.

Then, we introduce the main objects of study of this article, to wit the local
models

${\mathcal{M}}_{{\mathcal{G}},\mu}\subset{\operatorname{Gr}}_{{\mathcal{G}},O_{E}}$
(4.1)

defined for every $\mu\in X_{\ast}(T)$ via v-closures of Schubert diamonds in
a Beilinson–Drinfeld Grassmannian. We dedicate the rest of the section to
showing that ${\mathcal{M}}_{{\mathcal{G}},\mu}$ is an
$L^{+}_{O_{E}}{\mathcal{G}}$-stable flat, proper $p$-adic kimberlite with good
finiteness properties. In particular, its special fiber will be shown to be
representable by some connected Schubert perfect scheme
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}$.

###  4.1. The $B_{\mathrm{dR}}^{+}$-affine Grassmannian

In this section, we fix a complete discretely valued field
$F/{\mathbb{Q}}_{p}$ with perfect residue field $k$, ring of integers $O$ and
uniformizer $\pi$, a complete algebraic closure $C/F$ with ring of integers
$O_{C}$ and residue field $\bar{k}=k_{C}$. We denote by $\breve{F}\subset C$
the maximal unramified complete subextension with ring of integers $\breve{O}$
and the same residue field $\bar{k}=k_{\breve{F}}$. Further, we fix a
(connected) reductive $F$-group $G$ and a maximal $\breve{F}$-split $F$-torus
$S\subset G$ containing a maximal $F$-split torus, see [BT84, Proposition
5.1.10]. As $G$ is quasi-split over $\breve{F}$ by Steinberg’s theorem, the
centralizer $T$ of $S$ is a maximal torus. Also, we fix a Borel subgroup
$B\subset G_{\breve{F}}$ containing $T_{\breve{F}}$.

For any affinoid perfectoid space $\operatorname{{Spa}}(R,R^{+})$ in
characteristic $p$ equipped with a map to
$\operatorname{{Spd}}{\mathbb{Z}}_{p}$, let
$B_{{\mathrm{dR}}}^{+}(R^{\sharp})$, respectively
$B_{{\mathrm{dR}}}(R^{\sharp})$, be the rings of de Rham periods formed using
$O$-Witt vectors. For convenience, we set
$B^{+}_{{\mathrm{dR}}}:=B^{+}_{{\mathrm{dR}}}(C)$ and
$B_{{\mathrm{dR}}}:=B_{{\mathrm{dR}}}(C)$. The $B^{+}_{\mathrm{dR}}$-loop
group of $G$ is the group functor over $\operatorname{{Spd}}F$ given by

$LG\colon(R,R^{+})\mapsto G(B_{{\mathrm{dR}}}(R^{\sharp})),$ (4.2)

and the positive loop group is the subgroup functor

$L^{+}G\colon(R,R^{+})\mapsto G(B_{{\mathrm{dR}}}^{+}(R^{\sharp})).$ (4.3)

Their v-sheaf quotient

${\operatorname{Gr}}_{G}:=LG/L^{+}G$ (4.4)

is called the $B^{+}_{\mathrm{dR}}$-affine Grassmannian. Similarly to Section
3.2, ${\operatorname{Gr}}_{G}(R,R^{+})$ parametrizes $G$-torsors on the
spectrum ${\operatorname{Spec}}(B^{+}_{{\mathrm{dR}}}(R^{\sharp}))$ with a
trivialization over ${\operatorname{Spec}}(B_{\mathrm{dR}}(R^{\sharp}))$.
Here, we are primarily interested in the geometry and work therefore over
$\operatorname{{Spd}}C$. The base changes are denoted by $L_{C}G$,
$L_{C}^{+}G$ and ${\operatorname{Gr}}_{G,C}$, for convenience.

As an auxiliary first step, we study the affine flag variety and then
translate the results to the affine Grassmannian. For this, the Iwahori group
$B^{+}_{\mathrm{dR}}$-model $\mathcal{I}$ is given as the dilatation of
$G\otimes_{F}B^{+}_{\mathrm{dR}}$ along the subscheme $B_{C}\subset G_{C}$ of
its special fiber. Define

$L^{+}_{C}\mathcal{I}\colon(R,R^{+})\mapsto\mathcal{I}(B_{{\mathrm{dR}}}^{+}(R^{\sharp}))$
(4.5)

which is a subgroup v-sheaf of $L^{+}G$. It gives rise to the
$B^{+}_{\mathrm{dR}}$-affine flag variety

${{\mathcal{F}}\\!\ell}_{\mathcal{I},C}:=L_{C}G/L^{+}_{C}\mathcal{I},$ (4.6)

viewed as a v-sheaf over $\operatorname{{Spd}}C$.

We recall that ${\operatorname{Gr}}_{G,C}\to\operatorname{{Spd}}C$ is an
increasing union of proper, spatial diamonds by [SW20, Lecture XIX]. The same
holds for ${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}\to\operatorname{{Spd}}C$,
as the projection

${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}\to{\operatorname{Gr}}_{G,C}$ (4.7)

is a proper, cohomologically smooth $(G_{C}/B_{C})^{\diamondsuit}$-fibration.
The following discussion is parallel to parts of Section 3.2 but simplified by
the fact that we consider $G\otimes_{F}B^{+}_{\mathrm{dR}}$ which is a (split)
reductive group over $B^{+}_{\mathrm{dR}}$ (and not some parahoric group
scheme). The geometry of the affine flag variety
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$ or, better, the v-stack quotient

${\operatorname{Hk}}_{{\mathcal{I}},C}:=L^{+}_{C}\mathcal{I}\backslash{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}=L^{+}_{C}\mathcal{I}\backslash
L_{C}G/L^{+}_{C}\mathcal{I}$ (4.8)

is reflected in the Iwahori-Weyl group

$\tilde{W}:=N(B_{\mathrm{dR}})/T(B_{\mathrm{dR}}^{+}),$ (4.9)

where $N$ denotes the normalizer of $T$ in $G$. There is a canonical map
$\underline{\tilde{W}}\rightarrow{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$
because $T(B^{+}_{\mathrm{dR}})\subset\mathcal{I}(B^{+}_{\mathrm{dR}})$.

######  Lemma 4.1.

The map $\underline{\tilde{W}}\to{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$
induces a bijection

$\tilde{W}\cong|{\operatorname{Hk}}_{{\mathcal{I}},C}|.$ (4.10)

###### Proof.

Every point of $|{\operatorname{Hk}}_{{\mathcal{I}},C}|$ is represented by a
map $\operatorname{{Spa}}(K,K^{+})\to L_{C}G$ with $K$ algebraically closed
perfectoid. Two $K$-valued points have the same underlying element in
$|{\operatorname{Hk}}_{{\mathcal{I}},C}|$ if, v-locally, they lie in the same
double coset

$\mathcal{I}(B^{+}_{\mathrm{dR}}(K^{\sharp}))\backslash
G(B_{\mathrm{dR}}(K^{\sharp}))/\mathcal{I}(B^{+}_{\mathrm{dR}}(K^{\sharp})).$
(4.11)

The identification now follows from the Bruhat decomposition which is
independent of $K^{\sharp}$. ∎

Let ${\mathbf{a}}\subset{\mathscr{A}}(G,S,F)$ be the alcove defined by
$\mathcal{I}$ in the appartment fixed by $S$ of the Bruhat–Tits building. Let
$\mathbb{S}\subset\tilde{W}$ be the set of simple reflections along the walls
bounding ${\mathbf{a}}$. The affine Weyl group
$W_{{\operatorname{af}}}\subset\tilde{W}$ is the subgroup generated by the
elements in $\mathbb{S}$. Then $W_{{\operatorname{af}}}$ is a Coxeter group,
and depends only on the Bruhat–Tits building of $G$. There exists a canonical
short exact sequence

$1\to W_{{\operatorname{af}}}\to\tilde{W}\to\pi_{1}(G)\to 1,$ (4.12)

which is naturally split by taking the stabilizer
$\Omega_{{\mathbf{a}}}\subset\tilde{W}$ of the alcove ${\mathbf{a}}$. Thus, we
can write each $w\in\tilde{W}$ uniquely as $w=w_{{\operatorname{af}}}\tau$
with $\tau\in\Omega_{\mathbf{a}}$ and $w_{\operatorname{af}}\in
W_{\operatorname{af}}$.

######  Lemma 4.2.

Equip $\pi_{1}(G)$ with the discrete topology. The morphism

$|{\operatorname{Hk}}_{{\mathcal{I}},C}|\to\pi_{1}(G)$ (4.13)

is locally constant, thus underlies a morphism
${\operatorname{Hk}}_{{\mathcal{I}},C}\to\underline{\pi_{1}(G)}$ of small
v-stacks.

###### Proof.

Here, we follow the argument behind the proof of [PR08, Theorem 5.1]. If
$G=G_{\operatorname{sc}}$ is simply connected, then we see that, for every
algebraically closed perfectoid field $K/C$, the group $L_{C}G(K)$ is
generated by its affine root subgroups $L_{C}U_{a}(K)$.

But, since $L_{C}U_{a}$ is connected (choose a pinning), we conclude that
$L_{C}G$, hence also ${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$ and
${\operatorname{Hk}}_{{\mathcal{I}},C}$ are connected. If $G=T$ is a torus,
then we see easily that ${\operatorname{Gr}}_{T,C}$ equals the v-sheaf
$\underline{X_{\ast}(T)}$ compatibly with the map above.

Now, suppose that $G_{\operatorname{der}}=G_{\operatorname{sc}}$. Then,
$\pi_{1}(G)$ identifies with the fundamental group of the abelian quotient
$G/G_{\operatorname{der}}$, so the claim is clear. Finally, for a general
group $G$, consider the z-extension

$1\to T_{\operatorname{sc}}\to\tilde{G}\to G\to 1,$ (4.14)

where $\tilde{G}$ is given by the pushout of $(G_{\operatorname{sc}}\rtimes
T)$ along the morphism $\mathrm{ker}(T_{{\operatorname{sc}}}\to T)\to
T_{{\operatorname{sc}}}$, where $T_{{\operatorname{sc}}}$ is the preimage of
the maximal torus $T\subset G$ under the map $G_{{\operatorname{sc}}}\to G$.
Using the fact that $T_{\operatorname{sc}}$ is an induced torus, we see that
the Hecke stacks and the $\pi_{1}$’s lie in a similar exact sequence, which
yields the claim. ∎

For $\tau\in\pi_{1}(G)$, we denote by
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}^{\tau}$ the fiber over $\tau$ of the
morphism ${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}\to\underline{\pi_{1}(G)}$.
We note that right translation by a representative of $\tau$ in $L_{C}G$
induces an isomorphism

${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}^{1}\overset{\cong}{\longrightarrow}{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}^{\tau}.$
(4.15)

Moreover, ${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}^{1}$ is canonically
isomorphic to the affine flag variety
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}}_{\operatorname{sc}},C}$ of the simply
connected cover $G_{{\operatorname{sc}}}$. Namely, the transition morphism
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}}_{\operatorname{sc}},C}\rightarrow{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}^{1}$
is bijective by checking on geometric points ([Sch17, Lemma 12.5]) and using
the Bruhat decomposition, hence must be an isomorphism as both
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}}_{\operatorname{sc}},C},{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}^{1}$
are ind-proper over $\operatorname{{Spd}}C$.

######  Definition 4.3.

Let $w\in\tilde{W}$. The Schubert cell
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}^{\circ}\subset{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$
is the v-sheaf-theoretic image of the orbit map

$L^{+}_{C}{\mathcal{I}}\to{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C},\ i\mapsto
iw.$ (4.16)

The Schubert variety is the v-closure
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}:={{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}^{\circ,\mathrm{cl}}$
in the sense of Section 2.1.

By Proposition 2.8, we know that the underlying topological space of
${{\mathcal{F}}\\!\ell}_{I,w}$ is the weakly generalizing closure of
$|{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}^{\circ}|$ inside
$|{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}|$. But,
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}^{\circ}$ is possibly ill-behaved
because $L^{+}_{C}{\mathcal{I}}$ is not quasicompact. As we show in
Proposition 4.5 and Corollary 4.6 for the affine Grassmannian, our definition
is equivalent to the pointwise definition in [FS21, Definition VI.2.2]. We
start with the case of simple reflections:

######  Lemma 4.4.

Let $s\in\mathbb{S}$ be a simple reflection. Then there is an isomorphism
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,s}\simeq({\mathbb{P}}^{1}_{C})^{\diamondsuit}$
that restricts to
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,s}^{\circ}\simeq({\mathbb{A}}^{1}_{C})^{\diamondsuit}$.
In particular, ${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,s}^{\circ}$ is a
topologically dense open subset of
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,s}$.

###### Proof.

Let ${\mathcal{P}}_{s}$ be the parahoric group scheme over
$B_{\mathrm{dR}}^{+}$ associated to the wall of ${\mathbf{a}}$ defining $s$.
The maximal reductive quotient $H$ of its special fiber over $C$ has
semisimple rank $1$. Using [BT84, Théorème 4.6.33], we see that
$L^{+}_{C}{\mathcal{I}}$ is the preimage of $Q^{\diamondsuit}$ under
$L^{+}{\mathcal{P}}_{s}\rightarrow H^{\diamondsuit}$ for some Borel subgroup
$Q\subset H$. Thus, there are isomorphisms
$L^{+}_{C}\mathcal{P}_{s}/L^{+}_{C}\mathcal{I}\simeq(H/Q)^{\diamondsuit}\simeq(\mathbb{P}^{1}_{C})^{\diamondsuit}$
which can be made explicit via the choice of a pinning. This implies that the
monomorphism

$L^{+}_{C}\mathcal{P}_{s}/L^{+}_{C}\mathcal{I}\subset{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$
(4.17)

is a closed embedding, as ${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$ is
separated and $(\mathbb{P}^{1}_{C})^{\diamondsuit}$ is proper over
$\operatorname{{Spd}}C$. The isomorphism
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,s}^{\circ}\simeq({\mathbb{A}}^{1}_{C})^{\diamondsuit}$
is now clear, since this is the only non-trivial $Q^{\diamondsuit}$-orbit in
$(\mathbb{P}^{1}_{C})^{\diamondsuit}$. ∎

In order to treat more general $w=w_{\mathrm{af}}\tau\in\tilde{W}\cong
W_{\mathrm{af}}\rtimes\Omega_{\mathfrak{a}}$, we invoke Demazure resolutions
as follows. Let $\dot{w}=s_{1}\dots s_{n}$ be a reduced word for
$w_{\operatorname{af}}=w\tau^{-1}$ with $s_{i}\in\mathbb{S}$ and consider the
Demazure variety

${\mathcal{D}}_{C,\dot{w}}:=L^{+}_{C}{\mathcal{P}}_{1}\times^{L^{+}_{C}{\mathcal{I}}}\dots\times^{L^{+}_{C}{\mathcal{I}}}L^{+}_{C}{\mathcal{P}}_{n}/L^{+}_{C}{\mathcal{I}}$
(4.18)

which will also be denoted by
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,s_{1}}\tilde{\times}\dots\tilde{\times}{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,s_{n}}$.
It is connected and cohomologically smooth over $\operatorname{{Spd}}C$ (being
an iterated $\mathbb{P}^{1}_{C}$-fibration), and the twisted product

${\mathcal{D}}_{C,\dot{w}}^{\circ}={{\mathcal{F}}\\!\ell}^{\circ}_{{\mathcal{I}},C,s_{1}}\tilde{\times}\dots\tilde{\times}{{\mathcal{F}}\\!\ell}^{\circ}_{{\mathcal{I}},C,s_{n}}$
(4.19)

of the open cells is topologically dense by induction on $n$, starting with
Lemma 4.4 and using that $L^{+}_{C}{\mathcal{I}}$ is pro-(cohomologically
smooth) over $\operatorname{{Spd}}C$. It carries, moreover, a natural morphism
(induced by multiplication)

$\pi_{\dot{w}}\colon{\mathcal{D}}_{C,\dot{w}}\rightarrow{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$
(4.20)

which necessarily maps onto
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w_{\operatorname{af}}}$, by
properness, $L^{+}_{C}\mathcal{I}$-equivariance and the fact that $\dot{w}$
maps to $w$. After translation by $\tau$, we may regard this as a resolution
of ${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}$, which is thus in particular
connected.

For the next result, we note that $\tilde{W}$, in analogy to the discussion
following (3.8), is equipped with a length function and Bruhat partial order
induced from the quasi-Coxeter structure on $\tilde{W}\cong
W_{\mathrm{af}}\rtimes\Omega_{\mathfrak{a}}$.

######  Proposition 4.5.

Let $w\in\tilde{W}$. Then
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}^{\circ}$, respectively
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}$, agrees with the subfunctor of
all maps $S\rightarrow{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$ such that for
all geometric points $S^{\prime}=\operatorname{{Spa}}(K,K^{+})\rightarrow S$,
the induced point
$S^{\prime}\rightarrow{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}\rightarrow{\operatorname{Hk}}_{{\mathcal{I}},C}$
is given by $w$, respectively by $v$ for some $v\leq w$. In particular,
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}^{\circ}\subset{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}$
is a topologically dense open.

###### Proof.

Observe that the first assertion cannot be verified at geometric points
because $L^{+}_{C}{\mathcal{I}}$ is not quasicompact. However, we see from
Lemma 4.4 that the result holds for simple reflections. Indeed, we even have
by Bruhat–Tits combinatorics

$L^{+}_{C}{\mathcal{U}}_{\alpha_{s}}\cdot
s={{\mathcal{F}}\\!\ell}^{\circ}_{{\mathcal{I}},C,s},$ (4.21)

where $\alpha_{s}$ denotes the positive simple affine root associated to the
simple reflection $s$, and ${\mathcal{U}}_{\alpha_{s}}$ is the corresponding
$B_{\mathrm{dR}}^{+}$-model of the affine root group. Pulling across the
reflections $s_{i}$ appearing in the convolution product of
${\mathcal{D}}_{C,\dot{w}}$, we see that
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}^{\circ}$ surjects to the v-sheaf
image of ${\mathcal{D}}_{C,\dot{w}}^{\circ}$ along $\pi_{\dot{w}}$. This
v-sheaf image identifies with the pointwise description of
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}^{\circ}$ by quasicompactness of
$\pi_{\dot{w}}$ and bijectivity at geometric points.

Similarly, the v-sheaf image of ${\mathcal{D}}_{C,\dot{w}}$ along
$\pi_{\dot{w}}$ is a proper closed sub-v-sheaf of
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}$. By generalities of Tits system,
see [BT72, 1.2.6], this v-sheaf image coincides with the desired pointwise
description of ${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}$. Pulling back
again via the quotient map $\pi_{\dot{w}}$, we see that
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}^{\circ}\subset{{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}$
is a topologically dense open of the closed v-sheaf image of $\pi_{\dot{w}}$.
∎

As a corollary, we get that the bijection
$\lvert{\operatorname{Hk}}_{{\mathcal{I}},C}\rvert\cong\tilde{W}$ from Lemma
4.1 is a homeomorphism where $\tilde{W}$ is endowed with order topology via
its Bruhat order, and also that
$\pi_{0}({{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C})=\pi_{0}({\operatorname{Gr}}_{G,C})=\pi_{1}(G)$
via Lemma 4.2.

Now, we apply our results to the affine Grassmannian
${\operatorname{Gr}}_{G}$. Note that there is the group isomorphism

$X_{\ast}(T)\cong T(B_{\mathrm{dR}})/T(B_{\mathrm{dR}}^{+}),\
\chi\mapsto\chi(\xi)$ (4.22)

which is independent of the choice of uniformizer $\xi\in
B_{\mathrm{dR}}^{+}$. Then the Cartan decomposition induces a bijection

$\lvert{\operatorname{Hk}}_{G,C}\rvert\simeq X_{*}(T)_{+},$ (4.23)

where ${\operatorname{Hk}}_{G,C}=L^{+}_{C}G\backslash L_{C}G/L^{+}_{C}G$
denotes the Hecke stack. Therefore, we get a Schubert cell
${\operatorname{Gr}}_{G,C,\mu}^{\circ}\subset{\operatorname{Gr}}_{G,C}$
defined as the v-sheaf-theoretic image of the orbit map and the Schubert cell
${\operatorname{Gr}}_{G,C,\mu}$ defined as its closure, for each $\mu\in
X_{*}(T)_{+}$, compare with Definition 4.3.

######  Corollary 4.6.

Let $\mu\in X_{*}(T)_{+}$. Then ${\operatorname{Gr}}_{G,C,\mu}^{\circ}$,
respectively ${\operatorname{Gr}}_{G,C,\mu}$ agrees with the subfunctor of all
maps $S\rightarrow{\operatorname{Gr}}_{G,C,\mu}$ such that for all geometric
points $S^{\prime}=\operatorname{{Spa}}(K,K^{+})\rightarrow S$, the induced
point
$S^{\prime}\rightarrow{\operatorname{Gr}}_{G,C}\rightarrow{\operatorname{Hk}}_{G,C}$
is given by $\mu$, respectively by some ${\lambda}\leq\mu$ in the dominance
order on $X_{*}(T)_{+}$. In particular,
${\operatorname{Gr}}_{G,C,\mu}^{\circ}\subset{\operatorname{Gr}}_{G,C,\mu}$ is
a topologically dense open.

###### Proof.

This formally follows from Proposition 4.5 by using the projection
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}\rightarrow{\operatorname{Gr}}_{G,C}$
from (4.7) and noting that the dominance order on $X_{*}(T)_{+}$ is induced by
the Bruhat order, see [Ric13, Corollaries 1.8, 2.10] for similar arguments. We
leave the details to the reader. ∎

We also have the following fact which says that the
$\operatorname{{Spd}}C$-valued points are dense in
${\operatorname{Gr}}_{G,C,\mu}$ even for the constructible topology.

######  Corollary 4.7.

Let $\mu\in X_{*}(T)_{+}$. The spatial diamond ${\operatorname{Gr}}_{G,C,\mu}$
has enough $C$-facets in the sense of [Gle20, Definition 1.4.38].

###### Proof.

Taking the preimage under the projection
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C}\rightarrow{\operatorname{Gr}}_{G,C}$
from (4.7), this reduces to the analogous assertion for
${{\mathcal{F}}\\!\ell}_{{\mathcal{I}},C,w}$ for some $w\in\tilde{W}$. Since
the Demazure resolution is a v-cover, it is enough to prove that
${\mathcal{D}}_{C,\dot{w}}$ has enough $C$-facets. This in turn can be proved
inductively on the length of $\dot{w}$. If $\dot{w}=s\cdot\dot{v}$ then
${\mathcal{D}}_{C,\dot{w}}$ is a pro-étale
$({\mathbb{P}}^{1}_{C})^{\diamondsuit}$-bundle over
${\mathcal{D}}_{C,\dot{v}}$. We may find a pro-étale cover

$X\to{\mathcal{D}}_{C,\dot{v}}$ (4.24)

with
$X\times_{{\mathcal{D}}_{C,\dot{v}}}{\mathcal{D}}_{C,\dot{w}}=X\times_{\operatorname{{Spd}}C}({\mathbb{P}}^{1}_{C})^{\diamondsuit}$.
Following the arguments given in [Gle20, Lemma 2.2.24, Proposition 2.2.30], we
may even assume that $X$ has enough facets over $\operatorname{{Spd}}C$. By
[Gle20, Proposition 1.4.39, 2.], ${\mathcal{D}}_{C,\dot{w}}$ also has enough
facets. ∎

We conclude with some motivation for our later discussion of representability.

######  Proposition 4.8.

Let $\mu\in X_{*}(T)_{+}$. The v-sheaf ${\operatorname{Gr}}_{G,C,\mu}$ is
representable by a projective $C$-scheme ${\mathcal{F}}_{G,C,\mu}$ if and only
if $\mu$ is minuscule.

###### Proof.

If $\mu$ is minuscule, then the $L^{+}_{C}G$-action factors through
$G_{C}^{\diamondsuit}$ and the Bialynicki-Birula map gives an isomorphism
${\operatorname{Gr}}_{G,C,\mu}\simeq(G_{C}/P_{\mu})^{\diamondsuit}$, see
[SW20, Proposition 19.4.2].

Now suppose that $\mu$ is not minuscule, that is,
$\langle\mu,\theta\rangle\geq 2$ for some root $\theta$ of $G_{C}$. Then the
$L^{+}_{C}{\mathcal{I}}$-orbit of the point $\mu$ is not representable by a
rigid space: This orbit can be filtered by successive smooth rigid spaces, so
taking fiber products appropriately, we would eventually arrive at
representability of the Banach-Colmez space
${\mathcal{B}}{\mathcal{C}}(B^{+}_{\mathrm{dR}}/\xi^{2})$. However, this is a
non-split self-extension of ${\mathbb{G}}_{a}^{\diamondsuit}$ which does not
even split étale locally, so cannot be representable. Indeed, if the extension
were split étale locally, it would actually split on the nose, as
$H^{1}({\mathbb{A}}^{1}_{C},{\mathcal{O}})$ is trivial. However, if

$X:=\operatorname{{Spa}}(C\langle T^{\pm 1}\rangle,O_{C}\langle T^{\pm
1}\rangle)\subset{\mathbb{A}}^{1}_{C}$ (4.25)

is the affinoid torus, the element $T\in{\mathbb{A}}^{1}_{C}(X)$ does not
admit a lift to ${\mathcal{B}}{\mathcal{C}}(B^{+}_{\mathrm{dR}}/\xi^{2})(X)$
as we now show. Let $X^{\prime}=\operatorname{{Spa}}(C\langle T^{\pm
1/p^{\infty}}\rangle,O_{C}\langle T^{\pm 1/p^{\infty}}\rangle)$ be the usual
perfectoid $\mathbb{Z}_{p}(1)$-cover of $X$. Elements in
${\mathcal{B}}{\mathcal{C}}(B^{+}_{\mathrm{dR}}/\xi^{2})(X^{\prime})$ can be
represented by $[a]+[b]\xi$ with $a,b\in(C\langle T^{\pm
1/p^{\infty}}\rangle)^{\flat}$. Assume now that

$x:=[a]+[b]\xi$

maps to $T$, that means, $a^{\sharp}=T$ in ${\mathbb{A}}^{1}_{C}(X^{\prime})$
(cf. [SW20, Section 6.2] for the map $(-)^{\sharp}$), and assume that $x$ is
invariant under $\mathbb{Z}_{p}(1)$. Let $g\in\mathbb{Z}_{p}(1)$. Then $g$
acts on $[a]$ via multiplication with $[g]$ if we identify
$\mathbb{Z}_{p}(1)\subset C^{\flat}$. Using the
$({\operatorname{-}})^{\sharp}$-map, we get

$a^{\sharp}\left(\frac{[g]-1}{\xi}\right)^{\sharp}=b^{\sharp}-g(b^{\sharp}).$
(4.26)

But, if $g\in\mathbb{Z}_{p}(1)$ is a generator, then
$a^{\sharp}(\frac{[g]-1}{\xi})^{\sharp}=cT$ with $c\in C$ non-zero. Writing
$b^{\sharp}$ as a powerseries in the $g$-eigenvectors $T^{n}$ with
$n\in\mathbb{Z}[1/p]$, then shows that (4.26) can not hold because
$g\in\mathbb{Z}_{p}(1)$ fixes $T$. This finishes the argument. ∎

###  4.2. Local models

We continue with the notation of Section 4.1, and additionally let
${\mathcal{G}}$ be a parahoric $O$-model of $G$.

We work with the moduli space ${\operatorname{Gr}}_{\mathcal{G}}$ of
${\mathcal{G}}$-torsors over ${\operatorname{Spec}}(B^{+}_{\mathrm{dR}})$
trivialized over ${\operatorname{Spec}}(B_{\mathrm{dR}})$, see [SW20,
Definition 20.3.1], which is the Beilinson–Drinfeld Grassmannian over
$\operatorname{{Spd}}O$. A crucial result of Scholze–Weinstein concerns its
ind-properness, see [SW20, Theorems 19.3.4, 20.3.6, 21.2.1].

######  Theorem 4.9 (Scholze–Weinstein).

The structure morphism of the Beilinson–Drinfeld Grassmannian

${\operatorname{Gr}}_{\mathcal{G}}\rightarrow\operatorname{{Spd}}O$ (4.27)

is ind-proper and ind-representable in spatial diamonds.

In [Ans18, Section 12] and then later in [Gle20, Section 2.2], the first named
and second named author respectively constructed and studied the
specialization map for ${\operatorname{Gr}}_{\mathcal{G}}$, see also [Gle20,
Proposition 2.2.5].

Again, we have natural loop groups at hand, namely

$L_{O}{\mathcal{G}}\colon(R,R^{+})\mapsto{\mathcal{G}}(B_{\mathrm{dR}}(R^{\sharp}))$
(4.28)

and

$L^{+}_{O}{\mathcal{G}}\colon(R,R^{+})\mapsto{\mathcal{G}}(B_{\mathrm{dR}}^{+}(R^{\sharp})),$
(4.29)

where $(R^{\sharp},R^{\sharp+})$ denotes an untilt of $(R,R^{+})$ over $(O,O)$
and $B_{\mathrm{dR}}^{(+)}(R^{\sharp})$ the ring of de Rham periods formed
using $O$-Witt vectors. These define v-sheaves over $\operatorname{{Spd}}O$
and the base changes to $\operatorname{{Spd}}F$, respectively
$\operatorname{{Spd}}k$ recover the loop groups $L^{+}G\subset LG$ from
Section 4.1, respectively the v-sheaves
$(L^{+}_{k}{\mathcal{G}})^{\diamondsuit}\subset(L_{k}G)^{\diamondsuit}$
associated with the loop groups from Section 3.2. Their base changes to
$O_{C}$ are denoted $L_{O_{C}}{\mathcal{G}}$, $L^{+}_{O_{C}}{\mathcal{G}}$ and
${\operatorname{Gr}}_{{\mathcal{G}},O_{C}}$.

######  Lemma 4.10.

There is a natural isomorphism

$L_{O}{\mathcal{G}}/L^{+}_{O}{\mathcal{G}}\cong{\operatorname{Gr}}_{\mathcal{G}},$
(4.30)

where the left side is a quotient for the étale topology. In particular, on
geometric fibers

${\operatorname{Gr}}_{G}\cong{\operatorname{Gr}}_{\mathcal{G}}\times_{\operatorname{{Spd}}O}\operatorname{{Spd}}F,\;\;\;\;\;\;{{\mathcal{F}}\\!\ell}_{\mathcal{G}}^{\diamondsuit}\cong{\operatorname{Gr}}_{\mathcal{G}}\times_{\operatorname{{Spd}}O}\operatorname{{Spd}}k,$
(4.31)

where ${\operatorname{Gr}}_{G}$ is the affine Grassmannian from Section 4.1
and ${{\mathcal{F}}\\!\ell}_{\mathcal{G}}^{\diamondsuit}$ the v-sheaf attached
to the Witt vector partial affine flag variety from Section 3.2.

###### Proof.

For the uniformization (4.30), see [SW20, Proposition 20.3.2]. The
isomorphisms (4.31) are given by base change from (4.30) by unwinding the
definitions. ∎

Let $\mu$ be a conjugacy class of cocharacters of $G_{C}$, with field of
definition $E\subset C$. We denote by $O_{E}$ its ring of integers with
residue field $k_{E}$. We wish to construct a closed sub-v-sheaf

${\mathcal{M}}_{{\mathcal{G}},\mu}\subset{\operatorname{Gr}}_{{\mathcal{G}}}|_{\operatorname{{Spd}}O_{E}}$
(4.32)

prolonging the Schubert diamonds ${\operatorname{Gr}}_{G,\mu}$ which are the
descent to $\operatorname{{Spd}}E$ of the ones we studied in the previous
subsection.

######  Definition 4.11.

Let $\mu$ be a conjugacy class of cocharacters in $G_{C}$. The local model
${\mathcal{M}}_{{\mathcal{G}},\mu}$ is the v-closure of
${\operatorname{Gr}}_{G,\mu}$ inside
${\operatorname{Gr}}_{{\mathcal{G}}}|_{\operatorname{{Spd}}O_{E}}$.

A priori ${\mathcal{M}}_{{\mathcal{G}},\mu}$ does not admit a moduli problem
description for general parahoric ${\mathcal{G}}$, so its structure could be
harder to parse. Let us give some examples where the local model
${\mathcal{M}}_{{\mathcal{G}},\mu}$ is relatively well understood.

######  Example 4.12.

If ${\mathcal{G}}$ is reductive, then ${\mathcal{M}}_{{\mathcal{G}},\mu}$ is
the integral Schubert variety over $\operatorname{{Spd}}O_{E}$ and generalizes
the objects introduced in Section 4.1, see [SW20, Proposition 20.3.6] and
[FS21, VI.1]. If $G=T$ is a torus, then the explicit description of
${\operatorname{Gr}}_{{\mathcal{T}}}$ furnishes an identity
${\mathcal{M}}_{{\mathcal{T}},\mu}=\operatorname{{Spd}}O_{E}$, see [SW20,
Proposition 21.3.1].

We need to show permanence of the local model under the
$L^{+}_{O}{\mathcal{G}}$-action.

######  Proposition 4.13.

The natural action map
$L^{+}_{O}{\mathcal{G}}\times{\operatorname{Gr}}_{{\mathcal{G}}}\rightarrow{\operatorname{Gr}}_{{\mathcal{G}}}$
restricts, after base change to $\operatorname{{Spd}}O_{E}$, to a group action
on the closed sub-v-sheaf ${\mathcal{M}}_{{\mathcal{G}},\mu}$. Moreover, the
generic fiber of ${\mathcal{M}}_{{\mathcal{G}},\mu}$ is topologically dense.

###### Proof.

By embedding ${\mathcal{G}}$ in $\operatorname{GL}_{n}$, we may always find a
quasi-compact closed subsheaf
$X\subset{\operatorname{Gr}}_{{\mathcal{G}},O_{E}}$ with
${\mathcal{M}}_{{\mathcal{G}},\mu}\subset X$, stable under
$L^{+}_{O_{E}}{\mathcal{G}}$ whose action factors through a congruence
quotient $L^{\leq N}_{O_{E}}{\mathcal{G}}$, where $N$ is a sufficiently large
positive integer (necessarily at least $\langle 2\rho,\mu\rangle$ as one sees
by restricting to ${\operatorname{Gr}}_{G,\mu}$).

The structure map $L^{\leq
N}_{O_{E}}{\mathcal{G}}\to\operatorname{{Spd}}O_{E}$ is partially proper,
cohomologically smooth and consequently universally open, see [Sch17,
Proposition 23.11]. By Corollary 2.9,

$L^{\leq
N}_{O_{E}}{\mathcal{G}}\times_{\operatorname{{Spd}}O_{E}}{\mathcal{M}}_{{\mathcal{G}},\mu}=(L^{\leq
N}_{E}G\times_{\operatorname{{Spd}}E}{\operatorname{Gr}}_{G,\mu})^{\operatorname{cl}}$
(4.33)

as closed sub-v-sheaves of $L^{\leq
N}_{O_{E}}{\mathcal{G}}\times_{\operatorname{{Spd}}O_{E}}X$. Now, we have seen
that $L^{+}_{E}G$ respects ${\operatorname{Gr}}_{G,\mu}$, so the
multiplication map $L^{\leq
N}_{E}G\times_{\operatorname{{Spd}}E}{\operatorname{Gr}}_{G,\mu}\to X$ factors
through ${\operatorname{Gr}}_{G,\mu}$. This also implies that the integral
action map

$L^{\leq
N}_{O_{E}}{\mathcal{G}}\times_{\operatorname{{Spd}}O_{E}}{\mathcal{M}}_{{\mathcal{G}},\mu}\to
X$ (4.34)

factors through the closure ${\mathcal{M}}_{{\mathcal{G}},\mu}$, as we
desired.

For the last claim, we consider the restricted variant of the Hecke v-stack

${\operatorname{Hk}}^{\leq N}_{{\mathcal{G}},\mu}:=\big{[}L^{\leq
N}_{O_{E}}{\mathcal{G}}\backslash{\mathcal{M}}_{{\mathcal{G}},\mu}\big{]}.$
(4.35)

Its underlying topological space has the extra special property that every
subset is weakly generalizing, since for every perfectoid affinoid field
$(K,K^{+})$ the Bruhat decomposition over $B_{\mathrm{dR}}(K)$ is insensitive
to variation of $K^{+}$. Now, the projection map

$\operatorname{pr}\colon{\mathcal{M}}_{{\mathcal{G}},\mu}\to{\operatorname{Hk}}^{\leq
N}_{{\mathcal{G}},\mu}$ (4.36)

is cohomologically smooth and consequently open. Therefore, by the same
argument of Corollary 2.9, we see that both the usual topological and the
weakly generalizing closure commute with pullback along
$\lvert\operatorname{pr}\rvert$. It results that
$\lvert{\operatorname{Gr}}_{G,\mu}\rvert$ is a dense open of
$\lvert{\mathcal{M}}_{{\mathcal{G}},\mu}\rvert$. ∎

We can now prove the following structural properties of
${\mathcal{M}}_{{\mathcal{G}},\mu}$.

######  Proposition 4.14.

With notation as in Definition 2.35,
${\mathcal{M}}_{{\mathcal{G}},\mu}\in{\mathcal{K}}$. More specifically, the
local model ${\mathcal{M}}_{{\mathcal{G}},\mu}$ is a flat $p$-adic kimberlite
over $O_{E}$ with enough facets over $C$ and $O_{C}$-formalizable
$C$-sections. Moreover, the special fiber

${\mathcal{M}}_{{\mathcal{G}},\mu,k_{E}}:={\mathcal{M}}_{{\mathcal{G}},\mu}|_{\operatorname{{Spd}}(k_{E})}$
(4.37)

is of the form ${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}^{\diamondsuit}$ for a
connected perfect Schubert scheme ${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}$.
In particular,
${\mathcal{M}}_{{\mathcal{G}},\mu}^{\operatorname{red}}={{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}$
is perfectly proper and perfectly finitely presented perfect $k$-scheme.

###### Proof.

Choosing a closed embedding
${\mathcal{G}}\hookrightarrow\operatorname{GL}_{n}$, we may find a cocharacter
$\nu$ of $\operatorname{GL}_{n}$ giving rise to a closed immersion

${\mathcal{M}}_{{\mathcal{G}},\mu}\hookrightarrow{\mathcal{M}}_{\operatorname{GL}_{n},\nu},$
(4.38)

so that local models are proper v-sheaves and, in particular, quasi-compact.
By [Gle20, Proposition 1.4.30.(3), Proposition 2.2.5] to prove
${\mathcal{M}}_{{\mathcal{G}},\mu}$ is a kimberlite, it suffices to prove it
is $p$-adic. By [Gle20, Proposition 1.3.35], we may reduce to proving that the
special fiber of ${\mathcal{M}}_{{\mathcal{G}},\mu}$ is represented by a
v-sheaf of the form $X^{\diamond}$ for $X$ a perfect scheme.

By Proposition 4.13, ${\mathcal{M}}_{{\mathcal{G}},\mu,\bar{k}}$, is of the
form161616A priori this could be an infinite union. $\cup_{i\in
I}{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},W_{i}}^{\diamond}$ with finite
subsets $W_{i}\subset\tilde{W}$, where we have used the fact that
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}$ is perfectly proper, in order to
deduce
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}^{\diamondsuit}={{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}^{\diamond}$.
By quasi-compactness, we get
${\mathcal{M}}_{{\mathcal{G}},\mu,k_{E}}={{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}^{\diamond}$
for some finite subset $W\subset\tilde{W}$, which finishes the proof that
${\mathcal{M}}_{{\mathcal{G}},\mu}$ is a $p$-adic kimberlite.

That ${\mathcal{M}}_{{\mathcal{G}},\mu}$ has $O_{C}$-formalizable $C$-sections
is proved in [Gle20, Proposition 2.2.4], see also [Ans18, Section 12]. We
explained in Corollary 4.7 that ${\mathcal{M}}_{{\mathcal{G}},\mu}$ has enough
$C$-facets. Together with Proposition 4.13 and Lemma 2.37, flatness follows.
By Proposition 2.34,
$\lvert{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}\rvert=\operatorname{sp}(\lvert{\operatorname{Gr}}_{G,\mu}\rvert)$
and since ${\operatorname{Gr}}_{G,\mu}$ is connected
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},W}$ is also connected. ∎

######  Remark 4.15.

It follows that the base change
${\mathcal{M}}_{{\mathcal{G}},\mu}|_{\operatorname{{Spd}}O_{C}}$ is still
topologically flat, and hence agrees with the v-closure
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}$ of ${\operatorname{Gr}}_{G,C,\mu}$
inside ${\operatorname{Gr}}_{{\mathcal{G}},O_{C}}$. Indeed, repeating the
argument of Proposition 4.13 over $\operatorname{{Spd}}O_{C}$, we see that the
special fiber of ${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}$ is represented by a
Schubert perfect scheme. But a $\operatorname{{Spd}}k$-valued point of
${\mathcal{M}}_{{\mathcal{G}},\mu}$ is a specialization of some
$\operatorname{{Spd}}C$-valued point by Proposition 4.14, hence equality of
both closures is clear.

Next, we analyse some functoriality behavior of
${\mathcal{M}}_{{\mathcal{G}},\mu}$ in the pair $({\mathcal{G}},\mu)$. Here,
by definition, a map
$({\mathcal{G}}_{1},\mu_{1})\to({\mathcal{G}}_{2},\mu_{2})$ is a morphism of
$O$-group schemes ${\mathcal{G}}_{1}\to{\mathcal{G}}_{2}$ such that the image
of $\mu_{1}$ in $G_{2,C}$ lies in the same conjugacy class as $\mu_{2}$.

######  Proposition 4.16.

The association
$({\mathcal{G}},\mu)\mapsto{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}$, see
Remark 4.15, is functorial, preserves closed embeddings and direct products,
and induces isomorphisms
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}\cong{\mathcal{M}}_{{\mathcal{G}}_{\operatorname{ad}},O_{C},\mu_{\operatorname{ad}}}$,
where $\mu_{\mathrm{ad}}$ denotes the composite of $\mu$ with $G_{C}\to
G_{\mathrm{ad},C}$.

###### Proof.

Functoriality follows from that of ${\operatorname{Gr}}_{G,C,\mu}$ and the
definition using v-closures, see Remark 4.15. For the claim regarding closed
embeddings and central extensions, we refer to [Lou20, IV, Proposition 4.16,
Corollary 4.17]: one checks injectivity at geometric points, using Lemma 4.17.
As for direct products, it suffices to check equality at the level of
$\operatorname{{Spd}}k$-valued points by Proposition 4.14, and this is easy
because the generic fiber was already a product. ∎

######  Lemma 4.17.

The specialization map induces a bijection

$\pi_{0}({\operatorname{Gr}}_{{\mathcal{G}},\breve{O}})\overset{\cong}{\longrightarrow}\pi_{0}({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}})\cong\pi_{1}(G)_{I},$
(4.39)

where $I$ is the absolute Galois group of $\breve{F}$.

###### Proof.

For the final bijection, see [Zhu17a, Proposition 1.21]. The first is a
consequence of proper base change [Sch17, Theorem 19.2, Remark 19.3] applied
to
$f\colon{\operatorname{Gr}}_{{\mathcal{G}},\breve{O}}\rightarrow\operatorname{{Spd}}\breve{O}$
and the base change
$i\colon\operatorname{{Spd}}\bar{k}\to\operatorname{{Spd}}\breve{O}$, using
that the $0$-th cohomology group computes connected components: for some
coefficient ring, say, $\Lambda={\mathbb{Z}}/\ell$ with $\ell\neq p$, we apply
the proper base change $i^{*}R^{0}f_{*}\Lambda_{X}\cong
R^{0}(f^{\prime})_{*}(i^{\prime})^{*}\Lambda_{X}$ where $\Lambda_{X}$ is the
constant sheaf supported on increasing closed $\breve{O}$-proper sub-v-sheaves
$X\subset{\operatorname{Gr}}_{{\mathcal{G}},\breve{O}}$. Passing to global
sections, the second computes $\Lambda^{\pi_{0}(X_{\bar{k}})}$ by definition
whereas the first computes $\Lambda^{\pi_{0}(X)}$ by using the v-cover
$\operatorname{{Spd}}O_{C}\to\operatorname{{Spd}}\breve{O}$. Finally, we use
[Sch17, Section 27] to pass between
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}^{\diamondsuit}$ and
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}$. ∎

##  5\. Geometry of multiplicative group actions

Our approach to the Scholze–Weinstein conjecture requires determining the
special fiber of local models in terms of admissible loci. We follow the
general strategy of Haines and the fourth named author [HR21] of calculating
the support of nearby cycles using hyperbolic localization. This requires
translating the results from [HR21, Section 5] to the v-sheaf Beilinson-
Drinfeld Grassmannian. For basic facts pertaining to
${\mathbb{G}}_{m}^{\diamondsuit}$-actions on small v-stacks, the reader is
referred to [FS21, Chapter IV.6].

###  5.1. Over $O$

As in Section 4.1, we continue to fix a complete discretely valued field
$F/{\mathbb{Q}}_{p}$ with ring of integers $O$ and perfect residue field $k$,
a complete algebraic closure $C/F$ and a connected reductive $F$-group $G$
with parahoric model ${\mathcal{G}}$ over $O$ containing the connected Néron
model ${\mathcal{S}}$ of the maximally $F$-split maximal $\breve{F}$-split
$F$-torus $S$.

Fix a cocharacter
${\lambda}\colon{\mathbb{G}}_{m}\rightarrow{\mathcal{S}}\subset{\mathcal{G}}$
defined over $O$. After base change to $F$, this induces a Levi
$M=M_{\lambda}$ with Lie algebra
${\operatorname{Lie}}\,M=({\operatorname{Lie}}\,G)_{{\lambda}=0}$, a parabolic
subgroup $P=P_{\lambda}^{+}$ with Lie algebra
${\operatorname{Lie}}\,P=({\operatorname{Lie}}\,G)_{{\lambda}\geq 0}$ and an
unipotent subgroup $U=U^{+}_{\lambda}$ with Lie algebra
${\operatorname{Lie}}\,U=({\operatorname{Lie}}\,G)_{{\lambda}>0}$ fitting in a
semi-direct product decomposition $P=M\ltimes U$. Since ${\lambda}$ is defined
over $O$, the decomposition $P=M\ltimes U$ extends to $O$-models
${\mathcal{P}}={\mathcal{M}}\ltimes{\mathcal{U}}$, admitting analogous
descriptions for their Lie algebras and being equipped with homomorphisms

${\mathcal{M}}\longleftarrow{\mathcal{P}}\longrightarrow{\mathcal{G}}.$ (5.1)

The $O$-group schemes ${\mathcal{P}}$, ${\mathcal{M}}$, ${\mathcal{U}}$ are
smooth affine with connected fibers, and ${\mathcal{M}}$ is a parahoric
$O$-model of the Levi subgroup $M$, see [HR21, Lemma 4.5] and also [CGP15,
Section 2.1], [KP21, Section 6.2] for proofs of these claims.

By functoriality, (5.1) induces maps of ind-(spatial
$\operatorname{{Spd}}O$-diamonds)

${\operatorname{Gr}}_{\mathcal{M}}\longleftarrow{\operatorname{Gr}}_{\mathcal{P}}\longrightarrow{\operatorname{Gr}}_{\mathcal{G}},$
(5.2)

where ${\operatorname{Gr}}_{\mathcal{M}}\rightarrow\operatorname{{Spd}}O$ and
${\operatorname{Gr}}_{\mathcal{G}}\rightarrow\operatorname{{Spd}}O$ are ind-
proper by Theorem 4.9. On the other hand, the cocharacter ${\lambda}$ induces
a cocharacter

${\mathbb{G}}_{m}^{\diamondsuit}\overset{[{\operatorname{-}}]}{\longrightarrow}L^{+}_{O}{\mathbb{G}}_{m}\overset{L^{+}_{O}{\lambda}}{\longrightarrow}L^{+}_{O}{\mathcal{G}},$
(5.3)

where $[{\operatorname{-}}]$ denotes the Teichmüller lift. Thus, we obtain a
left action of ${\mathbb{G}}_{m}^{\diamondsuit}$ on
${\operatorname{Gr}}_{\mathcal{G}}$.

######  Lemma 5.1.

The ${\mathbb{G}}_{m}^{\diamondsuit}$-action on
${\operatorname{Gr}}_{\mathcal{G}}$ satisfies [FS21, Hypothesis IV.6.1].

###### Proof.

Choosing a closed immersion
${\mathcal{G}}\hookrightarrow{\operatorname{GL}}_{n,O}$ of group schemes, we
reduce to the case ${\mathcal{G}}={\operatorname{GL}}_{n,O}$, using Theorem
4.9 to see that the induced map
${\operatorname{Gr}}_{\mathcal{G}}\hookrightarrow{\operatorname{Gr}}_{{\operatorname{GL}}_{n,O}}$
is a closed immersion. Then the lemma is a special case of [FS21, Proposition
VI.3.1]. ∎

Consequently, we obtain a ${\mathbb{G}}_{m}^{\diamondsuit}$-equivariant
diagram

$({\operatorname{Gr}}_{\mathcal{G}})^{0}\longleftarrow({\operatorname{Gr}}_{\mathcal{G}})^{+}\longrightarrow{\operatorname{Gr}}_{\mathcal{G}},$
(5.4)

where
$({\operatorname{Gr}}_{\mathcal{G}})^{0}=({\operatorname{Gr}}_{\mathcal{G}})^{{\mathbb{G}}_{m}^{\diamondsuit}}$
denotes the fixed points and $({\operatorname{Gr}}_{\mathcal{G}})^{+}$ the
attractor classifying ${\mathbb{G}}_{m}^{\diamondsuit}$-equivariant maps
$({\mathbb{A}}^{1})^{{\diamondsuit},+}\rightarrow{\operatorname{Gr}}_{\mathcal{G}}$
over $\operatorname{{Spd}}O$, see also (6.3) below. The map
$({\operatorname{Gr}}_{\mathcal{G}})^{+}\rightarrow({\operatorname{Gr}}_{\mathcal{G}})^{0}$
is the Bialynicki-Birula map given by evaluating at the zero section. Our aim
is to understand the relation of (5.2) with (5.4). The following result is the
analogue of [HR21, Theorem 5.6, Theorem 5.19] in the context of ind-schemes:

######  Theorem 5.2.

The maps (5.2) and (5.4) fit into a commutative diagram of ind-(spatial
$\operatorname{{Spd}}O$-diamonds)

${{\operatorname{Gr}}_{\mathcal{M}}}$${{\operatorname{Gr}}_{{\mathcal{P}}}}$${{\operatorname{Gr}}_{\mathcal{G}}}$${({\operatorname{Gr}}_{{\mathcal{G}}})^{0}}$${({\operatorname{Gr}}_{{\mathcal{G}}})^{+}}$${{\operatorname{Gr}}_{{\mathcal{G}}},}$$\iota^{0}$$\iota^{+}$${\operatorname{id}}$
(5.5)

with the following properties.

  1. (1)

The maps $\iota^{0}$, $\iota^{+}$ are open and closed immersions.

  2. (2)

Their base changes $\iota^{0}_{F}$, $\iota^{+}_{F}$ are isomorphisms.

  3. (3)

If ${\mathcal{G}}$ is special parahoric (for example, reductive), then
$\iota^{0}$, $\iota^{+}$ are isomorphisms.

In particular, the complements
$({\operatorname{Gr}}_{\mathcal{G}})^{0}\setminus\iota^{0}({\operatorname{Gr}}_{\mathcal{M}})$,
$({\operatorname{Gr}}_{\mathcal{G}})^{+}\setminus\iota^{+}({\operatorname{Gr}}_{\mathcal{P}})$
are concentrated over $\operatorname{{Spd}}k$, that is, their base change to
$\operatorname{{Spd}}F$ is empty.

If ${\mathcal{G}}$ is reductive, then Theorem 5.2 is proved in [FS21,
Proposition VI.3.1]. Indeed, in this case $\iota^{0}$, $\iota^{+}$ are
isomorphisms. In general, we follow the strategy of [HR21]:

###### Proof of Theorem 5.2.

First, we construct the maps $\iota^{0},\iota^{+}$. The closed subgroup
${\mathcal{M}}\hookrightarrow{\mathcal{G}}$ induces a closed immersion
${\operatorname{Gr}}_{\mathcal{M}}\hookrightarrow{\operatorname{Gr}}_{\mathcal{G}}$
(using Theorem 4.9) that is ${\mathbb{G}}_{m}^{\diamondsuit}$-equivariant for
the trivial action on the source. Thus, the map factors though the fixed
points, defining the necessarily closed immersion
$\iota^{0}\colon{\operatorname{Gr}}_{\mathcal{M}}\hookrightarrow({\operatorname{Gr}}_{\mathcal{G}})^{0}$.
The construction of $\iota^{+}$ is more delicate and proceeds as follows. Pick
a closed immersion ${\mathcal{G}}\hookrightarrow{\operatorname{GL}}_{n,O}$ of
$O$-group schemes such that the fppf quotient
${\operatorname{GL}}_{n,O}/{\mathcal{G}}$ is quasi-affine, see [PR08,
Proposition 1.3]. The cocharacter
${\lambda}^{\prime}\colon{\mathbb{G}}_{m}\rightarrow^{\lambda}{\mathcal{G}}\hookrightarrow{\operatorname{GL}}_{n,O}$
induces parabolic subgroup ${\mathcal{P}}^{\prime}$ with Lie algebra
${\operatorname{Lie}}\,{\mathcal{P}}^{\prime}=({\operatorname{Lie}}\,{\operatorname{GL}}_{n,O})_{{\lambda}^{\prime}\geq
0}$. The induced map on fppf quotients
${\mathcal{P}}^{\prime}/{\mathcal{P}}\rightarrow{\operatorname{GL}}_{n,O}/{\mathcal{G}}$
is a monomorphism between finite type $O$-schemes, thus quasi-affine by
Zariski’s main theorem. By functoriality of the construction
${\mathcal{G}}\mapsto{\operatorname{Gr}}_{\mathcal{G}}$, we obtain a
commutative diagram of ind-(spatial $\operatorname{{Spd}}O$-diamonds):

${{\operatorname{Gr}}_{{\mathcal{P}}}}$${({\operatorname{Gr}}_{{\mathcal{G}}})^{+}}$${{\operatorname{Gr}}_{\mathcal{G}}}$${{\operatorname{Gr}}_{{\mathcal{P}}^{\prime}}}$${({\operatorname{Gr}}_{{\operatorname{GL}}_{n,O}})^{+}}$${{\operatorname{Gr}}_{{\operatorname{GL}}_{n,O}}}$$\iota^{+}$$\cong$
(5.6)

Using Theorem 4.9, the map
${\operatorname{Gr}}_{\mathcal{G}}\rightarrow{\operatorname{Gr}}_{{\operatorname{GL}}_{n,O}}$
is a closed immersion so that the square is Cartesian. This proves the
existence of $\iota^{+}$. Furthermore, the displayed map
${\operatorname{Gr}}_{{\mathcal{P}}^{\prime}}\to({\operatorname{Gr}}_{{\operatorname{GL}}_{n,O}})^{+}$
is an isomorphism by [FS21, Proposition VI.3.1]. As
${\mathcal{P}}^{\prime}/{\mathcal{P}}$ is quasi-affine, so
${\operatorname{Gr}}_{\mathcal{P}}\rightarrow{\operatorname{Gr}}_{{\mathcal{P}}^{\prime}}$
is a locally closed immersion (compare with the proof of [Zhu17b, Proposition
1.2.6] and [SW20, Lemma 19.1.5]), the map $\iota^{+}$ is necessarily a locally
closed immersion as well.

Now, part (2) is immediate from our construction and [FS21, Proposition
VI.3.1] applied over $\operatorname{{Spd}}F$. For part (3), we observe that
$\iota^{0},\iota^{+}$ are bijective on geometric points if ${\mathcal{G}}$ is
special parahoric: by (2) for geometric points lying over
$\operatorname{{Spd}}F$ and by the Iwasawa decomposition [KP21, Section 3.3]
for geometric points lying over $\operatorname{{Spd}}k$. As $\iota^{0}$,
$\iota^{+}$ are locally closed immersions, they must be isomorphisms, so (3)
follows.

For (1), it remains to prove that $\iota^{0}$, $\iota^{+}$ are open immersions
for general parahoric group schemes ${\mathcal{G}}$. For this, we may and do
assume that $k$ is algebraically closed and observe that there are bijections
of connected components

$\pi_{0}(({\operatorname{Gr}}_{\mathcal{G}})^{+})\overset{\cong}{\longrightarrow}\pi_{0}(({\operatorname{Gr}}_{\mathcal{G}})^{0})\overset{\cong}{\longrightarrow}\pi_{0}(({{\mathcal{F}}\\!\ell}_{\mathcal{G}})^{0}),$
(5.7)

where the first holds by general properties of Bialynicki-Birula maps (see the
proof of [Ric19, Corollary 1.12]) and the second by proper base change as in
the proof of Lemma 4.17. The fixed points
$({{\mathcal{F}}\\!\ell}_{\mathcal{G}})^{0}$ in the Witt vector partial affine
flag variety can be analyzed in analogy to [HR21, Section 4]: concretely, if
${\mathcal{P}}_{\operatorname{sc}}={\mathcal{M}}_{\operatorname{sc}}\ltimes{\mathcal{U}}$
for ${\mathcal{M}}_{\operatorname{sc}}$ being the corresponding parahoric
model of $M_{\operatorname{sc}}$, then there is a disjoint union (on points)
into connected locally closed sub-ind-schemes

${{\mathcal{F}}\\!\ell}_{\mathcal{G}}=\bigcup_{[w]}{\mathcal{S}}_{w},\
{\mathcal{S}}_{w}=L_{k}{\mathcal{P}}_{\operatorname{sc}}\cdot w$ (5.8)

where $[w]$ runs through the double coset
$W_{M,{\operatorname{af}}}\backslash\tilde{W}/W_{\mathcal{G}}$ and $w$ denotes
the image of a representative under the embedding
$\tilde{W}/W_{\mathcal{G}}\hookrightarrow{{\mathcal{F}}\\!\ell}_{\mathcal{G}}$.
The image of
${{\mathcal{F}}\\!\ell}_{\mathcal{P}}\hookrightarrow{{\mathcal{F}}\\!\ell}_{\mathcal{G}}$
consists of those ${\mathcal{S}}_{w}$ for $[w]$ lying in
$W_{M,{\operatorname{af}}}\backslash\tilde{W}_{M}/W_{\mathcal{M}}$. Passing to
fixed points, the image of
${{\mathcal{F}}\\!\ell}_{\mathcal{M}}\hookrightarrow({{\mathcal{F}}\\!\ell}_{\mathcal{G}})^{0}$
is the union of the $L_{k}{\mathcal{M}}_{\operatorname{sc}}$-orbits for these
$[w]$. So the map
$\pi_{0}({{\mathcal{F}}\\!\ell}_{\mathcal{M}})\rightarrow\pi_{0}(({{\mathcal{F}}\\!\ell}_{\mathcal{G}})^{0})$
identifies with the injection

$W_{M,{\operatorname{af}}}\backslash\tilde{W}_{M}/W_{\mathcal{M}}\hookrightarrow
W_{M,{\operatorname{af}}}\backslash\tilde{W}/W_{\mathcal{G}}.$ (5.9)

We let ${\mathcal{C}}^{0}_{\mathcal{G}}$, respectively
${\mathcal{C}}^{+}_{\mathcal{G}}$, be the open and closed sub-v-sheaf of
$({\operatorname{Gr}}_{\mathcal{G}})^{0}$, respectively of
$({\operatorname{Gr}}_{\mathcal{G}})^{+}$, consisting of those components
belonging to
$\operatorname{im}(\pi_{0}({{\mathcal{F}}\\!\ell}_{\mathcal{M}})\hookrightarrow\pi_{0}({{\mathcal{F}}\\!\ell}_{\mathcal{G}}))$
under (5.7). Then the maps $\iota^{0}$, $\iota^{+}$ factor through
${\mathcal{C}}^{0}_{\mathcal{G}}$, respectively
${\mathcal{C}}^{+}_{\mathcal{G}}$ inducing locally closed immersions

${\operatorname{Gr}}_{\mathcal{M}}\hookrightarrow{\mathcal{C}}^{0}_{\mathcal{G}},\
{\operatorname{Gr}}_{\mathcal{P}}\hookrightarrow{\mathcal{C}}^{+}_{\mathcal{G}},$
(5.10)

that are bijective on geometric points, hence isomorphisms. The theorem is
thus proven. ∎

###  5.2. Semi-infinite orbits

We end this section with a study of the stratification (5.8). Throughout, we
assume that $k=\bar{k}$ is algebraically closed so that $F=\breve{F}$ and
$O=\breve{O}$. Note that the torus $S$ is then maximal $F$-split. The
following lemma simplifies some arguments of [HR21, Theorem 6.12] and is used
in the proof of Theorem 6.16 given in Section 6.5.

######  Lemma 5.3.

For every $w\in\tilde{W}/W_{{\mathcal{G}}}$, there is an $O$-cocharacter
${\mathbb{G}}_{m}\rightarrow{\mathcal{S}}\subset{\mathcal{G}}$ such that for
the induced strata
${\mathcal{S}}_{w}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},w}=\\{w\\}$.

###### Proof.

Up to changing the Iwahori $L_{k}^{+}{\mathcal{I}}\subset
L_{k}^{+}{\mathcal{G}}$, we may and do assume that the Iwahori–Schubert
variety ${{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),w}^{\circ}$ is
a dense open of ${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},w}$. Notice that the
closed complement of that dense open is stable under the
${\mathbb{G}}_{m}^{\operatorname{perf}}$-action, so it follows that the
connected component of the fixed point $w$ in the attractor
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}}}^{+}$ cuts
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},w}$ inside
${{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),w}^{\circ}$.

The reduced word $\dot{w}=s_{1}\dots s_{n}$ determines a minimal gallery
$\Gamma=({\mathbf{a}}_{0},{\mathbf{a}}_{1},\dots,{\mathbf{a}}_{n})$, where
${\mathbf{a}}_{i}=s_{1}\dots s_{i}({\mathbf{a}})$, going from the alcove
${\mathbf{a}}$ fixed by ${\mathcal{I}}(O)$ to its $\dot{w}$-conjugate. Let
$\alpha_{i}$ be the unique positive affine root such that $\partial\alpha_{i}$
is the wall separating ${\mathbf{a}}_{i-1}$ and ${\mathbf{a}}_{i}$. We claim
that

${{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),w}^{\circ}=L_{k}^{+}{\mathcal{U}}_{\alpha_{1}}\cdot\ldots\cdot
L_{k}^{+}{\mathcal{U}}_{\alpha_{n}}w.$ (5.11)

This follows by expanding the Demazure twisted product, pulling across the
simple reflections to the right, compare with Proposition 4.5. Indeed,
$s_{i-1}\dots s_{1}(\alpha_{i})$ is by construction the positive simple affine
root attached to $s_{i}$. We need to produce an $O$-cocharacter
${\mathbb{G}}_{m}\to{\mathcal{S}}$ whose induced
${\mathbb{G}}_{m}^{\operatorname{perf}}$-action repels every affine root group
$L^{+}{\mathcal{U}}_{\alpha_{i}}$, because then it would also repel
${{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),w}^{\circ}$ by (5.11).
This follows now by [HN02, Corollary 5.6] but we give below a quick proof for
the reader’s convenience.

Consider the subset $\Phi_{G,w}\subset\Phi_{G}$ of all euclidean roots
$a=\nabla\alpha$ which are gradients of the prescribed affine root
$\alpha_{i}$ attached to $\Gamma$. If $b$ denotes the barycenter of
${\mathbf{a}}$, then by definition $a(\dot{w}b-b)<0$ is stricly negative for
$a\in\Phi_{G,w}$. In particular, $\Phi_{G,w}$ consists entirely of negative
$B$-roots where $S\subset B\subset G$ is a Borel subgroup whose closed Weyl
chamber contains the vector $\dot{w}b-b$. So we may take any $B$-dominant
regular coweight ${\mathbb{G}}_{m}\to S$ which uniquely extends to the desired
${\mathbb{G}}_{m}\to{\mathcal{S}}$ because $S$ is $F$-split. ∎

Now assume that ${\lambda}$ is regular. Then $M_{\lambda}=T$ is a maximal
torus, $P_{\lambda}=B$ a Borel subgroup with unipotent radical $U$ defined
over $F$. The stratification (5.8) becomes

${{\mathcal{F}}\\!\ell}_{{\mathcal{G}}}=\bigcup_{w\in\tilde{W}/W_{\mathcal{G}}}{\mathcal{S}}_{w},\
{\mathcal{S}}_{w}=L_{k}U\cdot w$ (5.12)

and the strata ${\mathcal{S}}_{w}$ are called semi-infinite orbits, compare
with [FS21, Proposition VI.3.1]. Recall that there is a semi-infinite Bruhat
order $\leq^{\frac{\infty}{2}}$ on $\tilde{W}/W_{\mathcal{G}}$ defined by:

$v\leq^{\frac{\infty}{2}}w\;\;\iff\;\;\forall\;\nu_{I}\gg
0\colon\;{\nu_{I}}(\pi)\cdot v\leq{{\nu_{I}}(\pi)}\cdot w$ (5.13)

where $X_{\ast}(T)_{I}\subset\tilde{W}$, $\nu_{I}\mapsto\nu_{I}(\pi)$ is
viewed as a subgroup using the Kottwitz morphism, see (3.19), and where
$\nu_{I}\gg 0$ means that $\nu_{I}$ is sufficiently $B$-dominant. This order
was first introduced by Lusztig in [Lus80] and depends on the $F$-Borel
subgroup $B$ attracted by ${\lambda}$.

######  Proposition 5.4.

The ind-closure of ${\mathcal{S}}_{w}$ inside
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}}}$ is given by the perfect sub-ind-
scheme whose geometric points factor through some ${\mathcal{S}}_{v}$ with
$v\leq^{\frac{\infty}{2}}w$.

###### Proof.

Let ${\mathcal{I}}\to{\mathcal{G}}$ be an auxiliary Iwahori model, fixed for
the remainder of the proof. Suppose there is a curve
${\mathcal{C}}\subset{{\mathcal{F}}\\!\ell}_{{\mathcal{G}}}$ containing $v$
and whose complement ${\mathcal{C}}^{\circ}:={\mathcal{C}}\setminus\\{v\\}$ is
contained in ${\mathcal{S}}_{w}=L_{k}U\cdot w$. Now, notice that, for
sufficiently dominant $\nu_{I}$, one gets the inclusion

$\nu_{I}(\pi)\cdot{\mathcal{C}}^{\circ}\subset
L^{+}_{k}{\mathcal{I}}\cdot{\nu_{I}}(\pi)\cdot w,$ (5.14)

because conjugation by $\nu_{I}(\pi)$ moves any given perfect $k$-subscheme of
$L_{k}U$ inside the Iwahori loop group $L^{+}_{k}{\mathcal{I}}\subset
L^{+}_{k}{\mathcal{G}}$. This implies the inequality
${{\nu_{I}}(\pi)}v\leq{{\nu_{I}}(\pi)}w$, and therefore
$v\leq^{\frac{\infty}{2}}w$.

Conversely, assume that the inequality $v\leq^{\frac{\infty}{2}}w$ holds. By
definition, for all sufficiently dominant translations $\nu_{I}\in
X_{\ast}(T)_{I}$, the inequality $\nu_{I}(\pi)\cdot w\leq\nu_{I}(\pi)\cdot v$
holds in the Bruhat order. By enlarging $\nu_{I}$ if necessary, we may assume
the
${{\mathcal{F}}\\!\ell}^{\circ}_{({\mathcal{I}},{\mathcal{G}}),\nu_{I}(\pi)\cdot
w}$ is of the form
$\prod_{\alpha\in\Gamma}L_{k}^{+}{\mathcal{U}}_{\alpha}\cdot\nu_{I}(\pi)\cdot
w$ where all of the $\alpha\in\Gamma$ have positive gradient. There is a curve
${\mathcal{C}}$ in
${{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),\nu_{I}(\pi)\cdot w}$
joining $\nu_{I}(\pi)\cdot w$ to $\nu_{I}(\pi)\cdot v$ since
$\nu_{I}(\pi)\cdot v\leq\nu_{I}(\pi)\cdot w$. By our assumption on
$\nu_{I}(\pi)$, we have
${\mathcal{C}}^{\circ}\subset{\mathcal{S}}_{\nu_{I}(\pi)\cdot w}$ for
${\mathcal{C}}^{\circ}:={\mathcal{C}}\cap{{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),\nu_{I}(\pi)\cdot
w}^{\circ}$. Now, the map
$t_{\nu_{I}}\colon{{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),w}\to{{\mathcal{F}}\\!\ell}_{({\mathcal{I}},{\mathcal{G}}),\nu_{I}(\pi)\cdot
w}$ induced by left translation with $\nu_{I}(\pi)$ is an isomorphism and
hence induces ${\mathcal{S}}_{w}\cong{\mathcal{S}}_{\nu_{I}(\pi)\cdot w}$.
Then the curve $t_{\nu_{I}}^{-1}({\mathcal{C}})$ joins $w$ to $v$, and
$t_{\nu_{I}}^{-1}({\mathcal{C}}^{\circ})\subset{\mathcal{S}}_{w}$. ∎

Next, we extend the equi-dimensionality of Mirković–Vilonen cycles [MV07,
Theorem 3.2] (see also [Zhu17a, Corollary 2.8] and [FS21, Corollary VI.3.8])
from split groups to twisted groups as follows. We continue to assume that
${\lambda}$ is regular and, additionally, that ${\mathcal{G}}$ is special
parahoric. Then $X_{\ast}(T)_{I}=\tilde{W}/W_{\mathcal{G}}$ and (5.12) becomes

${{\mathcal{F}}\\!\ell}_{\mathcal{G}}=\bigcup_{\nu_{I}\in
X_{*}(T)_{I}}{\mathcal{S}}_{\nu_{I}},\
{\mathcal{S}}_{\nu_{I}}=L_{k}U\cdot{\nu_{I}}(\pi).$ (5.15)

If ${\mathcal{G}}$ is reductive, then ${{\mathcal{F}}\\!\ell}_{\mathcal{G}}$
is the Witt vector affine Grassmannian studied in [Zhu17a]. So, in general,
${{\mathcal{F}}\\!\ell}_{\mathcal{G}}$ can be regarded as a twisted version
when ${\mathcal{G}}$ is special parahoric. Indeed,
${{\mathcal{F}}\\!\ell}_{\mathcal{G}}=\operatorname{colim}{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}}$
where $\mu_{I}$ runs through $X_{*}(T)_{I,+}$, the image of the $B$-dominant
cocharacters under the projection $X_{*}(T)\rightarrow X_{*}(T)_{I}$ equipped
with the induced dominance order and
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}}:={{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}(\pi)}$,
as is usual notation for (twisted) affine Grassmannians. Also, the semi-
infinite Bruhat order on $X_{\ast}(T)_{I}$ specializes to the dominance
relation, that is, $\nu_{I}^{\prime}\leq^{\frac{\infty}{2}}\nu_{I}$ if and
only if $\nu_{I}-\nu_{I}^{\prime}$ is a sum of coinvariants of positive roots
with non-negative coefficients.

######  Lemma 5.5.

For any $\nu_{I}\in X_{*}(T)_{I}$, the intersection
${\mathcal{S}}_{\nu_{I}}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}}$ is
non-empty if and only if $\nu_{I}$ lies in the $W_{\mathcal{G}}$-orbit of some
$\mu_{I}^{\prime}\in X_{*}(T)_{I,+}$ with $\mu^{\prime}_{I}\leq\mu_{I}$. In
this case, it is affine and equidimensional of dimension
$\langle\rho_{G},\nu+\mu\rangle$.

Here $\rho_{G}\in X^{*}(T)$ denotes the half sum of the $B$-positive roots. We
note that the pairing $\langle\rho_{G},\nu+\mu\rangle$ is well-defined
independently of the choice of lifts $\nu,\mu\in X_{*}(T)$ of
$\nu_{I},\mu_{I}$ because $\rho_{G}$ is $I$-invariant.

###### Proof of Lemma 5.5.

The map
$({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}})^{+}\rightarrow{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}}$
induces an isomorphism (see Theorem 5.2)

$({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}})^{+}\overset{\cong}{\longrightarrow}\bigsqcup_{\nu_{I}\in
X_{*}(T)_{I}}{\mathcal{S}}_{\nu_{I}}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}}.$
(5.16)

Under
$({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}})^{+}\rightarrow({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}})^{0}$,
the component
${\mathcal{S}}_{\nu_{I}}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}}$
contracts to $\underline{\\{\nu_{I}\\}}={\operatorname{Spec}}(k)$. Thus, it is
affine because Bialynicki-Birula maps for schemes are affine by [Ric19,
Corollary 1.12]. Also, $({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}})^{0}$
identifies with the constant scheme associated with the subset of $\nu_{I}\in
X_{*}(T)_{I}$ lying in the $W_{\mathcal{G}}$-orbit of some
$\mu_{I}^{\prime}\in X_{*}(T)_{I,+}$ with $\mu^{\prime}_{I}\leq\mu_{I}$. So
only such $\nu_{I}$ contribute to (5.16), and as the Bialynicki-Birula map has
a section, the non-emptiness criterion for
${\mathcal{S}}_{\nu_{I}}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}}$
holds true. Furthermore, the union over all
${\mathcal{S}}_{\nu^{\prime}_{I}}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}}$
with $\nu_{I}^{\prime}\leq^{\frac{\infty}{2}}\nu_{I}$ is a closed perfect
subscheme by Proposition 5.4.

As noted in [FS21, Corollary VI.3.8], the affineness implies the dimension
formula once we show that
${\mathcal{S}}_{\mu_{I}^{\operatorname{anti}}}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\mu_{I}}$
is a point where $\mu_{I}^{\operatorname{anti}}$ is the antidominant element
in the $W_{\mathcal{G}}$-orbit of $\mu_{I}$. This follows from the proof of
Lemma 5.3. ∎

##  6\. Nearby cycles of étale sheaves

###  6.1. Recollections

In [Sch17], Scholze constructs a category of étale sheaves

${\operatorname{D}}(X,\Lambda):={\operatorname{D}}_{\textup{\'{e}t}}(X,\Lambda)$
(6.1)

for all small v-stacks $X$. As coefficients $\Lambda$, we allow prime-to-$p$
torsion rings or, by the adic formalism of [Sch17, Section 26], an
$\ell$-torsion free, complete $\ell$-adic ring for $\ell\neq p$, or a ring of
the form $\Lambda=\Lambda_{0}[\ell^{-1}]$ where $\Lambda_{0}$ is as in the
previous case. In the final case, as this is not covered in [Sch17], we define
the triangulated category

${\operatorname{D}}(X,\Lambda):={\operatorname{D}}(X,\Lambda_{0})\otimes_{\Lambda_{0}}\Lambda,$
(6.2)

in analogy to the classical definition for schemes, for example, see [KW01,
Appendix A]. The adic formalism of [Sch17, Section 26] carries over to the
categories (6.2). Finally, we also allow $\Lambda$ to be a filtered colimit of
the aforementioned rings, with the obvious definition for the categories. This
includes algebraic field extensions $L/{\mathbb{Q}}_{\ell}$ and their rings of
integers $O_{L}$.

The categories of étale sheaves are equipped with the usual six functors
formalism: the endofunctors $\otimes^{\mathbb{L}}$, $\operatorname{{{\it
R}\mathscr{H}\mkern-1.8muo\mkern-0.7mum\mkern-0.5mu}}$ and functors $Rf_{*}$,
$f^{*}$ for a morphism $f\colon X\rightarrow Y$ of small v-stacks. If $f$ is
compactifiable and representable in locally spatial diamonds with
$\operatorname{dim.\\!tr}f<\infty$, we dispose of the functors $Rf_{!}$,
$Rf^{!}$, completing the six functor formalism.

In general, the categories ${\operatorname{D}}(X,\Lambda)$ and the six
functors are rather inexplicit, constructed through v-descent using Lurie’s
$\infty$-categorical machinery. Nevertheless, whenever $f\colon X\rightarrow
Y$ is a morphism between locally spatial diamonds, then $X$ and $Y$ admit a
well-defined étale site and Scholze’s operations are very closely related to
the operations that one can construct site-theoretically, see [Sch17,
Proposition 14.15, Section 17].

When $X$ and $Y$ are locally spatial diamonds we say that an object
$A\in{\operatorname{D}}(X,\Lambda)$ is ULA (=universally locally acyclic) with
respect to $f$ if, for all locally spatial diamonds $Y^{\prime}\to Y$, the
pullback $A^{\prime}\in{\operatorname{D}}(X^{\prime},\Lambda)$ is
overconvergent along the fibers of
$f^{\prime}:X^{\prime}=X\times_{Y}Y^{\prime}\to Y^{\prime}$ and
$R(f^{\prime}\circ j^{\prime})_{!}j^{\prime}{}^{*}A$ is perfect-constructible
for all separated étale neighborhoods $j^{\prime}\colon U^{\prime}\rightarrow
X^{\prime}$ for which $f^{\prime}\circ j^{\prime}$ is quasi-compact, see
[FS21, Definition IV.2.1]. If $\Lambda$ is $\ell$-adic as above, then a
complex $A\in{\operatorname{D}}(X,\Lambda)$ is called perfect-constructible if
$A\otimes^{\mathbb{L}}_{\Lambda}\Lambda/\ell$ is étale locally perfect-
constant after passing to a constructible stratification, equivalently
$A\otimes^{\mathbb{L}}_{\Lambda}\Lambda/\ell^{n}$ are so for all $n\geq 1$.
Finally, if $\Lambda=\Lambda_{0}[\ell^{-1}]$ is as in (6.2), then an object in
${\operatorname{D}}(X,\Lambda)$ is called perfect-constructible if it admits a
$\Lambda_{0}$-lattice which is so. For $X$ and $Y$ more general v-stacks (and
$f\colon X\to Y$ representable in locally spatial diamonds), we call $A$ ULA
if it is ULA after any base change $S\to Y$ with $S$ a locally spatial
diamond.

Suppose $X$ is a small v-stack proper and representable in spatial diamonds
over a base $S$, and that $X$ is equipped with an action by
${\mathbb{G}}_{m,S}^{\diamondsuit}$ satisfying the conditions [FS21,
Hypothesis IV.6.1.]. One can consider the v-stacks

$X^{\pm}={\operatorname{Hom}}_{{\mathbb{G}}_{m}^{\diamondsuit}}\big{(}({\mathbb{A}}^{1})^{\pm,\diamondsuit},X\big{)}$
(6.3)

which (by hypothesis) are represented by a finite partition of $X$ into
locally closed subsets. This also induces a partition of the fixed-point
v-stack $X^{0}=X^{{\mathbb{G}}_{m}^{\diamondsuit}}$ into closed and open
subsets. We have inclusion maps $q^{\pm}\colon X^{\pm}\to X$ and projection
maps $p^{\pm}\colon X^{\pm}\to X^{0}$, from that we obtain the hyperbolic
localization functor

$L_{X/S}\colon{\operatorname{D}}(X/{\mathbb{G}}_{m,S}^{\diamondsuit},\Lambda)\rightarrow{\operatorname{D}}(X^{0},\Lambda),$
(6.4)

which can be expressed as $R(p^{+})_{!}(q^{+})^{*}$ or equivalently as
$R(p^{-})_{*}R(q^{-})^{!}$ by [FS21, Theorem IV.6.5]. This functor enjoys
plenty of compatibilities, in analogy to [Ric19], which we will exploit to
compute nearby cycles, see [FS21, Propositions IV.6.12, IV.6.13, IV.6.14].

###  6.2. Over $C$

We continue with the notation and denote by $F/{\mathbb{Q}}_{p}$ a complete
discretely valued field with ring of integers $O$ and perfect residue field
$k$ of characteristic $p>0$. Also, we fix a complete algebraic closure $C/F$,
and a connected reductive $F$-group $G$.

In this section, we recall the structure of the categories of monodromic
sheaves with bounded support
${\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd}}$
and
${\operatorname{D}}({\operatorname{Gr}}_{G,C},\Lambda)^{\operatorname{mon},\operatorname{bd}}$
studied in [FS21, Section VI]. As in Section 5, for any cocharacter
$\lambda\colon{\mathbb{G}}_{m}\to G_{C}$, we have the induced
${\mathbb{G}}_{m}^{\diamondsuit}$-action on ${\operatorname{Gr}}_{G,C}$, whose
attractors only depend on the attracting parabolic $P\subset G_{C}$.

In particular, hyperbolic localization gives a constant terms functor

$\operatorname{CT}_{P}\colon{\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd}}\rightarrow{\operatorname{D}}({\operatorname{Gr}}_{G,C},\Lambda)^{\operatorname{mon},\operatorname{bd}}\xrightarrow{L_{{\operatorname{Gr}}_{G}/_{C}}}{\operatorname{D}}({\operatorname{Gr}}_{M,C},\Lambda)$
(6.5)

providing the main tool to effectively study the category of derived étale
sheaves on ${\operatorname{Hk}}_{G,C}$ as in [FS21, Corollary VI.3.5]. One of
the crucial techniques is the following conservativity lemma [FS21,
Proposition VI.4.2] whose proof we sketch for convenience.

######  Lemma 6.1.

Let $T\subset B\subset G_{C}$ be an arbitrary maximal torus and a Borel
containing it. Then
$A\in{\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd}}$
vanishes if and only if
$\operatorname{CT}_{B}(A)\in{\operatorname{D}}({\operatorname{Gr}}_{T,C},\Lambda)$
does.

###### Proof.

The proof is done by considering a maximal strata where $A$ is concentrated.
This strata is of the form $[\operatorname{{Spd}}C/(L^{+}_{C}G)_{\mu}]$ for
the stabilizer of $\mu\in X_{*}(T)_{+}$. The attractor of
${\operatorname{Gr}}_{G,C,\mu}$ at the anti-dominant coweight $-\mu$ with
respect to $B$ is an isolated point. Using the
$R(p^{+})_{!}(q^{+})^{*}$-version of hyperbolic localization, we see that the
fiber of $\operatorname{CT}_{B}$ over $\mu\in{\operatorname{Gr}}_{T}$ agrees
with pullback to this point. ∎

This allows us to localize several properties of derived objects in
${\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd}}$.
For instance, $A$ is ULA if and only if $\operatorname{CT}_{B}(A)$ which, in
turn, is equivalent to $[\mu]^{*}A$ being a perfect object for all maps
$[\mu]\colon\operatorname{{Spd}}C\to{\operatorname{Hk}}_{G}$ with $\mu\in
X_{*}(T)$, see [FS21, Propositions VI.6.4, VI.6.5].

Next, we move to the natural perverse t-structure on
${\operatorname{D}}({\operatorname{Hk}}_{G,C,\mu},\Lambda)^{\operatorname{bd}}$,
see [FS21, Definition/Proposition VI.7.1]. This is given in terms of the
following subcategory

${}^{p}{\operatorname{D}}^{\leq
0}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd}}=\\{A\in{\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd}}\colon
j_{\mu}^{*}A\in{\operatorname{D}}^{\leq-\langle 2\rho,\mu\rangle}\\},$ (6.6)

which determines ${}^{p}{\operatorname{D}}^{\geq
0}({\operatorname{Hk}}_{G,C},\Lambda)$ uniquely. Intersecting these two, we
get the category of perverse sheaves
${\operatorname{Perv}}({\operatorname{Hk}}_{G,C},\Lambda)$.

Thanks to [FS21, Proposition VI.7.4], the t-structure is preserved under and
detected by $\operatorname{CT}_{B}[\deg_{B}]$. Here, for any $C$-parabolic
$B\subset P$ with Levi quotient $M$, the degree is the locally constant
function on ${\operatorname{Gr}}_{P,C}$ induced by

$\deg_{P}(\lambda)=\langle 2\rho_{G}-2\rho_{M},\lambda\rangle,$ (6.7)

where $\lambda\in X_{*}(T)$ is a coweight and $\rho_{M}$ is the half-sum of
all $B$-positive $M$-roots. The main geometric fact used in the proof of
[FS21, Proposition VI.7.4] is the equidimensionality of semi-infinite orbits.

When working with torsion coefficients, it is convenient to single out flat
perverse sheaves, which are those objects $A$ such that for every
$\Lambda$-module $M$ the complex $A\otimes_{\Lambda}^{\mathbb{L}}M$ is
perverse.

######  Definition 6.2.

The Satake category ${\operatorname{Sat}}({\operatorname{Hk}}_{G,C},\Lambda)$
is the full subcategory of flat ULA objects in
${\operatorname{Perv}}({\operatorname{Hk}}_{G,C},\Lambda)$.

The Satake category is endowed with a monoidal product

$\star\colon{\operatorname{Sat}}({\operatorname{Hk}}_{G,C},\Lambda)\times{\operatorname{Sat}}({\operatorname{Hk}}_{G,C},\Lambda)\rightarrow{\operatorname{Sat}}({\operatorname{Hk}}_{G,C},\Lambda)$
(6.8)

arising from the convolution Hecke stack
${\operatorname{Hk}}_{G,C}\widetilde{\times}{\operatorname{Hk}}_{G,C}$, see
[FS21, Proposition VI.8.1]. Due to the fusion interpretation [FS21,
Definition/Proposition VI.9.4], the monoidal structure is naturally symmetric
monoidal.

Taking cohomology of the affine Grassmannian furnishes a fiber functor

$F\colon{\operatorname{Sat}}({\operatorname{Hk}}_{G,C},\Lambda)\rightarrow{\operatorname{Rep}}(\Lambda)$
(6.9)

to the category of $\Lambda$-finite locally free modules and the Tannakian
formalism gives us an interpretation of these categories in terms of a group
of automorphisms.

######  Theorem 6.3 (Fargues–Scholze).

The automorphism group of $F$ is naturally isomorphic to the Langlands dual
group $\widehat{G}_{\Lambda}$ formed over $\Lambda$.

One may regard the dual group $\widehat{G}_{\Lambda}$ as combinatorially
defined in terms of root data (a priori), or as its correct natural definition
(a posteriori). The cyclotomic twist in [FS21, Theorem VI.11.1] is not needed,
because we are working over the algebraically closed field $C$.

###  6.3. Over $\bar{k}$

Let ${\mathcal{G}}$ be a parahoric $O$-model of $G$. In this section, we look
at what happens with the geometric special fiber Hecke v-stack
${\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}=L^{+}_{\bar{k}}{\mathcal{G}}^{\diamondsuit}\backslash
L_{\bar{k}}G^{\diamondsuit}/L^{+}_{\bar{k}}{\mathcal{G}}^{\diamondsuit}$. We
note that the categories of étale sheaves compare well to their scheme-
theoretic companions, see Proposition A.5.

We continue with the notation and, in addition, fix a maximal
$\breve{F}$-split $F$-torus $S\subset G$ containing a maximal $F$-split torus
(see [BT84, Proposition 5.1.10]) with centralizer $T$, which is a maximal
$F$-torus inside $G$, such that their connected Néron $O$-models
${\mathcal{S}}\subset{\mathcal{T}}$ embed into ${\mathcal{G}}$. Each parabolic
subgroup $P\subset G_{\breve{F}}$ with Levi $M$ containing $S_{\breve{F}}$
extends to a diagram of $O$-group schemes
${\mathcal{M}}\leftarrow{\mathcal{P}}\rightarrow{\mathcal{G}}_{\breve{O}}$ by
taking flat closures. Again, choosing a cocharacter
${\lambda}\colon{\mathbb{G}}_{m}\to{\mathcal{S}}_{\breve{O}}\subset{\mathcal{G}}_{\breve{O}}$
with $M=M_{\lambda}$ and $P=P^{+}_{\lambda}$, the formalism of Section 5
applies to define ${\mathbb{G}}_{m}^{\diamondsuit}$-actions on the pro-smooth
cover
${\operatorname{Gr}}_{{\mathcal{G}},\bar{k}}={{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}^{\diamondsuit}$.
This gives rise to constant term functors

$\operatorname{CT}_{\mathcal{P}}\colon{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd}}\rightarrow{\operatorname{D}}({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}^{0,\diamondsuit},\Lambda)^{\operatorname{bd}},$
(6.10)

not depending on the choice of ${\lambda}$ such that $M=M_{\lambda}$ and
$P=P^{+}_{\lambda}$. Here, the fixed points
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}^{0}$ contain
${{\mathcal{F}}\\!\ell}_{{\mathcal{M}},\bar{k}}$ as an open and closed sub-
ind-scheme by Theorem 5.2, but are strictly bigger unless
${\mathcal{G}}_{\breve{O}}$ is special parahoric.

As in the previous section, we analyse the key properties ULA, flatness and
perversity. The crucial step is the following conservativity result.

######  Proposition 6.4.

An object
$A\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd}}$
vanishes if and only if $\operatorname{CT}_{\mathcal{B}}(A)$ does for every
$\breve{F}$-Borel $S_{\breve{F}}\subset B\subset G_{\breve{F}}$.

###### Proof.

Just like in Lemma 6.1, we argue on a maximal strata of
${\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}$ where $A$ does not vanish, say
one indexed by some $w$. In this case, by Lemma 5.3 there is a choice of
$\breve{F}$-Borel $B$ for which the associated attractor intersects
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},w}$ in an isolated point. In
this case, $\operatorname{CT}_{\mathcal{B}}$ agrees with pullback to this
point. ∎

######  Definition 6.5.

An object
$A\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd}}$
is ULA whenever its pullback to
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},w}^{\diamondsuit}$ is ULA over
$\operatorname{{Spd}}\bar{k}$.

A priori, this notion depends on the choice of left or right trivialization,
but it follows a posteriori from Proposition 6.7 that it does not, see [FS21,
Proposition VI.6.2]. The ULA property interacts very well with constant terms:

######  Proposition 6.6.

If
$A\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd}}$
is ULA, then so is $\operatorname{CT}_{\mathcal{P}}(A)$. Conversely, if
$\operatorname{CT}_{\mathcal{B}}(A)$ is ULA for all Borel subgroups
$S_{\breve{F}}\subset B\subset G_{\breve{F}}$, then so is $A$.

###### Proof.

Abusing notation, we also call $A$ the pullback of this object to
${{\mathcal{F}}\\!\ell}^{\diamondsuit}_{{\mathcal{G}},\bar{k}}$. By [FS21,
Theorem IV.2.23], to prove that for $B=A$ or
$B=\operatorname{CT}_{\mathcal{P}}(A)$, the object $B$ is ULA it is enough to
show that

$p_{1}^{*}{\mathbb{D}}(B)\otimes p_{2}^{*}B\rightarrow\operatorname{{{\it
R}\mathscr{H}\mkern-1.8muo\mkern-0.7mum\mkern-0.5mu}}(p_{1}^{*}B,Rp_{2}^{!}B)$
(6.11)

is an isomorphism. Now, for any pair of flat closures of parabolics
${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$ a direct computation (using
properties of hyperbolic localization, cf. the proof of [FS21, Proposition
VI.6.4]) shows that

$\operatorname{CT}_{{\mathcal{P}}_{1}\times{\mathcal{P}}_{2}}(p_{1}^{*}{\mathbb{D}}(A)\otimes
p_{2}^{*}A)=p_{1}^{*}{\mathbb{D}}(\operatorname{CT}_{{{\mathcal{P}}^{-}_{1}}}(A))\otimes
p_{2}^{*}\operatorname{CT}_{{\mathcal{P}}_{2}}(A)$ (6.12)

and that

$\operatorname{CT}_{{\mathcal{P}}_{1}\times{\mathcal{P}}_{2}}(\operatorname{{{\it
R}\mathscr{H}\mkern-1.8muo\mkern-0.7mum\mkern-0.5mu}}(p_{1}^{*}A,Rp_{2}^{!}A))=\operatorname{{{\it
R}\mathscr{H}\mkern-1.8muo\mkern-0.7mum\mkern-0.5mu}}(p_{1}^{*}(\operatorname{CT}_{{\mathcal{P}}^{-}_{1}}(A),Rp_{2}^{!}\operatorname{CT}_{{\mathcal{P}}_{2}}(A)))$
(6.13)

where ${\mathcal{P}}_{1}^{-}$ is opposite to ${\mathcal{P}}_{1}$. In the
forward direction, it is enough to use this for
${\mathcal{P}}_{1}={\mathcal{P}}^{-}$ and ${\mathcal{P}}_{2}={\mathcal{P}}$.
For the converse, we let $K$ denote the cone of Equation 6.11. By the
conservativity of Proposition 6.4, it is enough to prove
$\operatorname{CT}_{{\mathcal{B}}_{1},{\mathcal{B}}_{2}}(K)=0$ for all
${\mathcal{B}}_{1}$, ${\mathcal{B}}_{2}$ since these exhaust the Borel
subgroups of of $G_{\breve{F}}\times G_{\breve{F}}$. But this follows from the
computation above, [FS21, Proposition IV.2.19] and the hypothesis that
$\operatorname{CT}_{{\mathcal{B}}_{1}}(A)$ is ULA. ∎

We prove that ULA sheaves admit an easy description in terms of restrictions
to Schubert strata:

######  Proposition 6.7.

The following are equivalent for an object
$A\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd}}$:

  1. (1)

$A$ is ULA.

  2. (2)

For all strata of $A$ pullback along
$[w]\colon\operatorname{{Spd}}\bar{k}\rightarrow{\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}$
is a perfect complex171717If $\Lambda=\Lambda_{0}[\ell^{-1}]$ as in (6.2),
then we require the pullback to arise as the $\ell$ localization of a perfect
complex over $\Lambda_{0}$. in

${\operatorname{D}}(\operatorname{{Spd}}\bar{k},\Lambda)={\operatorname{D}}(\Lambda).$
(6.14)

  3. (3)

The pullback to
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}^{\diamondsuit}$ lies in

${\operatorname{D}}_{\operatorname{cons}}({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd}}\subset{\operatorname{D}}({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}^{\diamondsuit},\Lambda)^{\operatorname{ula},\operatorname{bd}},$
(6.15)

where
${\operatorname{D}}_{\operatorname{cons}}({{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd}}$
is the category of perfect-constructible $\Lambda$-sheaves with bounded
support on the ind-scheme ${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}$ and
the inclusion is the one constructed in [Sch17, Section 27].

###### Proof.

The equivalence of (2) and (3) follows from Proposition A.5, and the fact that
the equivalence in Proposition A.5 is compatible with pullback to
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}}}$, respectively to
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}}}^{\diamondsuit}$ or to strata.

Let $j_{w}$ denote the inclusion of the strata in
${\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}$ corresponding to $w$. For
proving that (2) implies (1) one can, as in [FS21, Proposition VI.6.5], reduce
to showing that $R(j_{w})_{!}\Lambda$ is ULA (for each $w$). But their
pullback to ${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}$ are clearly
algebraic and by Proposition A.1 they are also ULA, see also [FS21,
Proposition IV.2.30]. Alternatively one can use the Demazure resolution,
compare with [FS21, Proposition VI.5.7].

For the converse implication, we induct on the number of strata where $A$ does
not vanish and consider the cone of $R(j_{w})_{!}j^{\ast}_{w}A\to A$ for a
maximal strata $w$. Indeed, pullback by open immersion preserves being ULA by
[FS21, Proposition VI.2.13.(i)]. Then we apply [FS21, Proposition VI.4.1] to
see that
$j^{\ast}_{w}A\in{\operatorname{D}}(\mathrm{Hk}_{{\mathcal{G}},\bar{k},\mu},\Lambda)$
has a perfect stalk, and use that $R(j_{w})_{!}(j_{w}^{\ast}A)$ is ULA, by the
proven (2) implies (1). Then we can conclude by induction.

∎

Arguing as in [FS21, Definition/Proposition VI.7.1], we can define a perverse
t-structure.

######  Definition 6.8.

The perverse t-structure on ${\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}$ is
the only such that

${}^{p}{\operatorname{D}}^{\leq
0}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)=\\{A\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)\colon
j_{w}^{*}A\in{\operatorname{D}}^{\leq-l(w)}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)\\},$
(6.16)

respectively

${}^{p}{\operatorname{D}}^{\geq
0}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)=\\{A\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)\colon
Rj_{w}^{!}A\in{\operatorname{D}}^{\geq-l(w)}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)\\}.$
(6.17)

Perverse sheaves

${\operatorname{Perv}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)=\,^{p}{\operatorname{D}}^{\leq
0}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)\cap\,^{p}{\operatorname{D}}^{\geq
0}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)$ (6.18)

are the heart of the t-structure. Such an $A$ is flat perverse if in addition
$A\otimes^{\mathbb{L}}_{\Lambda}M$ is in
${\operatorname{Perv}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)$
for all $\Lambda$-modules $M$.

We note that, in general, there cannot be any degree shifts such that
$\operatorname{CT}_{\mathcal{P}}[\deg_{\mathcal{P}}]$ preserves the perverse
t-structure, due to lack of parity. But, we define

$\deg_{\mathcal{P}}({\lambda}_{I})=\langle
2\rho_{G}-2\rho_{M},{\lambda}\rangle$ (6.19)

for translation elements ${\lambda}_{I}\in X_{*}(T)_{I}$. This is useful for
the following result:

######  Proposition 6.9.

Assume that ${\mathcal{G}}_{\breve{O}}$ is special parahoric, and let
$A\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)$.
Then $A$ is perverse if and only if
$\operatorname{CT}_{\mathcal{B}}(A)[\deg_{\mathcal{B}}]\in{\operatorname{Perv}}({{\mathcal{F}}\\!\ell}_{{\mathcal{T}},\bar{k}}^{\diamondsuit},\Lambda)$
for all Borel subgroups $S_{\breve{F}}\subset B\subset G_{\breve{F}}$. The
same applies to the flat objects.

###### Proof.

It suffices to follow the proof of [FS21, Proposition VI.7.4]. For preserving
the t-structure, we use the fact that the non-empty intersections
${\mathcal{S}}_{\bar{k},{\lambda}_{I}}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},\nu_{I}}$
are equidimensional of dimension $\langle 2\rho_{G},\lambda+\nu\rangle$, see
Lemma 5.5. The converse then follows from Proposition 6.4. ∎

In particular, it is now permitted to introduce the Satake category at special
level. Notice that there is no hope of such a well-behaved class of objects to
exist at arbitrary level, because the quotient $\tilde{W}/W_{\mathcal{G}}$
carries no natural abelian structure.

######  Definition 6.10.

Let ${\mathcal{G}}_{\breve{O}}$ be special parahoric. Then the Satake category
${\operatorname{Sat}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)$ is
the full subcategory of
${\operatorname{Perv}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)$
comprised of flat ULA objects.

This category lies within the category of perverse sheaves
${\operatorname{Perv}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}^{\operatorname{sch}},\Lambda)$
on the schematic Hecke stack
${\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}^{\operatorname{sch}}=L^{+}_{\bar{k}}{\mathcal{G}}\backslash
L_{\bar{k}}G/L_{\bar{k}}^{+}{\mathcal{G}}$ by Proposition 6.7 and Appendix A
for the comparison with sheaves on schematic v-stacks. It carries moreover a
monoidal structure given by convolution $\star$.

###  6.4. Over $O_{C}$

Let $f\colon X\to{\operatorname{Spec}}(O_{C})$ be a scheme of finite
presentation over $O_{C}$ and denote by $j$ the inclusion
$X_{\eta}\hookrightarrow X$ of the generic fiber. In [HS21, Theorem 1.7],
Hansen and Scholze prove that the pullback functor

$j^{*}\colon{\operatorname{D}}(X,\Lambda)\to{\operatorname{D}}(X_{\eta},\Lambda)$
(6.20)

restricts to an equivalence between $f$-ULA and $f_{\eta}$-ULA objects. In the
setup of diamonds, the argument for full faithfulness is the same as was
explained to us by Scholze, and it consists of proving the adjunction map
$A\to Rj_{*}j^{*}A$ is an isomorphism.

######  Lemma 6.11.

Let $X$ be a small v-sheaf over $\operatorname{{Spd}}(O_{C})$ representable in
locally spatial diamonds, compactifiable and of finite trascendence degree.
Let $A\in{\operatorname{D}}(X,\Lambda)$ be ULA for the structure map to
$\operatorname{{Spd}}(O_{C})$. Then $A\to Rj^{\prime}_{\ast}j^{\prime\ast}A$
is a isomorphism, where $j^{\prime}:X_{\eta}\to X$ and
$j:\operatorname{{Spd}}(C)\to\operatorname{{Spd}}(O_{C})$ denote the inclusion
of generic fibers.

###### Proof.

By hypothesis $j^{\prime\ast}A$ is ULA with respect to
$\operatorname{{Spd}}(C)$. In particular, by [FS21, Proposition IV.2.19] the
map

$j^{\prime\ast}A\otimes_{\Lambda}^{\mathbb{L}}\Lambda\cong\operatorname{{{\it
R}\mathscr{H}\mkern-1.8muo\mkern-0.7mum\mkern-0.5mu}}(\mathbb{D}_{X_{\eta}/\operatorname{{Spd}}(C)}(j^{\prime\ast}A),Rf_{\eta}^{!}\Lambda)$
(6.21)

is an isomorphism. Since $j^{\prime}$ is an open immersion
$j^{\prime\ast}=Rj^{\prime!}$ and
$\mathbb{D}_{X_{\eta}/\operatorname{{Spd}}(C)}(j^{\prime\ast}A)=j^{\prime\ast}\mathbb{D}_{X/\operatorname{{Spd}}(O_{C})}(A)$
as follows from [Sch17, Theorem 1.8.(v)]. We get

$\displaystyle Rj^{\prime}_{\ast}j^{\prime\ast}A$ $\displaystyle\cong$
$\displaystyle Rj^{\prime}_{\ast}\operatorname{{{\it
R}\mathscr{H}\mkern-1.8muo\mkern-0.7mum\mkern-0.5mu}}(j^{\prime\ast}\mathbb{D}_{X/\operatorname{{Spd}}(O_{C})}(A),Rf_{\eta}^{!}\Lambda)$
(6.22) $\displaystyle\cong$ $\displaystyle\operatorname{{{\it
R}\mathscr{H}\mkern-1.8muo\mkern-0.7mum\mkern-0.5mu}}(\mathbb{D}_{X/\operatorname{{Spd}}(O_{C})}(A),Rj^{\prime}_{\ast}Rf_{\eta}^{!}\Lambda)$
$\displaystyle\cong$ $\displaystyle\operatorname{{{\it
R}\mathscr{H}\mkern-1.8muo\mkern-0.7mum\mkern-0.5mu}}(\mathbb{D}_{X/\operatorname{{Spd}}(O_{C})}(A),Rf^{!}Rj_{\ast}\Lambda)$

the result now follows from the identity $\Lambda=Rj_{\ast}\Lambda$ and double
duality for ULA sheaves. ∎

In particular, a $f_{\eta}$-ULA object $A$ comes from a $f$-ULA object if and
only if $Rj_{*}j^{\ast}A$ is $f$-ULA.

Below, we prove essential surjectivity for
${\operatorname{Hk}}_{{\mathcal{G}},O_{C}}$, the Hecke stack over
$\operatorname{{Spd}}O_{C}$. For hyperspecial parahoric ${\mathcal{G}}$, this
is [FS21, Corollary VI.6.7]. Before doing this, recall that hyperbolic
localization allows us to define again a constant term functors

$\operatorname{CT}_{\mathcal{P}}\colon{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},O_{C}},\Lambda)^{\operatorname{bd}}\rightarrow{\operatorname{D}}({\operatorname{Gr}}_{{\mathcal{G}},O_{C}}^{0},\Lambda)^{\operatorname{bd}}.$
(6.23)

By [FS21, Proposition IV.6.12], there is a natural equivalence

$\operatorname{CT}_{\mathcal{P}}\circ Rj_{{\mathcal{G}},*}\cong
Rj_{{\mathcal{M}},*}\circ\operatorname{CT}_{P},$ (6.24)

with $j_{\mathcal{G}},j_{\mathcal{M}}$ denoting the inclusion of the
respective generic fibers. Now, we can probe integral ULA objects.

######  Proposition 6.12.

Consider the inclusion of Hecke stacks
$j\colon{\operatorname{Hk}}_{G,C}\to{\operatorname{Hk}}_{{\mathcal{G}},O_{C}}$.
There is an equivalence

$j^{*}\colon{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},O_{C}},\Lambda)^{\operatorname{bd},\operatorname{ula}}\to{\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd},\operatorname{ula}},$
(6.25)

whose inverse functor is $Rj_{*}$.

###### Proof.

Suppose
$A\in{\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd},\operatorname{ula}}$,
it suffices to prove
$Rj_{*}A\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},O_{C}},\Lambda)^{\operatorname{bd},\operatorname{ula}}$.
Let $B$ denote the pullback of $A$ to
${\operatorname{Gr}}_{{\mathcal{G}},O_{C}}$, which by definition is ULA. By
smooth base change, $Rj_{*}A$ pulls back to $Rj_{*}B$ (here we implicitly use
[FS21, Proposition VI.4.1]). By [FS21, Theorem IV.2.23], we must show

$p_{1}^{*}{\mathbb{D}}(Rj_{*}B)\otimes
p_{2}^{*}Rj_{*}B\rightarrow\operatorname{{{\it
R}\mathscr{H}\mkern-1.8muo\mkern-0.7mum\mkern-0.5mu}}(p_{1}^{*}Rj_{*}B,Rp_{2}^{!}Rj_{*}B)$
(6.26)

is an isomorphism. Let $K$ denote the cone of this map. By assumption, and
since $j^{*}=Rj^{!}$, this map is an isomorphism on the generic fiber.
Consequently, $K=i_{*}L$ for some $L\in
D({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd}}$
and the inclusion
$i\colon{\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}\to{\operatorname{Hk}}_{{\mathcal{G}},O_{C}}$.
We may use the conservativity result Proposition 6.4 to prove $L=0$. This
reduces us to proving that
$\operatorname{CT}_{\mathcal{B}}(Rj_{*}A)=Rj_{*}\operatorname{CT}_{\mathcal{B}}(A)$
is ULA for all Borel subgroups $S_{\breve{F}}\subset B\subset G_{\breve{F}}$.
Now, the fixed-point locus ${\operatorname{Gr}}_{{\mathcal{G}},O_{C}}^{0}$ of
the action induced by ${\mathcal{B}}$ is ind-representable by a locally finite
type scheme over $O_{C}$ of relative dimension $0$. We call this scheme $X$
and let $h\colon X_{\eta}\to X$ the inclusion of generic fibers. By
inspection,
${\operatorname{D}}({\operatorname{Gr}}_{{\mathcal{G}},O_{C}}^{0},\Lambda)\cong{\operatorname{D}}(X,\Lambda)$
and
${\operatorname{D}}({\operatorname{Gr}}_{{\mathcal{G}},C}^{0},\Lambda)\cong{\operatorname{D}}(X_{\eta},\Lambda)$.
In particular $Rj_{*}\cong c_{X}^{*}Rh_{*}Rc_{X_{\eta},*}$ with notation as in
[Sch17, Section 27]. By [HS21, Theorem 1.7], $Rh_{*}$ preserves ULA objects,
which allows us to conclude the same holds for $Rj_{*}$. ∎

###  6.5. Nearby cycles

We can now look at the nearby cycles functor

$\Psi_{{\mathcal{G}}}:=i^{*}Rj_{*}\colon{\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd}}\rightarrow{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd}},$
(6.27)

Arising from the diagram

${\operatorname{Hk}}_{G,C}\xrightarrow{j}{\operatorname{Hk}}_{{\mathcal{G}},O_{C}}\xleftarrow{i}{\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}$
(6.28)

of geometric fibers inclusions of the integral Hecke stack.

######  Proposition 6.13.

The functor of nearby cycles lies in a natural equivalence

$\operatorname{CT}_{\mathcal{P}}[\deg_{\mathcal{P}}]\circ\Psi_{{\mathcal{G}}}\cong\Psi_{{\mathcal{M}}}\circ\operatorname{CT}_{P}[\deg_{P}],$
(6.29)

that is, it commutes with shifted constant term functors.

###### Proof.

Without the shift, this is a direct consequence of [FS21, Proposition
IV.6.12]. Using Theorem 5.2, this also shows that
$\operatorname{CT}_{\mathcal{P}}\circ\Psi_{{\mathcal{G}}}$ is supported on the
open and closed sub-v-sheaf
${{\mathcal{F}}\\!\ell}_{{\mathcal{M}},\bar{k}}\subset{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k}}^{0}$.
So the shifts agree by definition. ∎

Surprisingly, this commutativity property delivers us a lot of control on the
values assumed by $\Psi_{\mathcal{G}}$ on the Satake category.

######  Corollary 6.14.

Nearby cycles $\Psi_{\mathcal{G}}$ restrict to a functor

${\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)^{\operatorname{bd},\operatorname{ula}}\rightarrow{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd},\operatorname{ula}}$
(6.30)

and, if ${\mathcal{G}}$ is furthermore special parahoric, then it even
restricts to

${\operatorname{Sat}}({\operatorname{Hk}}_{G,C},\Lambda)\rightarrow{\operatorname{Sat}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)$
(6.31)

between the Satake categories.

###### Proof.

This follows from Proposition 6.6 and Proposition 6.9. ∎

Let us examine the nearby cycles $\Psi_{\mathcal{G}}({\operatorname{Sat}}(V))$
applied to a Satake object
${\operatorname{Sat}}(V)\in{\operatorname{Sat}}({\operatorname{Hk}}_{G,C},\Lambda)$
corresponding to a $\widehat{G}_{\Lambda}$-representation $V$ with $\mu$ as
its highest weight. Given an $F$-Borel $B\subset G$, the commutativity of
Proposition 6.13 yields

$\operatorname{CT}_{\mathcal{B}}[\deg_{\mathcal{B}}]\big{(}\Psi_{\mathcal{G}}({\operatorname{Sat}}(V))\big{)}=\bigoplus_{\lambda_{I}}V(\lambda_{I})\cdot{\lambda}_{I},$
(6.32)

where now the $\widehat{G}_{\Lambda}$-representation is regarded as a
$\widehat{T}^{I}_{\Lambda}$-representation by restriction. Here, we use that
(by construction) the constant term functor corresponds via geometric Satake
to restriction of representations, see [FS21, Section VI.11]. In particular,
we get:

######  Corollary 6.15.

For a $\widehat{G}_{\Lambda}$-representation $V$ with highest weight $\mu$,
the compactly supported cohomology groups

$\operatorname{H}^{l}_{c}\\!\big{(}{\mathcal{S}}_{\bar{k},w}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},w},\Psi_{\mathcal{G}}({\operatorname{Sat}}(V))\big{)}$
(6.33)

vanish for all $l\in{\mathbb{Z}}$ unless
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},w}\subset{\mathcal{A}}_{{\mathcal{G}},\bar{k},\mu}$.

###### Proof.

This follows from Lemma 3.12. ∎

We are now finally ready to compute the special fiber of the local model.

######  Theorem 6.16.

There is an equality
${\mathcal{A}}_{{\mathcal{G}},\mu}^{\diamondsuit}={\mathcal{M}}_{{\mathcal{G}},\mu,k_{E}}$
as sub-v-sheaves of
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},k_{E}}^{\diamondsuit}$.

###### Proof.

By specializing the orbit of $\mu$ under the finite Weyl group, it is easy to
see that ${\mathcal{A}}_{{\mathcal{G}},\mu}^{\diamondsuit}$ is contained in
the special fiber of ${\mathcal{M}}_{{\mathcal{G}},\mu}$. By Corollary 6.15,
it is thus enough to prove that for a maximal stratum in
${\mathcal{M}}_{{\mathcal{G}},\bar{k},\mu}$ enumerated by $w$, we have
$\operatorname{H}^{l}_{c}({\mathcal{S}}_{\bar{k},w}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},w},A)\neq
0$ for some $l\in{\mathbb{Z}}$ and for
$A:=\Psi_{\mathcal{G}}({\operatorname{Sat}}(V))$ for some
$\widehat{G}_{\Lambda}$-representation $V$ with highest weight $\mu$. Using
induction on $\mu$, we may and do assume that $w$ lies in the open complement
of the closed union of ${\mathcal{M}}_{{\mathcal{G}},O_{C},{\lambda}}$ for all
${\lambda}<\mu$. Indeed, if ${\lambda}<\mu$, then
${\mathcal{A}}_{{\mathcal{G}},{\lambda}}^{\diamondsuit}\subset{\mathcal{A}}_{{\mathcal{G}},\mu}^{\diamondsuit}$.
By Lemma 5.3, our Borel subgroup $S_{\breve{F}}\subset B\subset G_{\breve{F}}$
can always be chosen such that $w$ is an isolated point of the attractor
${{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},w}^{+}$ so that
$\operatorname{H}^{*}_{c}({\mathcal{S}}_{\bar{k},w}\cap{{\mathcal{F}}\\!\ell}_{{\mathcal{G}},\bar{k},w},A)=\operatorname{H}^{*}(\\{w\\},A)=:A_{w}$
is the stalk of $A$ at $w$.

Consider
$X={\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}\times_{\operatorname{{Spd}}O_{C}}U$
where $U$ denotes the analytic locus of the open unit ball
$\mathbb{D}_{O_{C}}^{\diamondsuit}$. Let $g\colon X_{C}\hookrightarrow X$ be
the inclusion of the generic fiber. Let $K$ denote a completed algebraic
closure of $k(({t}))$. We may choose a $\operatorname{{Spd}}(K)$-valued point
$\overline{w}$ of $X$ that lies over $w$. It suffices to prove
$A_{\overline{w}}$ is not identically $0$. Since $U$ is smooth over
$\operatorname{{Spd}}O_{C}$, the smooth base-change theorem and our inductive
assumption on $w$ allows us to compute

$A_{\overline{w}}=(Rg_{*}\Lambda[\langle 2\rho,\mu\rangle])_{\overline{w}},$
(6.34)

where $V$ is chosen to have weight multiplicity $1$ at $\mu$. Since $X_{C}$,
$X$ and $K^{\diamondsuit}$ are locally spatial diamonds, we may compute the
right-side term site-theoretically. Letting $l:=-\langle 2\rho,\mu\rangle$, we
have

$\operatorname{H}^{l}(A_{\overline{w}})=\operatorname{lim}_{W}\operatorname{H}^{0}(W_{C},\Lambda)$
(6.35)

where $W$ ranges over étale neighborhoods of $\overline{w}$ in $X$. By Remark
4.15 and openness of $X\to{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}$, the
generic fiber $X_{C}$ is dense in $X$ which proves that the above expression
does not vanish. ∎

###  6.6. Centrality of nearby cycles

In the classical theory, say, over function fields, it is known that nearby
cycles on Hecke stacks give central perverse sheaves on partial affine flag
varieties, see [Gai01]. Centrality holds true in our context as well:

######  Proposition 6.17.

For every $A\in{\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)$ and
$B\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)$
with bounded support, there is a canonical isomorphism

$\Psi_{\mathcal{G}}(A)\star B\cong B\star\Psi_{\mathcal{G}}(A)$ (6.36)

in ${\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)$.

###### Proof.

We can repeat the proof of [Zhu14, Proposition 7.4] in our context:

Similar to [FS21, Definition/Proposition VI.9.4], we work with the convolution
integral Hecke stack

${\operatorname{Hk}}_{{\mathcal{G}}}^{I;I_{1},\ldots,I_{k}}\to(\operatorname{{Spd}}O)^{I},$
(6.37)

where $I=I_{1}\sqcup\ldots\sqcup I_{k}$ is a finite partitioned index set. It
parametrizes ${\mathcal{G}}$-bundles
${\mathcal{E}}_{0},\ldots,{\mathcal{E}}_{k}$ over $B^{+}_{\mathrm{dR}}$
together with isomorphisms of ${\mathcal{E}}_{j-1}$ and ${\mathcal{E}}_{j}$
outside the reunion of the divisors $\xi_{i}$ for all $i\in I_{j}$. We fix
$I:=\\{1,2\\}$ and drop it from the notation. There are three ordered
partitions $\\{1\\}\sqcup\\{2\\}$, $\\{2\\}\sqcup\\{1\\}$ and $\\{1,2\\}$,
leading to the diagram of v-sheaves:

${{\operatorname{Hk}}_{{\mathcal{G}}}^{\\{1\\},\\{2\\}}|_{\operatorname{{Spd}}O_{C}}}$${{\operatorname{Hk}}_{{\mathcal{G}}}^{\\{1,2\\}}|_{\operatorname{{Spd}}O_{C}}}$${{\operatorname{Hk}}_{{\mathcal{G}}}^{\\{2\\},\\{1\\}}|_{\operatorname{{Spd}}O_{C}}}$${{\operatorname{Hk}}_{{\mathcal{G}},O_{C}}\times{\operatorname{Hk}}_{{\mathcal{G}},\bar{k}}}$$m$$n$$p$
$q$ (6.38)

The diagram arises by base change along the map
$\operatorname{{Spd}}O_{C}\to(\operatorname{{Spd}}O)^{2}$ induced by the
divisor $\pi=0$ in the second coordinate. The maps $m,n$ are the natural
projections given by remembering ${\mathcal{E}}_{0}$ and ${\mathcal{E}}_{2}$,
and are ind-proper, as one sees by pulling back to the convolution affine
Grassmannian, combine Theorem 4.9 with the proof of [SW20, Proposition
20.4.1]. The maps $p,q$ are given by sending
$({\mathcal{E}}_{0},{\mathcal{E}}_{1},{\mathcal{E}}_{2})$ to the ordered pair
$(({\mathcal{E}}_{0},{\mathcal{E}}_{1}),({\mathcal{E}}_{1},{\mathcal{E}}_{2}))$,
respectively
$(({\mathcal{E}}_{1},{\mathcal{E}}_{2}),({\mathcal{E}}_{0},{\mathcal{E}}_{1}))$,
and are pro-(cohomologically smooth) because
$L^{+}_{O}{\mathcal{G}}\to\operatorname{{Spd}}O$ is so. More precisely, one
passes to a bounded part and factors the action of $L^{+}_{O}{\mathcal{G}}$
through a congruence quotient.

Furthermore, the maps $m,n$ are convolution maps in the special fiber and
induce isomorphisms over the generic fiber such that (6.38) commutes. The
commutativity yields a canonical isomorphism by using adjunctions

$Rm_{\eta,*}p^{*}_{\eta}(A\boxtimes B)\cong Rn_{\eta,*}q_{\eta}^{*}(A\boxtimes
B),$ (6.39)

which will induce the desired isomorphism (6.36) upon applying the nearby
cycles for the family
${\operatorname{Hk}}_{\mathcal{G}}^{\\{1,2\\}}|_{\operatorname{{Spd}}O_{C}}$.
Indeed, since $(Rj_{*}A)\boxtimes B$ is ULA by Proposition 6.12 and [FS21,
Corollary IV.2.25] for outer tensor products, it is still ULA after
cohomologically smooth pullback along $p,q$ and proper pushforward along $m,n$
(here, we use that the support of $A,B$ is bounded). Thus, (6.39) canonically
extends integrally yielding (6.36) after restriction to the special fiber. ∎

The following would be a natural reinforcement of the previous proposition to
also preserving perversity. For schemes, nearby cycles always preserve
perversity [HS21, Lemma 6.3]. In our setting, this is not immediate and would
transport Gaitsgory’s central functor [Gai01] to the $p$-adic context.

######  Conjecture 6.18.

For every $A\in{\operatorname{D}}({\operatorname{Hk}}_{G,C},\Lambda)$, the
$\Lambda$-flat central sheaf
$\Psi_{\mathcal{G}}(A)\in{\operatorname{D}}({\operatorname{Hk}}_{{\mathcal{G}},\bar{k}},\Lambda)^{\operatorname{bd},\operatorname{ula}}$
is perverse.

By a combination of Corollary 6.14 and Proposition 6.17, we know that 6.18
holds true whenever ${\mathcal{G}}_{\breve{O}}$ is special parahoric. Also,
using the representability in Theorem 1.1 and a comparison with schematic
nearby cycles (see [Sch17, Proposition 27.6.]), the conjecture holds true
whenever $\mu$ is minuscule. In general, we lack tools to verify 6.18 –
shifted constant terms appear to be insufficient – but one still expects some
form of Artin vanishing to hold in this very particular context of the Hecke
stack.

##  7\. Minuscule means representable

Our goal in this section is to prove the Scholze–Weinstein conjecture on
minuscule181818Recall that this is sharp, due to the theory of Banach-Colmez
spaces, see Proposition 4.8. local models [SW20, Conjecture 21.4.1] as stated
in Theorem 1.1. Its main feature is the representability part, see [Lou20,
Conjecture IV.4.18], which we verify without any assumption on the prime $p$
or the pair $({\mathcal{G}},\mu)$, thereby showing the existence of weakly
normal projective $O_{E}$-schemes
${\mathcal{M}}_{{\mathcal{G}},\mu}^{\operatorname{sch}}$ with natural,
equivariant isomorphisms
$({\mathcal{M}}_{{\mathcal{G}},\mu}^{\operatorname{sch}})^{\diamondsuit}\cong{\mathcal{M}}_{{\mathcal{G}},\mu}$
in all cases.

As for their geometry, we show under 1.9 and 1.13 that the special fiber is
given by ${\mathcal{A}}_{{\mathcal{G}},\mu}^{\operatorname{can}}$, in
particular reduced and even weakly normal. This implies the geometry part of
the Scholze–Weinstein conjecture under those assumptions, see [Lou20,
Conjecture IV.4.19].

Recall that our strategy for representability involves specializations
triples. Since explicitly calculating the specialization map seems very hard,
we need to consider convolutions of local models, so as to partially resolve
${\mathcal{M}}_{{\mathcal{G}},\mu}$ and understand their integral sections
better.

###  7.1. Convolution

We continue to denote by $F/{\mathbb{Q}}_{p}$ a complete discretely valued
field with ring of integers $O$ and perfect residue field $k$ of
characteristic $p>0$. Fix a completed algebraic closure $C/F$, and a connected
reductive $F$-group $G$ with parahoric $O$-model ${\mathcal{G}}$. Also, we fix
an auxiliary maximal $\breve{F}$-split $F$-torus $S\subset G$ whose connected
Néron model ${\mathcal{S}}$ embeds in ${\mathcal{G}}$, see [BT84, Proposition
5.10], and denote by $T$ its centralizer with connected Néron model
${\mathcal{T}}\subset{\mathcal{G}}$. Additionally, we fix an auxiliary
$\breve{F}$-Borel $T_{\breve{F}}\subset B\subset G_{\breve{F}}$.

When proving the representability of the v-sheaf local models, it is not
difficult to reduce to the case that $G$ is the Weil restriction of a split
group, see proof of Theorem 7.21. In this case, it will be helpful to
partially resolve the local model via convolution. We recall that the
Beilinson–Drinfeld Grassmannian admits the following convolution variant

${\operatorname{Gr}}_{{\mathcal{G}}}\widetilde{\times}\dots\widetilde{\times}{\operatorname{Gr}}_{{\mathcal{G}}}:=L_{O}{\mathcal{G}}\times^{L^{+}_{O}{\mathcal{G}}}\cdots\times^{L^{+}_{O}{\mathcal{G}}}{\operatorname{Gr}}_{{\mathcal{G}}}\to\operatorname{{Spd}}O,$
(7.1)

which, in terms of torsors, parametrizes successive modifications of
${\mathcal{G}}$-torsors together with a generic trivialization of the last. It
admits natural closed sub-v-sheaves

${\mathcal{M}}_{{\mathcal{G}},\mu_{\bullet}}:={\mathcal{M}}_{{\mathcal{G}},\mu_{1}}\widetilde{\times}\dots\widetilde{\times}{\mathcal{M}}_{{\mathcal{G}},\mu_{n}},$
(7.2)

for any sequence $\mu_{\bullet}=(\mu_{1},\dots,\mu_{n})$ of $B_{C}$-dominant
coweights $\mu_{i}$ of $T_{C}$, after base change to
$\operatorname{{Spd}}O_{E}$, where $E$ is the reflex field of $\mu_{\bullet}$.
We will still call them (convolution) local models for simplicity. More
precisely, denote by $\widetilde{{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{i}}}$
the preimage in $L_{O_{C}}{\mathcal{G}}$ of
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{i}}\subset{\operatorname{Gr}}_{{\mathcal{G}},O_{C}}$,
which is an $L_{O_{C}}^{+}{\mathcal{G}}$-torsor over
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{i}}$. Then

${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}=\widetilde{{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{1}}}\times^{L^{+}_{O_{C}}{\mathcal{G}}}\cdots\times^{L^{+}_{O_{C}}{\mathcal{G}}}\widetilde{{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{2}}}\times^{L^{+}_{O_{C}}{\mathcal{G}}}{\mathcal{M}}_{{\mathcal{G}},\mu_{n}}.$
(7.3)

This presentation is not “minimal” in the following sense: Namely, given a
contracted product $X\times^{H}Y$ in any topos and a normal subgroup $N\subset
H$ acting trivially on $Y$, then the natural map

$X\times^{H}Y\to X/N\times^{H/N}Y$

is an isomorphism. Hence, in (7.3) we may replace $L^{+}_{O_{C}}{\mathcal{G}}$
by some sufficiently large congruence quotient, and accordingly the torsors
$\widetilde{{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{i}}}$ by their
pushforwards to these congruence quotients. Let us note that the
multiplication

${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}\to{\operatorname{Gr}}_{{\mathcal{G}},O_{C}}$

has image ${\mathcal{M}}_{{\mathcal{G}},O_{C},|\mu_{\bullet}|}$ with
$|\mu_{\bullet}|:=\mu_{1}+\ldots+\mu_{n}$, and can therefore be regarded as a
(partial) resolution of the latter. Regarding the structure of the convoluted
local models, we can record the following.

######  Lemma 7.1.

The convoluted local model ${\mathcal{M}}_{{\mathcal{G}},\mu_{\bullet}}$ is a
proper, flat $p$-adic kimberlite over $\operatorname{{Spd}}O_{E}$ with
topologically dense generic fiber.

###### Proof.

In order to prove that ${\mathcal{M}}_{{\mathcal{G}},\mu_{\bullet}}$ is a
proper flat $p$-adic kimberlite, we must first show that this proper v-sheaf
is v-formalizing. Replace the universal $L^{+}_{O_{C}}{\mathcal{G}}$-torsors
by the corresponding $W^{+}_{O_{C}}{\mathcal{G}}$-torsors, see [Gle20,
Definition 2.2.14] and denote by
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}^{\operatorname{for}}$ the
corresponding convolution. The natural map

${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}^{\operatorname{for}}\to{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}$
(7.4)

is an isomorphism because both v-sheaves are qcqs and have the same geometric
points, compare with [Gle20, Proposition 2.2.27]. It is now easy to show that
it is formally separated and formally adic, with representable special fiber,
see Proposition 7.2. Via projection to the first factor we have a map

${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}\to{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{1}},$
(7.5)

which splits after pullback to
$\widetilde{{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{1}}}$ into the projection
of the product
$\widetilde{{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{1}}}\times_{\operatorname{{Spd}}O_{C}}{\mathcal{M}}_{{\mathcal{G}},O_{C},(\mu_{2},\ldots,\mu_{n})}$.
Hence, after replacing $L^{+}_{O_{C}}{\mathcal{G}}$ (and hence
$\widetilde{{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{1}}}$) by a sufficiently
large congruence quotient, we can deduce that
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}$ has dense generic fiber by
induction on $n$, Proposition 4.14 and preservation of closures under open
maps. ∎

From now on, we will always consider sequences of minuscule dominant coweights
$\mu_{\bullet}=(\mu_{1},\dots,\mu_{n})$ whose sum

$\lvert\mu_{\bullet}\rvert=\mu_{1}+\dots+\mu_{n}$ (7.6)

is still minuscule. Basically, this means that the support of each $\mu_{i}$
lies in disjoint irreducible components of the Dynkin diagram. We say that a
coweight is tiny if it is minuscule and its support is contained in at most
one irreducible component.

######  Proposition 7.2.

Both fibers of
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}\to\operatorname{{Spd}}O_{C}$
are representable. More precisely, we have isomorphisms

${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}|_{\operatorname{{Spd}}C}\cong{\mathcal{F}}_{G,C,\mu_{\bullet}}^{\diamondsuit}\cong{\mathcal{F}}_{G,C,\mu_{1}}^{\diamondsuit}\times\dots\times{\mathcal{F}}_{G,C,\mu_{n}}^{\diamondsuit},$
(7.7)

and also

${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}|_{\operatorname{{Spd}}\bar{k}}\cong{\mathcal{A}}_{{\mathcal{G}},\mu_{\bullet}}^{\diamondsuit},$
(7.8)

where on the right we mean the convolution
${\mathcal{A}}_{{\mathcal{G}},\mu_{1}}\widetilde{\times}\dots\widetilde{\times}{\mathcal{A}}_{{\mathcal{G}},\mu_{n}}$.

###### Proof.

The description of the generic fiber via convolution is formal, and it is
formal (using Theorem 6.16) that
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}|_{\operatorname{{Spd}}k}$
is the convolution of the
${\mathcal{A}}_{{\mathcal{G}},\mu_{i}}^{\diamondsuit}$. Using Lemma A.3 this
convolution identifies with
${\mathcal{A}}_{{\mathcal{G}},\mu_{\bullet}}^{\diamondsuit}$. ∎

We are aiming to carefully write down certain “minimal”
${\mathcal{G}}_{\operatorname{ad},O_{C}}^{>i,\diamondsuit}$-torsors
${\mathcal{M}}^{\operatorname{tor}}_{{\mathcal{G}},O_{C},\mu_{i}}\to{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{i}}$
for some associated smooth connected groups
${\mathcal{G}}_{\operatorname{ad},O_{C}}^{>i}$, such that

${\mathcal{M}}^{\operatorname{tor}}_{{\mathcal{G}},O_{C},\mu_{1}}\times^{{\mathcal{G}}_{\operatorname{ad},O_{C}}^{>1,\diamondsuit}}\dots\times^{{\mathcal{G}}_{\operatorname{ad},O_{C}}^{>i-1,\diamondsuit}}{\mathcal{M}}^{\operatorname{tor}}_{{\mathcal{G}},O_{C},\mu_{i}}\times^{{\mathcal{G}}_{\operatorname{ad},O_{C}}^{>i,\diamondsuit}}\dots\times^{{\mathcal{G}}_{\operatorname{ad},O_{C}}^{>n-1,\diamondsuit}}{\mathcal{M}}^{\operatorname{tor}}_{{\mathcal{G}},O_{C},\mu_{n}}$
(7.9)

recovers the convolution local model
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{\bullet}}$. We begin by introducing
the group schemes ${\mathcal{G}}_{\operatorname{ad},O_{C}}^{>i}$.

######  Lemma 7.3.

Let $\mu_{>i}=\mu_{i+1}+\dots+\mu_{n}$ and let $G_{\operatorname{ad},C}^{>i}$
be the quotient of $G_{C}$ by the intersection of all conjugates of
$P_{\mu_{>i}}^{-}$. Then, $G_{\operatorname{ad},C}^{>i}$ acts faithfully on
${\mathcal{F}}_{G,C,\mu_{>i}}$.

Furthermore, if we let ${\mathcal{G}}_{\operatorname{ad},O_{C}}^{>i}$ be the
unique fppf quotient191919Beware that the morphism of parahoric group schemes
${\mathcal{G}}\to{\mathcal{G}}_{\operatorname{ad}}$ is not always an fppf
surjection. of ${\mathcal{G}}_{\operatorname{ad},O_{C}}$ with generic fiber
$G^{>i}_{\mathrm{ad},C}$, then ${\mathcal{G}}_{\operatorname{ad},O_{C}}^{>i}$
acts on ${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{>i}}$, and its fibers are
smooth, affine, connected with trivial center.

###### Proof.

The claims on $G_{\operatorname{ad},C}^{>i}$ follow from Lemma 7.1. The smooth
group scheme quotient with the asserted properties obviously exist, due to
[BT84, Proposition 1.7.6], and it clearly inherits connected fibers. The
generic fiber is clearly adjoint: apply semi-simplicity of
$G_{\operatorname{ad}}$. However, it is more delicate to show that the special
fiber is adjoint.

Assume without loss of generality (Proposition 4.16) that
$G={\operatorname{Res}}_{F^{\prime}/F}G^{\prime}$ is an adjoint $F$-simple
group with $G^{\prime}$ absolutely simple, and $F^{\prime}$ a finite field
extension of $F$. Let ${\mathcal{G}}^{\prime}$ be that parahoric over
$F^{\prime}$ associated with ${\mathcal{G}}$. Then, a simple calculation
reveals that

${\mathcal{G}}^{>i}={\operatorname{Res}}_{A_{i}/O_{C}}{\mathcal{G}}_{A_{i}}^{\prime},$
(7.10)

where $A_{i}$ is the finite $O_{C}$-algebra obtained as the image of
$O_{F^{\prime}}\otimes_{O}O_{C}$ in the product of those copies of $C$ indexed
by the support of $\mu_{>i}$. Indeed, this smooth connected group scheme has
the desired universal property and its special fiber is adjoint by [CGP15,
Proposition A.5.15 (1)]. (Notice that the reducedness hypothesis is
superfluous for calculating the center.) ∎

Now, we come to the definition of the torsors.

######  Definition 7.4.

The v-sheaf ${\mathcal{M}}^{\operatorname{tor}}_{{\mathcal{G}},O_{C},\mu_{i}}$
is defined as the pushforward to
$({\mathcal{G}}_{\operatorname{ad},O_{C}}^{>i})^{\diamondsuit}$ of the natural
$L^{+}_{O_{C}}{\mathcal{G}}$-torsor
$\widetilde{{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{i}}}$ over
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu_{i}}$.

One might expect, just like in Theorem 7.21, that these torsors have natural
algebraic models over the ring of integers of some reflex field and this will
be an integral part of our strategy. Since the fibers are relatively easy to
understand by the Demazure resolution, we now focus on the behavior over a
certain ${\mathcal{G}}_{O_{C}}$-semi-orbit.

Recall that for any ${\lambda}\in W_{0}\cdot\mu$, where $W_{0}$ is the Weyl
group of $(G,S)$, the induced point
$[{\lambda}]\colon\operatorname{{Spd}}C\to{\operatorname{Gr}}_{G,C,\mu}$
uniquely extends to a point
$[{\lambda}]\colon\operatorname{{Spd}}O_{C}\to{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}$
by properness of v-sheaf local models.

######  Definition 7.5.

Let
${\mathcal{M}}_{{\mathcal{G}},\mu}^{\circ}\subset{\mathcal{M}}_{{\mathcal{G}},\mu}$
be the unique sub-v-sheaf whose base change
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}^{\circ}\subset{\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}$
to $\operatorname{{Spd}}O_{C}$ is given by the finite (non-disjoint) reunion

${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}^{\circ}=\bigcup_{\lambda\in
W_{0}\cdot\mu}{\mathcal{G}}_{O_{C}}^{\diamondsuit}\cdot[{\lambda}].$ (7.11)

We recall that the elements ${\lambda}\in W_{0}\cdot\mu$ are the rational
conjugates of $\mu$ in $X_{*}(T)$ and correspond to the open Schubert orbits
in the $\mu$-admissible locus, see the discussion after Definition 3.11. Also,
it is easy to see and left to the reader that the definition of
${\mathcal{M}}_{{\mathcal{G}},\mu}^{\circ}$ does not depend on the choice of
the auxiliary maximal $\breve{F}$-split $F$-torus $S\subset G$ whose connected
Néron model ${\mathcal{S}}$ embeds in ${\mathcal{G}}$. The stabilizer of
rational conjugates $[{\lambda}]$ is actually representable and well behaved.

######  Lemma 7.6.

Let ${\mathcal{P}}_{\lambda}^{-}$ be the flat closure in
${\mathcal{G}}_{O_{C}}$ of the repeller parabolic $P_{\lambda}^{-}\subset
G_{C}$ defined by ${\lambda}$. Then

  1. (1)

${\mathcal{P}}_{\lambda}^{-,\diamondsuit}$ is the
${\mathcal{G}}^{\diamondsuit}_{O_{C}}$-fixer of ${\lambda}$ inside
${\mathcal{M}}_{{\mathcal{G}},O_{C},\mu}$.

  2. (2)

${\mathcal{P}}_{\lambda}^{-}\to{\operatorname{Spec}}(O_{C})$ is smooth affine
with connected fibers.

###### Proof.

By topological flatness, it is clear that
${\mathcal{P}}_{\lambda}^{-,\diamondsuit}$ fixes ${\lambda}$. For dimension
reasons, the special fiber of ${\mathcal{P}}_{\lambda}^{-}$ is equal to the