We write $T=\underline{T}(\mathbb{Q}_{p})\simeq\left(\prod_{v\in
S_{p}}F_{v}^{\times}\right)^{n}$ and let $T_{0}\simeq\left(\prod_{v\in S_{p}}
\mathcal{O}_{F_{v}}^{\times}\right)^{n}\subset T$ denote its maximal compact
subgroup. We denote by $\widehat{T}$ (resp. $\widehat{T}_{0}$) the rigid
analytic spaces over $L$ parametrizing the continuous characters of $T$ (resp.
of $T_{0}$) and recall from 5 that there is a weight map

${\rm wt}:\widehat{T}_{0}\rightarrow\mathfrak{t}^{\ast}$

with values in the dual Lie algebra $\mathfrak{t}^{\ast}$ of $\underline{T}$
(considered as a rigid space over $L$). We will often, by abuse of notation,
also write ${\rm wt}$ for the composition of ${\rm wt}$ with the canonical
projection $\widehat{T}\rightarrow\widehat{T}_{0}$. Recall that we had
identified $X^{\ast}(\underline{T})$ with a $\mathbb{Z}$-lattice in
$\mathfrak{t}^{\ast}$. Often we will identify $X^{\ast}(\underline{T})$ with
$\mathbb{Z}^{n[F:\mathbb{Q}]}$.

######  Definition 5.1.

Let $\delta\in\widehat{T}$ (resp. $\in\widehat{T_{0}}$) be a character.
(i) The _weight_ of $\delta$ is the image ${\rm wt}(\delta)$ under the weight
map.
(ii) The character $\delta$ is called _of algebraic weight_ if ${\rm
wt}(\delta)\in X^{\ast}(\underline{T})\subset\mathfrak{t}^{\ast}$.
(iii) The character $\delta$ is called _algebraic_ if it is of the form

$\delta_{\underline{k}}:(z_{1}\otimes 1,\dots,z_{n}\otimes
1)\longmapsto\prod_{\tau}\left(\tau(z_{1})^{k_{1}^{\tau}}\cdots\tau(z_{n})^{k_{n}^{\tau}}\right)$

for some
$\underline{k}=(k_{1}^{\tau},\dots,k_{n}^{\tau})_{\tau:F\hookrightarrow
L}\in\mathbb{Z}^{n[F:\mathbb{Q}]}$. It is called _dominant algebraic_ if
$\underline{k}\in X^{\ast}(\underline{T})^{+}$, i.e. if
$k_{1}^{\tau}\geqslant\dots\geqslant k_{n}^{\tau}$ for all $\tau$.

Note that $\underline{k}\mapsto\delta_{\underline{k}}$ defines a section of
the weight map over the algebraic weights, and we use this map to identify
$X^{\ast}(\underline{T})$ with a subset of $\widehat{T}$ (resp.
$\widehat{T}_{0}$).

Let $K^{p}\subset U(\mathbb{A}^{\infty,p})$ be a compact open subgroup, called
a _tame level_ that we assume to be of the form $\prod_{\ell\neq p}K_{\ell}$
where $K_{\ell}$ is a compact open subgroup of $U(\mathbb{Q}_{\ell})$. Let
$I_{p}$ be the Iwahori subgroup of
$G=\underline{G}(\mathbb{Q}_{p})=U(\mathbb{Q}_{p})$ with respect to our choice
of $\underline{B}$. For any compact open $K_{p}\subset U(\mathbb{Q}_{p})$ we
consider the Shimura set

$Sh_{K^{p}K_{p}}\coloneqq U(\mathbb{Q})\backslash
U(\mathbb{A}^{\infty})/K^{p}K_{p}.$

As $U(\mathbb{R})$ is compact, this is indeed a finite set of points.

######  Definition 5.2.

The _completed cohomology_ of the tower $(Sh_{K^{p}K_{p}})_{K_{p}\subset
U(\mathbb{Q}_{p})}$ of Shimura sets is:

$\Pi\coloneqq\Pi^{\circ}\otimes_{\mathcal{O}_{L}}L,\quad\text{with}\quad\Pi^{\circ}\coloneqq\varprojlim_{n}\varinjlim_{K_{p}}H^{0}(Sh_{K^{p}K_{p}},\mathcal{O}_{L}/\pi_{L}^{n}),$

see [Eme06b].

The completed cohomology is an $L$-Banach space endowed with a continuous
action of $U(\mathbb{Q}_{p})$. This space is naturally identified with the
space of continuous functions

$f:U(\mathbb{Q})\backslash U(\mathbb{A}^{\infty})/K^{p}\longrightarrow L.$ (7)

We denote $\Pi^{\mathrm{la}}$ the subspace of locally analytic vectors in
$\Pi$ for $U(\mathbb{Q}_{p})$. This is the subspace of functions in (7) which
are locally analytic. As $\Pi^{\rm la}$ is a locally analytic representation,
there is a natural $U(\mathfrak{g})$-action on $\Pi^{\rm la}$ obtained by
deriving the $G=\underline{G}(\mathbb{Q}_{p})$-action. Here, as above, we
write $\mathfrak{g}$ for the Lie algebra of $G$, and
$\mathfrak{b},\mathfrak{t},\mathfrak{n}$ for the Lie algebras of the Borel
$B$, of the torus $T$ and of the unipotent radical $N$ of $B$.

######  Definition 5.3.

The space of _overconvergent $p$-adic automorphic forms of tame weight
$K^{p}$_ is the space

$S^{\dagger}(K^{p})=(\Pi^{\mathrm{la}})^{\mathfrak{n}}=\varinjlim_{N_{0}\subset\underline{N}(\mathbb{Q}_{p})}(\Pi^{\mathrm{la}})^{N_{0}},$

where $N_{0}$ varies among the compact open subgroups of
$\underline{N}(\mathbb{Q}_{p})$. Given a weight
$\kappa\in\mathfrak{t}^{\ast}$, the space of overconvergent $p$-adic
automorphic forms of tame weight $K^{p}$ and _weight_ $\kappa$ is the
eigenspace

$S^{\dagger}_{\kappa}(K^{p})\subset S^{\dagger}(K^{p})$

of eigenvalue $\kappa$ for the $U(\mathfrak{t})$-action.

Denote by $\mathbb{T}(K^{p})\coloneqq\mathbb{Z}[K^{p}\backslash
U(\mathbb{A}^{\infty,p})/K^{p}]$ the Hecke algebra of Hecke operators over
$\mathbb{Z}$ of tame level $K^{p}$. Then $\mathbb{T}(K^{p})$ acts by
convolution on $S^{\dagger}(K^{p})$ and $S^{\dagger}_{\kappa}(K^{p})$. Let $S$
be a finite set of prime numbers containing $p$ and all the $\ell$ such that
$K_{\ell}$ is not hyperspecial. The subalgebra
$\mathbb{T}^{S}\coloneqq\bigotimes_{\ell\notin S}
\mathbb{T}_{\ell}\subset\mathbb{T}(K^{p})$ is commutative.

######  Definition 5.4.

Let

$\underline{T}(\mathbb{Q}_{p})^{+}\coloneqq\\{\operatorname{\mathrm{diag}}(a_{1}^{v},\dots,a_{n}^{v})_{v}\in\underline{T}(\mathbb{Q}_{p})\mid
v(a_{1}^{v})\geqslant\dots\geqslant v(a_{n}^{v}),\ \forall v\in S_{p}\\}.$

The _Atkin–Lehner ring_ $\mathcal{A}(p)$ is the sub-algebra of
$\mathbb{Z}[\underline{T}(\mathbb{Q}_{p})]$ generated by the elements
$t\in\underline{T}(\mathbb{Q}_{p})^{+}$.

Let $\delta:T\rightarrow L^{\times}$ be a continuous character. Then we can
extend $\delta$ to a character $\mathcal{A}(p)\rightarrow L$ whose restriction
to $T^{+}$ is given by $\delta$. By abuse of notation we still write $\delta$
for this character of $\mathcal{A}(p)$.

Note that there is a cofinal system of compact open subgroups $N_{0}\subset
N=\underline{N}(\mathbb{Q}_{p})$ such that $tN_{0}t^{-1}\subset N_{0}$ for all
$t\in T^{+}$. We hence can define a Hecke action of $\mathcal{A}(p)$ on
$S^{\dagger}(K^{p})=(\Pi^{\mathrm{la}})^{\mathfrak{n}}$ by letting
$t\in\underline{T}(\mathbb{Q}_{p})^{+}$ act on
$f\in(\Pi^{\mathrm{la}})^{N_{0}}$ via

$[t]f\coloneqq\left(x\mapsto\frac{1}{[N_{0}:tN_{0}t^{-1}]}\sum_{n\in
N_{0}/tN_{0}t^{-1}}f(xnt)\right),$

where $N_{0}$ is a sufficiently small compact open subgroup of $N$ such that
$f\in(\Pi^{\mathrm{la}})^{N_{0}}$ and such that $tN_{0}t^{-1}\subset N_{0}$.

Let $\mathbb{T}$ be the commutative algebra
$\mathbb{T}^{S}\otimes_{\mathbb{Z}}\mathcal{A}(p)$. Definition 5.4 provides a
structure of $\mathbb{T}$-module on $S^{\dagger}(K^{p})$ and
$S^{\dagger}_{\kappa}(K^{p})$.

######  Definition 5.5.

An overconvergent $p$-adic automorphic form $f\in
S^{\dagger}(K^{p})=(\Pi^{\mathrm{la}})^{\mathfrak{n}}$ is called a _finite
slope eigenvector_ for the $\mathcal{A}(p)$-action if, for any
$t\in\underline{T}(\mathbb{Q}_{p})^{+}$, there exists $a_{t}\in L^{\times}$
such that

$[t]f=a_{t}f.$

More generally $f$ is of _finite slope_ for the $\mathcal{A}(p)$-action if for
all $t\in\underline{T}(\mathbb{Q}_{p})^{+}$, there exists a polynomial $P\in
L[X]$ such that $P(0)\neq 0$ and $P([t])f=0$.

Given a continuous character $\delta:T\rightarrow L^{\times}$, we write
$S^{\dagger}(K^{p})[\delta]$ for the eigenspace with respect to the
$\mathcal{A}(p)$-action of eigensystem $\delta:\mathcal{A}(p)\rightarrow L$.
Note, that by definition this eigensystem is automatically of finite slope and
of weight $\kappa={\rm wt}(\delta)$. Moreover, the $\mathcal{A}(p)$-action on
$S^{\dagger}(K^{p})[\delta]$ uniquely extends to an action of
$\mathbb{Z}[T(\mathbb{Q}_{p})]$.

######  Remark 5.6.

An overconvergent automorphic form of tame level $K^{p}$ with eigenvalue
$\delta:T\rightarrow L^{\times}$ for the Hecke-action at $p$ (i.e. for the
action of the Atkin–Lehner ring) is thus the same as a locally analytic
function

$f:U(\mathbb{Q})\backslash U(\mathbb{A}^{\infty})/K^{p}\longrightarrow L,$

such that there exists a compact open subgroup
$N_{0}\subset\underline{N}(\mathbb{Q}_{p})$ so that, for all $g\in
U(\mathbb{A}^{\infty}),t\in T_{0},n\in N_{0}$,

$f(gtn)=\delta(t)f(g),$

and such that moreover, for all $t\in\underline{T}(\mathbb{Q}_{p})^{+}$,
$[t]f=\delta(t)f$.

######  Definition 5.7.

The space of _classical automorphic forms of tame level $K^{p}$_ is the
subspace $S^{\rm cl}(K^{p})=(\Pi^{\rm cl})^{\mathfrak{n}})$ of
$S^{\dagger}(K^{p})=(\Pi^{\rm la})^{\mathfrak{n}}$ of elements which are
$K_{p}$-finite for some (resp. any) compact open $K_{p}\subset
U(\mathbb{Q}_{p})$.

We note that this subspace is stable under the action of $\mathbb{T}$.

For any character $\chi^{S}:\mathbb{T}^{S}\longrightarrow L$, we let
$\Pi[\chi^{S}]$ (resp. $S^{\dagger}(K^{p})[\chi^{S}]$, resp. $S^{\rm
cl}(K^{p})[\chi^{S}]$) denote the subspace of $\chi^{S}$-eigenvectors for
$\mathbb{T}^{S}$ in $\Pi$ (resp. $S^{\dagger}(K^{p})$, resp. $S^{\rm
cl}(K^{p})$). If $\delta:T\rightarrow L$ is a character of $T$ (defining a
character of $\mathcal{A}(p)$) and if $\chi=\chi^{S}\otimes\delta$ is the
corresponding character of
$\mathbb{T}=\mathbb{T}^{S}\otimes_{\mathbb{Z}}\mathcal{A}(p)$, we write
$S^{\dagger}(K^{p})[\chi]$ etc. for the corresponding eigenspace.

Let $\mathfrak{m}$ be a maximal ideal in $\mathbb{T}^{S}$. We then define

$\Pi_{\mathfrak{m}}\coloneqq\Pi^{\circ}_{\mathfrak{m}}\otimes_{\mathcal{O}_{L}}L,\quad\text{where}\quad\Pi^{\circ}_{\mathfrak{m}}\coloneqq\varprojlim_{n}(\Pi^{\circ}/\pi_{L}^{n}\Pi^{\circ})_{\mathfrak{m}}.$

As there are only finitely many maximal ideals $\mathfrak{m}$ of
$\mathbb{T}^{S}$ such that $(\Pi^{\circ}/\pi_{L}\Pi^{\circ})_{\mathfrak{m}}$
is nonzero, the space $\Pi_{\mathfrak{m}}$ is a topological direct summand of
$\Pi$ stable under the actions of $U(\mathbb{Q}_{p})$ and $\mathbb{T}$.

Recall that if $\mathfrak{m}$ is a maximal ideal (whose residue field is
assumed to equal $k_{L}$) such that $\Pi_{\mathfrak{m}}$ is non zero, then we
may associate to $\mathfrak{m}$ a continuous representation
$\overline{\rho}:\operatorname{\mathrm{Gal}}_{E}\longrightarrow\operatorname{\mathrm{GL}}_{n}(k_{L})$
which is conjugate autodual, and unramified away from $S$. Such
representations $\overline{\rho}$ are called modular (see for example [BHS17b,
§2.4]).

###  5.2 Patching the completed cohomology

We fix a maximal ideal $\mathfrak{m}\subset\mathbb{T}^{S}$ such that
$\Pi_{\mathfrak{m}}\neq 0$ is non-zero and denote by
$\overline{\rho}:\operatorname{\mathrm{Gal}}_{E}\longrightarrow\operatorname{\mathrm{GL}}_{n}(k_{L})$
the corresponding modular Galois representation. For each place $v$ of $F$
which splits in $E$ we write

$\overline{\rho}_{v}:=\overline{\rho}_{|\operatorname{\mathrm{Gal}}_{E_{\tilde{v}}}},$

for a choice of $\widetilde{v}|v$ of $E$. From now on we assume that, for
$v\in S$, the place $v$ splits in $E/F$, we make a fixed choice
$\widetilde{v}|v$ as before such that $\widetilde{v}\in\Sigma$ if $v| p$, and
denote $\widetilde{S}=\\{\widetilde{v}|v\in S\\}$ so that $\widetilde{S}$ is
in bijection with $S$ and contains $\Sigma$. For $v\in S$ we write
$R_{\overline{\rho}_{v}}^{\Box}$ for the universal lifting (i.e. framed
deformation) ring of $\overline{\rho}_{v}$ and define

$R_{\overline{\rho}_{v}}^{\Box}\twoheadrightarrow\overline{R}_{\overline{\rho}_{v}}^{\Box}$

to be the maximal reduced $\mathbb{Z}_{p}$-flat quotient.

######  Remark 5.8.

If $v|p$ we have in fact, by the main results of [BIP23],
$\overline{R}_{\overline{\rho}_{v}}^{\Box}=R_{\overline{\rho}_{v}}^{\Box}$.
Using the main result of [DHKM24] we find that the same applies to places
$v\nmid p$, as the deformation rings $R_{\overline{\rho}_{v}}^{\Box}$ may be
identified with versal rings to the moduli space of L-parameters. We still
keep the notations introduced above in order to be consistent with the
notations from the references for the patching construction below.

We denote by $R_{\overline{\rho},\mathcal{S}}$ the quotient of
$R_{\overline{\rho}}$ corresponding to the deformation problem

$\mathcal{S}=(E/F,S,\widetilde{S},\mathcal{O}_{L},\overline{\rho},\varepsilon^{1-n}\delta^{n}_{E/F},\\{\overline{R}_{\overline{\rho}_{v}}^{\Box}\\}_{v\in
S})$

in the notations of [CHT08, §2.3], where
$\delta_{E/F}:\operatorname{\mathrm{Gal}}_{F}\longrightarrow\\{\pm 1\\}$ is
the quadratic character associated to $E/F$, and

$R^{\rm loc}\coloneqq\widehat{\bigotimes_{v\in
S}}\overline{R}_{\overline{\rho}_{v}}^{\Box}.$

We assume the following (strong) Taylor-Wiles hypothesis

######  Hypothesis 5.9.

  1. 1.

$p>2$ ;

  2. 2.

the extension $E/F$ is unramified and $E$ does not contain a (non-trivial)
$p$-th root $\zeta_{p}$ of $1$ ;

  3. 3.

the group $U$ is quasi-split at all finite places of $\mathbb{Q}$ ;

  4. 4.

the level $K^{p}$ is chosen such that $K_{v}$ is hyperspecial whenever the
finite place $v$ of $F$ is inert in $E$ ;

  5. 5.

the representation
$\overline{\rho}_{|\operatorname{\mathrm{Gal}}_{E(\zeta_{p})}}$ is adequate.

By [CEG+16] sections 2.7,2.8, (see also [BHS17b, Théorème 3.5]), we have the
following data.

######  Proposition 5.10.

There exist

  1. 1.

an integer $g\geqslant 1$ ;

  2. 2.

a continuous, admissible, unitary $R_{\infty}$-representation $\Pi_{\infty}$
of $U(\mathbb{Q}_{p})$ over $L$, where

$R_{\infty}\coloneqq R^{\rm loc}[[x_{1},\dots,x_{g}]];$

  3. 3.

a local map of local rings
$S_{\infty}:=\mathcal{O}_{L}[[y_{1},\dots,y_{t}]]\longrightarrow R_{\infty}$
with

$t=g+\dim R^{loc}-[F^{+}:\mathbb{Q}]\frac{n(n+1)}{2}$

and a local map of local rings $R_{\infty}\longrightarrow
R_{\overline{\rho},\mathcal{S}}$ such that

    1. (i)

there exists an $\mathcal{O}_{L}$-lattice
$\Pi_{\infty}^{0}\subset\Pi_{\infty}$ stable by $U(\mathbb{Q}_{p})$ and
$R_{\infty}$ such that

$
(\Pi_{\infty}^{0})^{\prime}=\operatorname{\mathrm{Hom}}_{\mathcal{O}_{L}}(\Pi_{\infty}^{0},\mathcal{O}_{L}),$

is a projective $S_{\infty}[[K_{p}]]$-module of finite type (via
$S_{\infty}\longrightarrow R_{\infty}$) for some (equivalently all) compact
open subgroup $K_{p}\subset U(\mathbb{Q}_{p})$ ;

    2. (ii)

the map $R_{\infty}\rightarrow R_{\overline{\rho},\mathcal{S}}$ induces an
isomorphism

$ R_{\infty}/\mathfrak{a}R_{\infty}\simeq R_{\overline{\rho},\mathcal{S}},$

of local noetherian $\mathcal{O}_{L}$-algebras and an isomorphism of
continuous admissible unitary
$R_{\infty}/\mathfrak{a}R_{\infty}$-representations of $U(\mathbb{Q}_{p})$ on
$L$

$ \Pi_{\infty}[\mathfrak{a}]\simeq\Pi_{\mathfrak{m}},$

where $\mathfrak{a}=(y_{1},\dots,y_{t})$ denotes the augmentation ideal of
$S_{\infty}$,

It is a direct consequence of this proposition that the
$R_{\infty}$-representation $\Pi_{\infty}$ of $U(\mathbb{Q}_{p})$ satisfies
Hypothesis 3.15. We note that the same applies to a slightly more general
context:

######  Lemma 5.11.

Let $V$ be a finite dimensional algebraic representation of
$U(\mathbb{Q}_{p})$ over $L$. Then the $R_{\infty}$-Banach representation
$\Pi_{\infty}\otimes_{L}V$ satisfies Hypothesis 3.15.

###### Proof.

As $\Pi_{\infty}$ satisfies Hypothesis 3.15, for any open pro-$p$-subgroup $H$
of $U(\mathbb{Q}_{p})$ there exists an isomorphism of
$\mathbb{Z}_{p}^{t}\times H$-representations
$\Pi_{\infty|\mathbb{Z}_{p}^{t}\times
H}\simeq\mathcal{C}(\mathbb{Z}_{p}^{t}\times H,L)^{m}$ for some $m\geqslant
1$. But then

$(\Pi_{\infty}\otimes_{L}V)_{|\mathbb{Z}_{p}^{t}\times
H}\simeq\mathcal{C}(\mathbb{Z}_{p}^{t}\times
H,V)^{m}\simeq\mathcal{C}(\mathbb{Z}_{p}^{t}\times H,L)^{m\dim_{L}V}.\qed$

In the reminder of this paper we will use the following notations: we set

$\mathcal{X}^{p}\coloneqq\operatorname{\mathrm{Spf}}(\widehat{\bigotimes}_{v\in
S\backslash
S_{p}}\overline{R}_{\overline{\rho}_{v}}^{\Box})^{\mathrm{rig}}\simeq\prod_{v\in
S\backslash
S_{p}}\operatorname{\mathrm{Spf}}(\overline{R}_{\overline{\rho}_{v}}^{\Box})^{\mathrm{rig}},$

where
$\mathbb{U}^{g}\coloneqq\operatorname{\mathrm{Spf}}(\mathcal{O}_{L}[[x_{1},\dots,x_{g}]])^{\mathrm{rig}}$
is an open polydisc. Moreover, we set

$\displaystyle\mathcal{X}_{\overline{\rho}_{p}}$
$\displaystyle\coloneqq\operatorname{\mathrm{Spf}}(\widehat{\bigotimes}_{v\in
S_{p}}R_{\overline{\rho}_{v}}^{\Box})^{\mathrm{rig}},$
$\displaystyle\mathcal{X}_{\infty}$
$\displaystyle\coloneqq\operatorname{\mathrm{Spf}}(R_{\infty})^{\mathrm{rig}}\simeq\mathcal{X}^{p}\times\mathcal{X}_{\overline{\rho}_{p}}\times\mathbb{U}^{g}.$

By construction the space $\mathcal{X}_{\infty}$ contains
$\mathcal{X}_{\overline{\rho},\mathcal{S}}=(\operatorname{\mathrm{Spf}}R_{\overline{\rho},\mathcal{S}})^{\mathrm{rig}}$
as a closed subspace. For a point
$x=(x^{p},x_{p},z)\in\mathcal{X}_{\infty}(L)$ and a place $v$ of $F$ dividing
$p$, we denote by $\rho_{x,v}$ the framed representation
$\operatorname{\mathrm{Gal}}_{F_{v}}\rightarrow\operatorname{\mathrm{GL}}_{n}(L)$
associated to $x$. Finally we write $\rho_{x,p}$ for the the family of
representations $(\rho_{x,v})_{v\mid p}$.

##  6 Patching functors

In this section, we keep notations and conventions of section 5. In
particular, we have $\underline{G}\simeq\prod_{v\in
S_{p}}(L\times_{\mathbb{Q}_{p}}\operatorname{\mathrm{Res}}_{F_{v}/\mathbb{Q}_{p}}\operatorname{\mathrm{GL}}_{n,F_{v}})$
which is an algebraic group over $L$ and we consider the associated categories
$\mathcal{O},\mathcal{O}_{\mathrm{alg}}^{\infty}$ and
$\widetilde{\mathcal{O}}_{\mathrm{alg}}$ as in section 2 (for the choice of
the upper triangular Borel subgroup $\underline{B}$).

We fix once and for all a point $x\in\mathcal{X}_{\infty}(L)$ such that $x$
maps to the origin in
$(\operatorname{\mathrm{Spf}}\,S_{\infty})^{\mathrm{rig}}$ (i.e. the point
defined by the augmentation ideal of $S_{\infty}$) and we denote by
$\widehat{R}_{\infty,x}$ the completed local ring of $\mathcal{X}_{\infty}$ at
$x$.

###  6.1 Locally analytic patching functors

We fix a smooth and unramified character
$\varepsilon:\underline{T}(\mathbb{Q}_{p})\rightarrow L^{\times}$ and consider
$\varepsilon$ as a point of $\widehat{T}$.

By Lemma 5.11, we can apply Corollary 3.18 to the admissible locally analytic
representation $\Pi_{\infty}^{\mathrm{la}}$, and obtain a functor

$\displaystyle\mathcal{O}_{\mathrm{alg}}^{\infty}$
$\displaystyle\rightarrow{\rm Coh}(\mathcal{X}_{\infty}\times\widehat{T})$
$\displaystyle M$ $\displaystyle\mapsto\mathcal{M}_{\Pi_{\infty}}(M).$

######  Definition 6.1.

For $M\in\mathcal{O}_{\mathrm{alg}}^{\infty}$ we define

$\mathcal{M}_{\infty,x,\varepsilon}(M)\coloneqq\mathcal{M}_{\Pi_{\infty}}(M)_{x,\varepsilon}$

to be the stalk of $\mathcal{M}_{\Pi_{\infty}}(M)$ at $(x,\varepsilon)$.

It follows from Proposition 3.17 that $\mathcal{M}_{\infty,x,\varepsilon}(M)$
is a Cohen–Macaulay $\widehat{R}_{\infty,x}$-module and is follows from
Theorem 3.14 that the functor $M\mapsto\mathcal{M}_{\infty,x,\varepsilon}(M)$
is exact.

######  Remark 6.2.

We also have the following description:

$\mathcal{M}_{\infty,x,\varepsilon}(M)\simeq\left(\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M,\Pi^{\mathrm{la}}_{\infty}[\mathfrak{m}_{x}^{\infty}])^{N_{0}}[\mathfrak{m}_{\varepsilon}^{\infty}]\right)^{\prime}$

where $\mathfrak{m}_{\varepsilon}$ is the maximal ideal of
$\mathcal{A}(p)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}=\mathbb{Q}_{p}[\underline{T}(\mathbb{Q}_{p})^{+}]$
corresponding to the character $\varepsilon$ and $\mathfrak{m}_{x}$ is the
maximal ideal of $R_{\infty}[1/p]$ corresponding to $x$.

######  Remark 6.3.

Note that we have two $U(\mathfrak{t})$-module structures on
$\mathcal{M}_{\infty,x,\varepsilon}(M)$: The first one comes from the
nilpotent $U(\mathfrak{t})$-module structure on $M$ as in section 2.2. The
second one comes from the action of $U(\mathfrak{t})$ induced from the locally
analytic $T$-structure on $\Pi_{\infty}^{\mathrm{la}}$. It is a tautological
consequence of the construction, but we point out that these two actions
coincide.

######  Definition 6.4.

Let $I\subset\Delta$ be a finite subset of simple roots and let $M$ be an
object of $\widetilde{\mathcal{O}}_{\mathrm{alg}}^{I}$. Then we define

$\mathcal{M}_{\infty,x,\varepsilon}(M)\coloneqq\varprojlim_{n}\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_{I}^{n}).$

######  Proposition 6.5.

The functor $M\mapsto\mathcal{M}_{\infty,x,\varepsilon}(M)$ is exact on
$\widetilde{\mathcal{O}}_{\mathrm{alg}}^{I}$ and for each
$M\in\widetilde{\mathcal{O}}_{\mathrm{alg}}^{I}$ the
$\widehat{R}_{\infty,x}$-module $\mathcal{M}_{\infty,x,\varepsilon}(M)$ is
finitely generated and Cohen–Macaulay of dimension
$t+\dim_{K}\mathfrak{z}_{I}$. Moreover $\mathcal{M}_{\infty,x,\varepsilon}(M)$
is flat over $U(\mathfrak{z}_{I})$.

###### Proof.

Let $\widehat{S}_{\infty}$ be the completion of $S_{\infty}[1/p]$ along the
maximal ideal generated by the augmentation ideal $\mathfrak{a}$ of
$S_{\infty}$. Moreover, we write $\widehat{U}_{I}$ for the completion of
$U(\mathfrak{z}_{I})$ at the maximal ideal $\mathfrak{m}_{I}$.

By exactness of the functor $\mathcal{M}_{\infty,x,\varepsilon}$, we have

$\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_{I}^{n+1})/\mathfrak{m}_{I}^{n}\simeq\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_{I}^{n})$

for any $n\geqslant 1$. It follows from Theorem 3.17 that
$\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_{I})$ is a finite
projective $\widehat{S}_{\infty}$-module. We denote its rank by $r\geqslant
0$. The exactness of $\mathcal{M}_{\infty,x,\varepsilon}$ implies that
$\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_{I}^{n})$ is a finite
projective
$\widehat{S}_{\infty}\otimes_{L}U(\mathfrak{z}_{I})/\mathfrak{m}_{I}^{n}$-module
of rank $r$ and it follows that $\mathcal{M}_{\infty,x,\varepsilon}(M)$ is a
finite projective
$\widehat{S}_{\infty}\hat{\otimes}_{L}\widehat{U}_{I}$-module of rank $r$. As
the action of $\widehat{S}_{\infty}\hat{\otimes}_{L}\widehat{U}_{I}$ factors
through $\widehat{R}_{\infty,x}$ we deduce the result. The exactness of the
functor $\mathcal{M}_{\infty,x,\varepsilon}$ is a consequence of the exactness
of $\mathcal{M}_{\infty,x,\varepsilon}$ restricted to
$\mathcal{O}_{\mathrm{alg}}^{\infty}$ and the fact that each system
$(\mathcal{M}_{\infty,x,\varepsilon}(M/\mathfrak{m}_{I}^{n}))_{n}$ satisfies
the Mittag-Leffler condition.

Let $\underline{t}=(t_{1},\dots,t_{m})$ be a regular sequence generating the
maximal ideal of $U(\mathfrak{z}_{I})_{\mathfrak{m}_{I}}$. This is also a
regular sequence generating the maximal ideal of the completion
$\widehat{U}_{I}$. By exactness of the functor
$\widehat{S}_{\infty}\otimes_{L}-$ on strict exact sequences of Fréchet
$L$-algebras, the sequence $\underline{t}$ is
$\widehat{S}_{\infty}\widehat{\otimes}_{L}\widehat{U}_{I}$-regular. As
$\mathcal{M}_{\infty,x,\varepsilon}(M)$ is a finite free
$\widehat{S}_{\infty}\widehat{\otimes}_{L}\widehat{U}_{I}$-module, the
sequence $\underline{t}$ is $\mathcal{M}_{\infty,x,\varepsilon}(M)$-regular.
This is equivalent to flatness over $U(\mathfrak{z}_{I})_{\mathfrak{m}_{I}}$.
∎

###  6.2 A factorization property

We use the spaces and notations introduced in section 4. A point
$x\in\mathcal{X}_{\infty}(L)$ is said to be crystalline $\varphi$-generic and
Hodge–Tate regular if for all $v|p$ the representation $\rho_{x,v}$ is
crystalline $\varphi$-generic and Hodge–Tate regular. Let
$x=(\rho^{p},\rho_{p},z)\in\mathcal{X}_{\infty}(L)$ be such a
$\varphi$-generic Hodge–Tate regular point. We fix a refinement $\mathcal{R}$
of $\rho_{p}$.

Recall that $\underline{G}\simeq\prod_{v\in
S_{p}}(L\times_{\mathbb{Q}_{p}}\operatorname{\mathrm{Res}}_{F_{v}/\mathbb{Q}_{p}}\operatorname{\mathrm{GL}}_{n,F_{v}})$.
If $I$ is a set of simple roots of $\underline{G}$, we set

$\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}\coloneqq\widehat{\mathcal{X}^{p}}_{\rho^{p}}\times\mathcal{X}_{\rho_{p},\mathcal{R}}^{I-\mathrm{qtri}}\times\widehat{\mathbb{U}^{g}}.$

This is a closed subscheme of $(\widehat{\mathcal{X}_{\infty}})_{x}$ and we
write $\widehat{R}_{\infty,x}\twoheadrightarrow
R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$ the corresponding quotient map.
Moreover, for $w\in W$, we set

$\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},w}\coloneqq\widehat{\mathcal{X}^{p}}_{\rho^{p}}\times\mathcal{X}_{\rho_{p},\mathcal{R}}^{I-\mathrm{qtri},w}\times\widehat{\mathbb{U}^{g}}.$

If
$\mathcal{R}=(\varphi_{1,v},\dots,\varphi_{n,v})_{v|p}\in\prod_{v|p}(L^{\times})^{n}$,
we define $\delta_{\mathcal{R}}$ to be the smooth unramified character of $T$
defined by

$(x_{1,v},\dots,x_{n,v})_{v|p}\mapsto\prod_{v|p}\prod_{i}(\varphi_{i,v}^{v_{F_{v}}(x_{i,v})}q_{v}^{i-n})$

where $q_{n}$ denotes the cardinality of the residue field of $F_{v}$. We use
the notation
$\mathcal{M}_{\infty,x,\mathcal{R}}\coloneqq\mathcal{M}_{\infty,x,\delta_{\mathcal{R}}}$.
The goal of this section is to prove the following result.

######  Theorem 6.6.

Let $x\in\mathcal{X}_{\infty}(L)$ be a $\varphi$-generic Hodge–Tate regular
crystalline point and let $\mathcal{R}$ be a refinement of $x$. Then, for any
$M\in\mathcal{O}_{\mathrm{alg}}^{\infty,I}$, the
$\widehat{R}_{\infty,x}$-module $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is
killed by the kernel of the map $\widehat{R}_{\infty,x}\twoheadrightarrow
R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$. Equivalently its support is
contained in $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$.

###### Proof.

This is a consequence of Proposition 3.20, Proposition 2.14 and Corollary 6.11
which will be proved below. ∎

We will prove the auxiliary statements in (the proof of) this theorem by
making use of variants of the construction of eigenvarieties. More precisely,
for a subset $I\subset\Delta$, a character $\lambda\in
X^{\ast}(\underline{T})^{+}_{I}$ (dominant with respect to
$\underline{P}_{I}$) and an algebraic representation $V$ of $\underline{G}$ we
will consider the scheme-theoretic supports

$\displaystyle\mathcal{E}_{\infty}^{I}(\lambda)={\rm
supp}(\mathcal{M}_{\Pi_{\infty}}^{I,\lambda})$
$\displaystyle\subset\mathcal{X}_{\infty}\times\widehat{T}$
$\displaystyle\mathcal{E}_{\infty}^{I}(\lambda,V)={\rm
supp}(\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V})$
$\displaystyle\subset\mathcal{X}_{\infty}\times\widehat{T},$

where $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ respectively
$\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ are the coherent sheaves associated
to $J_{I,\lambda}(\Pi_{\infty}^{\mathrm{la}})^{\prime}$ respectively to
$J_{I,\lambda}((\Pi_{\infty}\otimes_{L}V)^{\mathrm{la}})^{\prime}$ (see
section 3.4 for the notation). We will link the completions of
$\mathcal{E}_{\infty}^{I}(\lambda)$ resp.
$\mathcal{E}_{\infty}^{I}(\lambda,V)$ at points
$(x,\delta)\in\mathcal{X}_{\infty}\times\widehat{T}$ to the quasi-trianguline
deformation rings of section 4. This is done in two steps: we first show that
the set-theoretic support of $\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ resp. of
$\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ is contained in the
(quasi-)trianguline locus (see the proof of Proposition 6.7). We then prove
that $\mathcal{E}_{\infty}^{I}(\lambda)$ resp.
$\mathcal{E}_{\infty}^{I}(\lambda,V)$ is reduced (see the proof of Proposition
6.9). The proof of the latter statement follows the usual argument in the case
of eigenvarieties, see e.g. [BHS17b, Corollaire 3.12 and Corollaire 3.20]: the
general properties of eigenvarieties (deduced from the fact that the sheaves
$\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ resp.
$\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ are locally finite projective over
$(\operatorname{\mathrm{Spf}}S_{\infty})^{\mathrm{rig}}\times\widehat{T}_{0}$
imply that $\mathcal{E}_{\infty}^{I}(\lambda)$ resp.
$\mathcal{E}_{\infty}^{I}(\lambda,V)$ have no embedded components. Hence it is
enough to produce on each of their irreducible components a point $y$ such
that $\mathcal{E}_{\infty}^{I}(\lambda)$ resp.
$\mathcal{E}_{\infty}^{I}(\lambda,V)$ are reduced in a neighborhood of $y$. By
the same projectivity argument as above, the point $y$ can be chosen so that
the weight map to $\widehat{T}_{0}$ is smooth at this point. Reducedness then
boils down to checking that the Hecke operators (that generate the local ring
of $\mathcal{E}_{\infty}^{I}(\lambda)$ resp.
$\mathcal{E}_{\infty}^{I}(\lambda,V)$ at $y$) act semi-simply on the fiber of
$\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ resp.
$\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ over $\widehat{T}_{0}$ which in
turn follows from the fact that Hecke-operators act semi-simply on spaces of
classical automorphic forms. We now give the details of these arguments.

Let $\delta=(\delta_{1,v},\dots,\delta_{n,v})_{v|p}\in\hat{T}(L)$ be a
parameter for a quasi-triangulation of $x$ at $p$, i.e. the trianguline
filtration of the $(\varphi,\Gamma)$-module $D^{\dagger}_{\rm
rig}(\rho_{v})[1/t]$ over $\mathcal{R}_{K,L}[1/t]$ has graded pieces
$\mathcal{R}_{K,L}(\delta_{i,v})[1/t]$. As $x$ is Hodge–Tate regular, there is
a natural map

$
\omega_{\delta}:\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}\longrightarrow\widehat{T}^{\wedge}_{\delta},$

mapping a deformation at $p$ of the $(\varphi,\Gamma)$-module
$D^{\dagger}_{\rm rig}(\rho_{v})[1/t]$, equipped with its trianguline
filtration, to its parameter (see e.g. [BHS19, eq (3.15)]). If $\delta$ is
locally algebraic of the form $\delta=\lambda\delta_{\mathcal{R}}$ for
$\lambda\in X^{*}(T)$ and some smooth character
$\delta_{\mathcal{R}}\in\widehat{T}(L)$, we shift the previous map to get

$
\omega=t_{-\lambda}\omega_{\delta}:\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}\longrightarrow\widehat{T}^{\wedge}_{\delta_{\mathcal{R}}}$

which only depends on the chosen refinement. This induces a map

$
i\times\omega:\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}\longrightarrow\widehat{\mathcal{X}_{\infty,x}}\times\widehat{T}^{\wedge}_{\delta_{\mathcal{R}}},$

or equivalently, a homomorphism
$\widehat{R}_{\infty,x}\otimes\mathcal{O}^{\wedge}_{\hat{T},\delta_{\mathcal{R}}}\longrightarrow
R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$.

######  Proposition 6.7.

Let $\lambda\in X^{*}(\underline{T})_{I}^{+}$ be a weight dominant with
respect to $\underline{P}_{I}$. The $\widehat{R}_{\infty,x}$-module
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(\lambda))$ is
annihilated by the kernel of $\widehat{R}_{\infty,x}\rightarrow
R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$. More precisely,
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(\lambda))$ is an
$\widehat{R}_{\infty,x}\otimes\mathcal{O}^{\wedge}_{\hat{T},\delta_{\mathcal{R}}}$-module
and annihilated by the kernel of

$\widehat{R}_{\infty,x}\otimes\mathcal{O}^{\wedge}_{\hat{T},\delta_{\mathcal{R}}}\longrightarrow
R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}.$

###### Proof.

It follows from Proposition 3.20 and the definition of
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(\lambda))$ that

$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(\lambda))=(t_{\lambda}^{*}\mathcal{M}_{\Pi_{\infty}}^{I,\lambda})^{\wedge}_{(x,\delta_{\mathcal{R}})}$

as an
$\widehat{R}_{\infty,x}\otimes\mathcal{O}^{\wedge}_{\hat{T},\delta_{\mathcal{R}}}$-module.
It is thus enough to show that the completion of
$\mathcal{M}_{\Pi_{\infty}}^{I,\lambda}$ at the point
$(x,\lambda\delta_{\mathcal{R}})\in\mathcal{X}_{\infty}(L)\times\widehat{T}(L)$
is supported at the closed subspace

$i\times
w_{\delta}:\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}\longrightarrow\widehat{\mathcal{X}_{\infty,x}}\times\widehat{T}^{\wedge}_{\delta_{\lambda}\delta_{\mathcal{R}}}.$

We closely follow the proof of [Wu, Prop. 5.13]. Let us write
$\mathcal{E}_{\infty}\subset\mathcal{X}\times\widehat{T}$ for the scheme-
theoretic support of the coherent sheaf defined by
$J_{B}(\Pi_{\infty})^{\prime}$. By [Wu, 5.4] this contains
$\mathcal{E}_{\infty}^{I}(\lambda)$ as a closed subspace. As in the proof of
[Wu, Prop. 5.13] we consider a proper birational map
$f:\mathcal{E}_{\infty}^{\prime}\rightarrow\mathcal{E}_{\infty}$ such that the
universal $(\varphi,\Gamma)$-module over $\mathcal{E}_{\infty}^{\prime}$ has a
quasi-triangulation, and write $\mathcal{E}_{\infty}^{\prime\prime}$ for the
preimage of $\mathcal{E}_{\infty}^{I}(\lambda)$ in
$\mathcal{E}_{\infty}^{\prime}$. Let
$Y\subset\mathcal{E}_{\infty}^{\prime\prime}$ be the Zariski closed reduced
subspace of $\mathcal{E}_{\infty}^{\prime\prime}$ whose points are exactly the
points of $\mathcal{E}_{\infty}^{\prime\prime}$ where the universal filtered
$(\varphi,\Gamma)$-module over $\mathcal{R}[1/t]$ is $P_{I}$-de Rham. As in
[Wu], the existence of $Y$ is a consequence of [Wu, Prop. A.10]. It follows
that for any $y\in Y$ lying above $(x,\delta_{\mathcal{R}})$ the map

$\widehat{Y}_{y}\rightarrow\mathcal{X}_{\infty}\times\widehat{T}$

factors through $\mathcal{X}_{\infty,y,\mathcal{R}_{y}}^{I-\mathrm{qtri}}$.
Let $U\subset\mathcal{E}_{\infty}^{I}(\lambda)$ be an affinoid open subset
containing $x$ and a Zariski dense subset of points which are de Rham (and in
particular $P_{I}$-de Rham) and trianguline with parameter given by
$\mathcal{E}_{\infty}^{I}(\lambda)\longrightarrow\widehat{T}$. Such a
neighborhood exists by [Wu, Prop. 5.11 & 5.12]. We deduce that $f(Y)\supset U$
and hence $f^{-1}(U)\subset Y$ and we conclude as in the proof of [BHS19,
Prop. 3.7.2] (see the erratum in [BD]) that the map

$\widehat{U}_{x,\lambda\delta_{\mathcal{R}}}\rightarrow\mathcal{X}_{\infty}\times\widehat{T}$

factors through $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$. ∎

######  Corollary 6.8.

Let $V$ be an algebraic representation of $\underline{G}$, then

$\mathcal{M}=\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(\lambda)\otimes_{L}V)$

is annihilated by some power of the kernel of
$\widehat{R}_{\infty,x}\otimes\mathcal{O}^{\wedge}_{\hat{T},\delta_{\mathcal{R}}}\longrightarrow
R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$.

###### Proof.

We recall that

$\widetilde{M}_{I}(\lambda)\otimes_{L}V=U(\mathfrak{g})\otimes_{U(\mathfrak{p}_{I})}(L_{I}(\lambda))\otimes_{L}V\cong
U(\mathfrak{g})\otimes_{U(\mathfrak{p}_{I})}(L_{I}(\lambda)\otimes
V_{|P_{I}})$

and that $V_{|P_{I}}$ is an extension of algebraic irreducible representations
of $\underline{L}_{I}$. Exactness of $\mathcal{M}_{\infty,x,\mathcal{R}}$ (see
Proposition 6.5) implies that the $\widehat{R}_{\infty,x}$-module
$\mathcal{M}$ is an extension of $\widehat{R}_{\infty,x,\mathcal{R}}$-module
of the form $\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(\mu))$ for
$\mu\in X^{*}(\underline{T})_{I}^{+}$. We deduce the result from Proposition
6.7. ∎

######  Proposition 6.9.

Let $V$ be an algebraic representation of $\underline{G}$. Then the schematic
support $\mathcal{E}_{\infty}^{I}(\lambda,V)$ of the coherent sheaf associated
to $J_{I,\lambda}((\Pi_{\infty}\otimes_{L}V)^{\mathrm{la}})^{\prime}$ is
reduced.

###### Proof.

We follow closely the proof of [BHS17b, Cor. 3.20] replacing, where it is
nedeed, some arguments by results of [Wu]. To simplify notations we just write
$\mathcal{E}=\mathcal{E}_{\infty}^{I}(\lambda,V)$ and
$\mathcal{M}=\mathcal{M}_{\Pi_{\infty}}^{I,\lambda,V}$ for the reminder of
this proof.

Let $\mathcal{N}$ be the radical ideal of $\mathcal{O}_{\mathcal{E}}$. Assume
that $\mathcal{N}\neq 0$ and let $x\in\mathcal{E}$ be a point in the support
of $\mathcal{N}$. Let $\widehat{T}^{\circ}_{\lambda}$ be the preimage of
$\lambda_{|\mathfrak{t}\cap\mathfrak{l}_{I}^{\mathrm{ss}}}\in(\mathfrak{t}\cap\mathfrak{l}_{I}^{\mathrm{ss}})^{*}$
under the map

$\widehat{T}\rightarrow\mathfrak{t}^{*}\rightarrow(\mathfrak{t}\cap\mathfrak{l}_{I}^{\mathrm{ss}})^{*},$

where the first map is the weight map (5). According to [Wu, §5.4] there
exists an open affinoid neighborhood $U$ of $x$ and an open affinoid subset
$W\subset\widehat{T}^{\circ}_{\lambda}\times\operatorname{\mathrm{Spf}}(S_{\infty})^{\mathrm{rig}}$
such that $\Gamma(U,\mathcal{M})$ is a finite free $\mathcal{O}(W)$-module
(such a data exists according to the results of [Wu, §5.4]). Then
$\Gamma(U,\mathcal{N})$ is the radical ideal of $\mathcal{O}(U)$. Moreover, as
$\mathcal{O}(U)=\Gamma(U,\mathcal{O}_{\mathcal{E}})$ is a
sub-$\mathcal{O}(W)$-module of
$\operatorname{\mathrm{End}}(\Gamma(U,\mathcal{M}))$ (by the same argument as
in the proof of Theorem 3.17 respectively of [BHS17b, Prop. 3.11]), the same
is true for $\Gamma(U,\mathcal{N})$. Therefore $\Gamma(U,\mathcal{N})$ is a
torsion free $\mathcal{O}(W)$-module and its support has the same dimension as
$W$ and hence contains an irreducible component $U_{0}$ of $U$. As a
consequence the support of $\mathcal{N}$ contains an admissible open subset of
$\mathcal{E}$. As the support of $\mathcal{N}$ is also a closed analytic
subset of $\mathcal{E}$, it follows from [Con99, Lemm. 2.2.3] that the support
of $\mathcal{N}$ contains an irreducible component of $\mathcal{E}$. It hence
suffices to produce on each irreducible component of $\mathcal{E}$ a point $y$
such that $\mathcal{E}$ is reduced in a neighborhood of $y$.

By [Wu, Prop. 5.11] every irreducible component of $\mathcal{E}$ contains a
point with algebraic weight.

Therefore we fix a point $x\in\mathcal{E}(L)$ with integral weight
$\lambda^{\prime}\in\widehat{T}^{\circ}_{\lambda}$. Let $U$ be an open
affinoid neighborhood of $x$ and
$W\subset\widehat{T}^{\circ}_{\lambda}\times\operatorname{\mathrm{Spf}}(S_{\infty})^{\mathrm{rig}}$
an open affinoid open subset such that $M=\Gamma(U,\mathcal{M})$ is a direct
factor of
$\mathcal{O}(W)\hat{\otimes}_{L}J_{B_{I}}(J_{P_{I}}(\Pi_{\infty}\otimes_{L}V)_{\lambda})^{\prime}$.
Let $A=\mathcal{O}(W)$ and $B=\mathcal{O}(U)$. Then $M$ is a finitely
generated $B$-module and a finite projective $A$-module. Let $C>0$ and
$C^{\prime}>0$ as in the proof of [Wu, Prop. 5.11]. We set $Z\subset W$ be the
subset of algebraic character $\delta_{\lambda^{\prime}}$ such that, for any
simple root $\alpha\notin I$,
$\langle\lambda^{\prime}+\nu,\alpha\rangle>C^{\prime}$ for any $\nu$ weight of
$V^{\vee}$. This is a Zariski dense subset of $W$. Then for
$z=\delta_{\lambda^{\prime}}\delta_{\mathrm{sm}}$ with $\delta_{\mathrm{sm}}$
a smooth character, using Proposition 3.20, we see that the $B$-module
$M_{z}=M\otimes k(z)$ is a direct factor of
$J_{B}(\operatorname{\mathrm{Hom}}(M_{I}(\lambda^{\prime}),\Pi_{\infty}\otimes_{L}V))^{\prime}$.
Let $(x,\delta)\in U$ be a point above $z$, i.e.
$\delta=\delta_{\lambda^{\prime}}\delta_{\mathrm{sm}}$, then arguing as in
_loc. cit._ , we have
$\operatorname{\mathrm{Hom}}_{G}(\mathcal{F}_{\overline{B}}^{G}(N\otimes_{L}V^{\vee},\delta_{\mathrm{sm}}\delta_{B}^{-1}),\Pi_{\infty}[\mathfrak{p}_{x}])=0$
for any subquotient $N$ of $M_{I}(\lambda^{\prime})$ different from
$L(\lambda^{\prime})$. This implies that $M_{z}$ is actually a quotient of
$J_{B}(\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(L(\lambda^{\prime})\otimes_{L}V^{\vee},\Pi_{\infty}))$
which is isomorphic to a finite direct sum of
$J_{B}(\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(L(\mu),\Pi_{\infty}))$
with $\mu$ dominant. The proof of [BHS17b, Cor. 3.20] shows that the global
sections of the coherent sheaf associated to each
$J_{B}(\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(L(\mu),\Pi_{\infty}))^{\prime}$
on $U\cap\kappa^{-1}(\\{\delta_{\lambda^{\prime}}\\})$ is a semisimple
$B$-module. This concludes the proof. ∎

######  Corollary 6.10.

The rigid analytic space $\mathcal{E}_{\infty}^{I}(\lambda)$ is reduced.

###### Proof.

This is Proposition 6.9 with $V$ the trivial representation. ∎

######  Corollary 6.11.

Let $V$ be an irreducible algebraic representation of $\underline{G}$. Then
the
$\widehat{R}_{\infty,x}\otimes\mathcal{O}_{\widehat{T}^{\wedge}_{\delta_{\mathcal{R}}}}$-module
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(\lambda)\otimes_{L}V)$
is killed by the kernel of the map

$\widehat{R}_{\infty,x}\otimes\mathcal{O}_{\widehat{T}^{\wedge}_{\delta_{\mathcal{R}}}}\rightarrow
R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}.$

###### Proof.

By Proposition 6.9, the support of the module
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(\lambda)\otimes_{L}V)$
is reduced for any $\lambda\in X^{*}(\underline{T})$ dominant with respect to
$\underline{P}_{I}$ and any algebraic representation $V$ of $\underline{G}$.
Therefore the result follows from Corollary 6.8. ∎

###  6.3 Bi-module structure on the patched functor

Let $M$ be an object of $\mathcal{O}_{\mathrm{alg}}^{\infty}$ or
$\widetilde{\mathcal{O}}_{\mathrm{alg}}^{I}$. As seen in section 2.2, there is
a natural structure of $A=U(\mathfrak{t})_{\mathfrak{m}}$-module on $M$ which
provides, by functoriality, the structure of an $A$-module on
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$. This $A$-module structure extends to
an action of the completion $\widehat{A}$ of $A$ with respect to the maximal
ideal $\mathfrak{m}$. We recall from Remark 6.3 that this action coincides
with the structure of an $\widehat{A}$-module on
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ induced from the $T$-action on
$\Pi_{\infty}$.

On the other hand, the ring $R_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$ also
carries a structure of an $\widehat{A}$-module induced from the map
$\kappa_{1}$ defined in section 4.1. This gives a further structure of an
$\hat{A}$-module on the $R_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$-module
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$. We will show that these
$\hat{A}$-module structures agree.

For $a\in\widehat{A}$, we denote by $a$ (resp. $\widetilde{a}$) the
endomorphism of $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ defined by the first
(resp. second) action. Note that if $M$ is an object of
$\widetilde{\mathcal{O}}_{\mathrm{alg}}$, then
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is a finite free
$\widehat{A}\widehat{\otimes}_{L}\widehat{S}_{\infty}$-module for the first
$\hat{A}$-module structure by the proof of Proposition 6.5. Thus it is
$A$-torsion free (since $\widehat{A}$ is domain).

######  Lemma 6.12.

For any $a\in\widehat{A}$ and any $M$ in $\mathcal{O}_{\mathrm{alg}}^{\infty}$
or $\widetilde{\mathcal{O}}_{\mathrm{alg}}$, there is an equality

$a=\widetilde{a}\in{\rm End}(\mathcal{M}_{\infty,x,\mathcal{R}}(M)).$

###### Proof.

If $M=\widetilde{M}(\mu)\otimes_{U(\mathfrak{t})}U(\mathfrak{t})/m^{n}$ for
some $\mu\in X^{*}(\underline{T})$, this is a consequence of [BHS17b, Thm.
3.21], the commutative diagram [BHS19, (3.30)] and Remark 6.3. This implies
that for any $\mu\in X^{*}(\underline{T})$, we have $a=\widetilde{a}$ on
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}(\mu))$.

Now we consider the general case. It follows from Proposition 2.14 that it is
sufficient to prove the equality $\widetilde{a}=a$ when
$M=\widetilde{M}(\mu)\otimes_{L}V$ for $\mu\in X^{*}(\underline{T})$ dominant
and $V$ a finite dimensional $U(\mathfrak{g})$-module. Let
$(\operatorname{\mathrm{Fil}}_{i})$ be an increasing filtration of
$\widetilde{M}(\mu)\otimes_{L}V$ such that
$\operatorname{\mathrm{Fil}}_{i}/\operatorname{\mathrm{Fil}}_{i-1}\simeq\widetilde{M}(\mu_{i})$
where $\mu_{1},\dots,\mu_{d}\in X^{*}(\underline{T})$ and $d=\dim_{L}V$ (such
a filtration exists by [Soe92, Lem. 8]. Let $K$ denote the fraction field of
$A$. It follows from Proposition 2.12 that we have a decomposition of
$U(\mathfrak{g})_{K}$-modules

$(\widetilde{M}(\mu)\otimes_{L}V)\otimes_{A}K\simeq\bigoplus_{i=1}^{d}\widetilde{M}(\mu_{i})_{K}$

splitting the filtration $(\operatorname{\mathrm{Fil}}_{i}\otimes_{A}K)$. Let
$p_{i}\in\operatorname{\mathrm{End}}_{U(\mathfrak{g})_{K}}((\widetilde{M}(\mu)\otimes_{L}V)\otimes_{A}K)$
be the projector on $\widetilde{M}(\mu_{i})_{K}$. As

$\operatorname{\mathrm{End}}_{U(\mathfrak{g})_{K}}((\widetilde{M}(\mu)\otimes_{L}V)\otimes_{A}K)\simeq\operatorname{\mathrm{End}}_{U(\mathfrak{g})}((\widetilde{M}(\mu)\otimes_{L}V))\otimes_{A}K$

by [Soe92, Thm. 5], there exists, for each $1\leqslant i\leqslant d$, a
nonzero element $q_{i}\in A$ such that $q_{i}p_{i}$ actually restricts to an
endomorphism of $\widetilde{M}(\mu)\otimes_{L}V$. We set $q=q_{1}\cdots q_{r}$
and $\alpha_{i}=qp_{i}$. Then the $\alpha_{i}$ are endomorphisms of
$\widetilde{M}(\mu)\otimes_{L}V$ that stabilize the filtration
$\operatorname{\mathrm{Fil}}_{\bullet}$. As each
$\operatorname{\mathrm{Fil}}_{i}/\operatorname{\mathrm{Fil}}_{i-1}$ is a free
$A$-module, the endomorphisms $\alpha_{i}$ induce the zero endomorphism of
$\operatorname{\mathrm{Fil}}_{i-1}$ and
$\widetilde{M}(\mu)\otimes_{L}V/\operatorname{\mathrm{Fil}}_{i}$ and the
multiplication by $q$ on
$\operatorname{\mathrm{Fil}}_{i}/\operatorname{\mathrm{Fil}}_{i-1}$.

In order to simplify notations we set

$\displaystyle M_{\infty}$
$\displaystyle=\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}(\mu)\otimes_{L}V),$
$\displaystyle\operatorname{\mathrm{Fil}}_{i}M_{\infty}$
$\displaystyle=\mathcal{M}_{\infty,x,\mathcal{R}}(\operatorname{\mathrm{Fil}}_{i}).$

By construction, for each $i$ the endomorphism $\alpha_{i}$ induces an
$R_{\infty,x}$-linear endomorphisms of
$\operatorname{\mathrm{Fil}}_{j}M_{\infty}$ for all $j$. By exactness of
$\mathcal{M}_{\infty,x,\mathcal{R}}$, the family
$(\operatorname{\mathrm{Fil}}_{i}M_{\infty})$ is a filtration of $M_{\infty}$
and
$\operatorname{\mathrm{Fil}}_{i}M_{\infty}/\operatorname{\mathrm{Fil}}_{i-1}M_{\infty}\simeq\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}(\mu_{i}))$
for any $i$, so that $a$ and $\widetilde{a}$ induces the same endomorphism of
$\operatorname{\mathrm{Fil}}_{i}M_{\infty}/\operatorname{\mathrm{Fil}}_{i-1}M_{\infty}$.
Finally, for $1\leqslant i\leqslant d$, we denote by
$M_{\infty}^{(i)}=\alpha_{i}(\operatorname{\mathrm{Fil}}_{i}M_{\infty})$ the
image of the $i$-th filtration step under $\alpha_{i}$. It follows from the
properties of $\alpha_{i}$ that

  * •

$M_{\infty}^{(i)}\subset\operatorname{\mathrm{Fil}}_{i}M_{\infty}$;

  * •

the quotient
$\operatorname{\mathrm{Fil}}_{i}M_{\infty}/(\operatorname{\mathrm{Fil}}_{i-1}M_{\infty}+M_{\infty}^{(i)})$
is killed by $q$;

  * •

$M_{\infty}^{(i)}$ is isomorphic to a quotient of
$\operatorname{\mathrm{Fil}}_{i}M_{\infty}/\operatorname{\mathrm{Fil}}_{i-1}M_{\infty}$.

Therefore, we have $\widetilde{a}=a$ on $M_{\infty}^{(i)}$ for any
$a\in\widehat{A}$ and the quotient of $M_{\infty}$ by the sum of the
$M_{\infty}^{(i)}$ is killed by $q^{d}$. As $M_{\infty}$ is $A$-torsion free
it follows that $\widetilde{a}=a$. ∎

Let $\xi:Z(\mathfrak{g})\rightarrow U(\mathfrak{t})$ be the Harish-Chandra map
as recalled in section 2.4. As in loc.cit. we write $t_{\nu}$ for the unique
endomorphism of $U(\mathfrak{t})$ mapping $x\in\mathfrak{t}$ to
$t_{\nu}(x)=x+\nu(x)$.

Let $h=(h_{1,\tau,v}<\cdots<h_{n,\tau,v})_{\tau,v}\in X^{*}(\underline{T})$ be
the weight corresponding to the Hodge–Tate weights of
$\rho_{x}=(\rho_{v})_{v|p}$ and let
$\delta^{\prime}_{G}=(0,-1,-2,\dots,1-n)_{\tau,v}\in X^{*}(\underline{T})$ be
fixed central shift of the half sum of the positive roots $\delta_{G}\in
X^{\ast}(\underline{T})\otimes\mathbb{Q}$. We have a map

$\kappa_{2}:\widehat{A}=\widehat{U(\mathfrak{t})}_{\mathfrak{m}}\rightarrow
R_{\rho_{p},\mathcal{R}}^{\mathrm{qtri}}$

induced from the map $\kappa_{2}$ of section 4.1 and we define the $L$-algebra
homomorphism

$\alpha=\kappa_{2}\circ
t_{h-\delta^{\prime}_{G}}\circ\xi:Z(\mathfrak{g})\rightarrow
R_{\rho_{p},\mathcal{R}}^{\mathrm{qtri}}.$

As in [DPS, Def. 4.23], we define, for any $v|p$, an $L$-algebra homomorphism

$\zeta_{\widetilde{\rho}_{v}^{C}}^{C}:Z({\rm
Lie}(\operatorname{\mathrm{Res}}_{F_{v}/\mathbb{Q}_{p}}\operatorname{\mathrm{GL}}_{n}))\longrightarrow
R_{\overline{\rho}_{v}}^{\Box,\mathrm{rig}}$

where $\widetilde{\rho}_{v}$ is the universal family of Galois representations
over $R_{\overline{\rho}_{v}}^{\Box,\mathrm{rig}}$. After completion at
$\rho_{v}$ and taking the tensor product over all $v|p$, we obtain an
$L$-algebra homomorphism

$\zeta^{C}:Z(\mathfrak{g})=\bigotimes_{v|p}Z({\rm
Lie}(\operatorname{\mathrm{Res}}_{F_{v}/\mathbb{Q}_{p}}\operatorname{\mathrm{GL}}_{n}))\longrightarrow
R_{\rho_{p}}^{\Box}\twoheadrightarrow
R_{\rho_{p},\mathcal{R}}^{\mathrm{qtri}}.$

Note that the definition of $\zeta_{\widetilde{\rho}_{v}^{C}}^{C}$ from
$\rho_{v}$ depends on a choice of a central shift of $\delta_{G}$ (see the
discussion ending [DPS, §4.7]). We choose it equal to $\delta^{\prime}_{G}$.
More concretely $\zeta^{C}$ is characterized by the following property. This
is the unique continuous homomorphism such that, for any local artinian
$L$-algebra and any local homomorphism
$f:R_{\rho_{p},\mathcal{R}}^{\mathrm{qtri}}\rightarrow A$, corresponding to
$\rho_{A}=(\rho_{A,v}:\operatorname{\mathrm{Gal}}_{F_{v}}\rightarrow\operatorname{\mathrm{GL}}_{n}(A))_{v|p}$,
the composition map
$Z(\mathfrak{g})\xrightarrow{\zeta^{C}}R_{\rho_{p},\mathcal{R}}^{\mathrm{qtri}}\rightarrow
A$ is
$Z(\mathfrak{g})\xrightarrow{\xi}U(\mathfrak{t})\xrightarrow{t_{\nu-\delta^{\prime}_{G}}}k(x)$
where

$\nu\in\operatorname{\mathrm{Hom}}_{L}(U(\mathfrak{t})^{W},A)\simeq\operatorname{\mathrm{Hom}}_{L}(U(\mathfrak{t}^{*})^{W},A)\simeq\operatorname{\mathrm{Hom}}_{L}(U(\mathfrak{g}^{*})^{\underline{G}_{L}},A)$

is the map induced by the conjugacy class of the Sen operators

$(\Theta_{\mathrm{Sen},\rho_{A,v}})_{v|p}\in(\mathfrak{g}\otimes_{L}A).$

######  Proposition 6.13.

The homomorphisms $\zeta^{C}$ and $\alpha$ defined above coincide.

###### Proof.

It is sufficient to prove that for any local artinian $L$-algebra $A$ and any
map $f:R_{\rho_{p},\mathcal{R}}^{\mathrm{qtri}}\rightarrow A$, we have
$f\circ\zeta^{C}=f\circ\alpha$. Note that the map $f$ gives rise to a family
$(\rho_{A,v})_{v|p}$ of local Galois representations. It follows from [BHS19,
Lem. 3.7.5] that, for any embedding $\tau:F_{v}\hookrightarrow L$, the
$\tau$-part of the Sen polynomial of $\rho_{v}$ is
$\prod_{i=1}^{n}(X-(h_{i,\tau}+\nu_{i,\tau}))$ where
$(\nu_{i,\tau})\in\operatorname{\mathrm{Hom}}_{L}(\mathfrak{t},A)$ corresponds
to $f\circ\kappa_{2}:U(\mathfrak{t})\rightarrow A$. The result is then a
direct comparison of the definitions of $\alpha$ and $\zeta^{C}$.∎

For each element $M$ of the category $\mathcal{O}_{\mathrm{alg}}^{\infty}$ or
$\widetilde{\mathcal{O}}_{\mathrm{alg}}$, there is a natural homomorphism of
$L$-algebras $Z(\mathfrak{g})\rightarrow\operatorname{\mathrm{End}}(M)$. By
functoriality of $\mathcal{M}_{\infty,x,\mathcal{R}}$, this gives a map

$z:Z(\mathfrak{g})\rightarrow\operatorname{\mathrm{End}}_{\widehat{R}_{\infty,x}}(\mathcal{M}_{\infty,x,\mathcal{R}}(M)).$

The following result tells us that this map factors through
$R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$.

######  Corollary 6.14.

For any $x\in Z(\mathfrak{g})$, the element $z(x)$ is the multiplication by
$\alpha(x)\otimes 1\in R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$.

###### Proof.

This is a consequence of Proposition 6.13 and of [DPS, Thm. 9.27]. ∎

######  Remark 6.15.

Recall that $h=(h_{1,\tau,v}<\cdots<h_{n,\tau,v})_{\tau,v}$ denotes the weight
corresponding to the Hodge–Tate weights of $\rho$. Let $\lambda\coloneqq
w_{0}(h)-\delta^{\prime}_{G}\in X^{*}(\underline{T})$, which is still a
dominant character. Recall that $t_{-\delta_{G}}\circ\xi$ has image contained
in $U(\mathfrak{t})^{W}$. Hence we have

$t_{h-\delta^{\prime}_{G}}\circ\xi=t_{h}\circ\mathrm{Ad}(w_{0})\circ
t_{-\delta^{\prime}_{G}}\circ\xi=\mathrm{Ad}(w_{0})\circ t_{w_{0}(h)}\circ
t_{-\delta^{\prime}_{G}}\circ\xi=\mathrm{Ad}(w_{0})\circ t_{\lambda}\circ\xi.$

Therefore

$\mathrm{Id}\otimes\alpha=(\mathrm{Id}\otimes\mathrm{Ad}(w_{0}))\circ
h_{\lambda}:A\otimes_{L}Z(\mathfrak{g})\rightarrow A\otimes_{A^{W}}A,$

where $h_{\lambda}$ is the map defined in section 2.4.

###  6.4 Computation of a support

Now we can prove our main result of this section concerning the support of the
patched functor applied to a generalized Verma module respectively applied to
its dual.

######  Theorem 6.16.

Let $x\in\mathcal{X}_{\infty}(L)$ be a point whose associated Galois
representation is crystalline, $\varphi$-generic and Hodge–Tate regular. Let
$\mathcal{R}$ be a refinement of $x$. Let
$h=(h_{1,\tau}<\cdots<h_{n,\tau})_{\tau:F\hookrightarrow L}\in
X^{*}(\underline{T})$ be the character given by the Hodge–Tate weights of
$\rho_{x}$. Let
$\delta^{\prime}_{G}=\det{}^{\frac{1-n}{2}}\delta_{G}=(0,-1,\dots,1-n)_{\tau:F\hookrightarrow
L}\in X^{*}(\underline{T})$, where $\delta_{G}$ is the half sum of the
positive roots, and define $\lambda\coloneqq w_{0}(h)-\delta^{\prime}_{G}\in
X^{*}(\underline{T})^{+}$.

Then, for $I\subset\Delta$ and $w\in W$, the schematic supports of
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w^{\rm
min}\cdot\lambda))$ and
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w^{\rm
min}\cdot\lambda)^{\vee})$ are either
$\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},w^{\rm min}w_{0}}$ or
empty.

###### Proof.

Let $M$ be $\widetilde{M}_{I}(w^{\rm min}\cdot\lambda)$ or
$\widetilde{M}_{I}(w^{\rm min}\cdot\lambda)^{\vee}$. As
$R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$ is generically reduced and equi-
dimensional by Lemma 4.4 and as $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is
Cohen–Macaulay of dimension $\dim R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$,
its schematic support is reduced and is a union of irreducible components of
$\operatorname{\mathrm{Spec}}R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$, i.e.
it is a union of
$\operatorname{\mathrm{Spec}}R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},w^{\prime}}$
for some $w^{\prime}\in W$.

By Proposition 2.15, the module $M$ is annihilated by $I_{w}\subset
A_{I}\otimes_{L}Z(\mathfrak{g})$. This implies in particular that the action
of $A_{I}\otimes_{L}Z(\mathfrak{g})$ on $M$ factors through $h_{\lambda}$. By
functoriality, this gives rise to a structure of an
$A_{I}\otimes_{A^{W}}A$-module on $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$.
Note that the map $(\kappa_{1},\kappa_{2})$ of section 4.1 provides a morphism
of $L$-algebras $A_{I}\otimes_{A^{W}}A\rightarrow
R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$ and, using Theorem 6.6, a second
structure of an $A_{I}\otimes_{A^{W}}A$-module on
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$. It follows from Lemma 6.12, Corollary
6.14 and Remark 6.15 that this two actions coincide up to composition with
$\mathrm{Id}\otimes\mathrm{Ad}(w_{0})$. We deduce that
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is killed by the ideal of
$R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$ defining the inverse image of
$T_{I,ww_{0}}\subset\mathfrak{z}_{I}\times_{\mathfrak{t}/W}\mathfrak{t}$.
Therefore Lemma 4.5 (see also Remark 2.16) implies that the action of
$R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri}}$ factors through
$R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},ww_{0}}$ so that the schematic
support of $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is
$\operatorname{\mathrm{Spec}}R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},ww_{0}}$.
∎

##  7 Main results

Let $x=(\rho_{p},\rho^{p},z)\in\mathcal{X}_{\infty}(L)$ corresponding to a
classical automorphic form of tame level $K^{p}$. Moreover, we assume that
(the Galois representation defined by) $x$ is crystalline, Hodge–Tate regular
and $\varphi$-generic (see section 6.2) at $p$. This means that
$x\in\mathcal{X}_{\overline{\rho},\mathcal{S}}(L)\subset\mathcal{X}_{\infty}(L)$
and that there exists an automorphic representation
$\pi=\pi_{\infty}\otimes_{\mathbb{C}}\pi_{f}$ of $U(\mathbb{A}_{\mathbb{Q}})$
whose associated Galois representation $\rho$ is the representation defined by
$x$ and such that $\pi_{f}\otimes W$ occurs in the locally algebraic vectors
of $\Pi_{\mathfrak{m}}$ for some algebraic representation $W$ depending on
$\rho$. In particular, the automorphic representation $\pi$ is finite slope at
$p$. It follows from the proof of [BHS17a, Cor. 3.12] that the image
$\rho^{p}$ of $x$ in $\operatorname{\mathrm{Spf}}(\bigotimes_{v\in S,p\nmid
v}\overline{R}^{\Box}_{\overline{\rho}_{v}})^{\mathrm{rig}}$ lies in the
smooth locus.

We fix a refinement $\mathcal{R}=(\varphi_{1,v},\dots,\varphi_{n,v})_{v}$ of
$x$. Let us denote the $\tau$-Hodge–Tate weights of $\rho_{x,v}$ for $v|p$ in
$F$ and $\tau:F_{v}\hookrightarrow L$ by
$h_{v,\tau}\coloneqq(h_{1,v,\tau}<\dots<h_{1,v,\tau})$. Given this collection
of Hodge–Tate weights we write $h=(h_{v,\tau})_{v,\tau}$ and
$h_{v}=(h_{v,\tau})_{\tau}$. We then define
$R_{\rho_{v}}^{\mathrm{cris},h_{v}}$ to be the crystalline deformation ring of
$\rho_{v}$ of labelled Hodge–Tate weight $h_{v}$ and set

$R_{\rho_{p}}^{\mathrm{cris},h}=\widehat{\bigotimes}_{v|p}R_{\rho_{v}}^{\mathrm{cris},h_{v}}.$

We further define

$\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{cris},h}=\widehat{\mathcal{X}^{p}}_{\rho^{p}}\times(\operatorname{\mathrm{Spf}}R_{\rho_{v}}^{\mathrm{cris},h_{v}})\times\widehat{\mathbb{U}^{g}}.$

Note that is follows from the definitions that
$\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{cris},h}$ embeds into
$\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},w_{0}}$ for any choice of a
refinement $\mathcal{R}$.

We set

$\mu_{v,\tau}=(h_{1,v,\tau},h_{2,v,\tau}+1,\dots,h_{n,v,\tau}+(n-1))=h_{v,\tau}-\delta^{\prime}_{G,v,\tau},$

and $\mu=(\mu_{v,\tau})_{v,\tau}$, which is thus antidominant (for the upper
Borel), and $\lambda=w_{0}(h)-\delta_{G}^{\prime}=w_{0}\cdot\mu\in
X^{*}(\underline{T})^{+}$. For all $v|p$ in $F$, we denote by $W_{v}$ the Weyl
group of ${\rm GL}_{n}(F_{v})$, which we identify with $\mathfrak{S}_{n}$ and
denote by $s_{1,v},\dots,s_{n-1,v}$ the simple reflections with respect to the
choice of the upper Borel $B_{v}\subset\operatorname{\mathrm{GL}}_{n,F_{v}}$.
Moreover, $w_{0,v}=s_{n-1,v}\dots s_{2,v}s_{1,v}s_{2,v}\dots s_{n-1,v}$ will
denote the longest element of $W_{v}$. We then write $W= \prod_{v}W_{v}$ the
Weyl group of
$\underline{G}_{\mathbb{Q}_{p}}\simeq\prod_{v|p}\operatorname{\mathrm{GL}}_{n,F_{v}}$
with respect to the Borel $B=\prod_{v|p}B_{v}$. Because of the product
structure, we will sometimes abuse notations and simply write $s_{i}$ for the
simple reflections and $w_{0}$ for the longest element.

For a scheme $X$ of dimension $d$ we write $Z^{0}(X)=Z_{d}(X)$ for the free
abelian group on the irreducible components of $X$. Moreover, for
$d^{\prime}\leqslant d$ we write $Z_{d^{\prime}}(X)$ for the free abelian
group on the irreducible and reduced closed subschemes of dimension
$d^{\prime}$. We recall that a coherent sheaf $\mathcal{F}$ on $X$ with
$d^{\prime}$-dimensional support defines a class $[\mathcal{F}]\in
Z_{d^{\prime}}(X)$, see e.g. [BHS19, Equation (2.13)].

###  7.1 Sheaves and supports.

Let $\lambda=w_{0}\cdot\mu\in X^{*}(\underline{T})^{+}$ dominant, integral.

We moreover write

$m_{x}=\dim\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))\otimes k(x).$ (8)

It follows from [BHS19, Thm. 5.1.3] that $m_{x}\geqslant 1$ and the proof of
[BHS19, Thm. 5.3.3] implies that $m_{x}$ does not depend on the choice of a
refinement $\mathcal{R}$. To $x$ and $\mathcal{R}$ we associate a permutation

$w_{x,\mathcal{R}}=(w_{x,\mathcal{R}_{v}})_{v\in\Sigma}=(w_{x,\mathcal{R}_{v},\tau})_{v,\tau}\in
W$

defined as in [HMS14, § 3.7]. We recall that these permutations encode the
relative position of the Hodge–Tate flags with respect to the full flag
corresponding to the refinement $\mathcal{R}$. We recall that, for any object
$M$ of $\mathcal{O}^{\infty}_{\mathrm{alg}}$ or
$\widetilde{\mathcal{O}}_{\mathrm{alg}}$, the sheaf
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is zero or Cohen–Macaulay of dimension
$d$.

######  Lemma 7.1.

Let $R$ be a Cohen–Macaulay noetherian local ring of dimension $d^{\prime}$
and let $M$ and $M^{\prime}$ be two finitely generated Cohen–Macaulay modules.
Let $(t_{1},\dots,t_{m})$ be a regular sequence of elements of the maximal
ideal of $R$ which is also $M$ and $M^{\prime}$-regular. Assume that
$[M]=[M^{\prime}]$ in $Z_{d^{\prime}}(\operatorname{\mathrm{Spec}}R)$. Then

$[M/(t_{1},\dots,t_{m})M]=[M^{\prime}/(t_{1},\dots,t_{m})M^{\prime}]\in
Z_{d^{\prime}-m}(R).$

###### Proof.

By induction it is sufficient to prove the result when $m=1$. Set $t=t_{1}$.
Let $\mathfrak{p}$ be a prime ideal of $R$ which is a generic point of
$\operatorname{\mathrm{Supp}}(M)$ or
$\operatorname{\mathrm{Supp}}(M^{\prime})$. It is sufficient to prove that
$[M_{\mathfrak{p}}/tM_{\mathfrak{p}}]=[M^{\prime}_{\mathfrak{p}}/tM^{\prime}_{\mathfrak{p}}]$
in $Z_{d^{\prime}-1}(\operatorname{\mathrm{Spec}}R_{\mathfrak{p}}/(t))$, i.e.
that $M_{\mathfrak{p}}/tM_{\mathfrak{p}}$ and
$M^{\prime}_{\mathfrak{p}}/tM^{\prime}_{\mathfrak{p}}$ are two
$R_{\mathfrak{p}}/(t)$-modules of the same length. This is a consequence of
[Sta24, Lemma 02QG]. ∎

Let $\mathcal{N}\subset\mathfrak{g}$ be the nilpotent cone and let
$\widetilde{\mathcal{N}}\rightarrow\mathcal{N}$ be the Springer resolution.
Similarly to the definition of the closed subschemes $X_{w}\subset X$ in 4.1
we define

$Z_{w}\subset\widetilde{\mathcal{N}}\times_{\mathcal{N}}\widetilde{\mathcal{N}}\subset
X$

to be the Zariski closure of preimage under
$\widetilde{\mathcal{N}}\times_{\mathcal{N}}\widetilde{\mathcal{N}}\rightarrow\underline{G}_{L}/\underline{B}\times\underline{G}_{L}/\underline{B}$
of the orbit
$\underline{G}_{L}(1,w)\subset\underline{G}_{L}/\underline{B}\times\underline{G}_{L}/\underline{B}$.
Set

$\mathcal{Z}_{w}=g(f^{-1}(Z_{w}\cap\widehat{X}_{I,w,x_{\mathrm{dR}}}))\times\widehat{\mathcal{X}}^{p}_{\rho^{p}}\times\widehat{\mathbb{U}}^{g}\subset\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}},$

where $f$ and $g$ are the maps from Theorem 4.7.

In the following we will make use of the following abusive notation for
(local) formal schemes: Let $\operatorname{\mathrm{Spf}}R$ be a (local) affine
formal scheme. Then we will say that $\operatorname{\mathrm{Spf}}R$ is
reduced, if $R$ is reduced. Moreover, we will say that
$\operatorname{\mathrm{Spf}}R$ is irreducible if ${\rm Spec}\,R$ is
irreducible. More generally, for a given irreducible component ${\rm
Spec}\,R/\mathfrak{a}\subset{\rm Spec}\,R$, we will refer to the formal
subscheme ${\rm Spf}\,R/\mathfrak{a}\subset{\rm Spf}\,R$ as an irreducible
component of ${\rm Spf}\,R$. Similarly, we will write
$Z^{0}({\operatorname{\mathrm{Spf}}}\,R)=Z^{0}({\operatorname{\mathrm{Spec}}}\,R)$
for the free abelian group on the irreducible components of
$\operatorname{\mathrm{Spf}}\,R$ to which we also refer as the irreducible
components of $\operatorname{\mathrm{Spf}}\,R$, etc.

######  Proposition 7.2.

Let $w\in W$. Then the following properties hold:

  1. 1)

For all $I\subset\Delta$ and all $\overline{w}\in W_{I}\backslash W$
satisfying $w^{\min}w_{0}\geqslant w_{x,\mathcal{R}}$ , the formal subscheme
$\mathcal{X}^{I-\mathrm{qtri},ww_{0}}_{\infty,x,\mathcal{R}}$ is reduced and
irreducible and coincides with an irreducible component of
$\mathcal{X}^{I-\mathrm{qtri}}_{\infty,x,\mathcal{R}}$.

  2. 2)

The schematic support of
$\mathcal{M}_{\infty,x,{\mathcal{R}}}(M(w\cdot\lambda))$, for $w\in W$, is
contained in
$\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},ww_{0}}$ if
$ww_{0}\geqslant w_{x,\mathcal{R}}$, and this sheaf is zero otherwise.
Moreover,

$[\mathcal{M}_{\infty,x,{\mathcal{R}}}(M(w\cdot\lambda))]=m_{x}[\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},ww_{0}}]\in
Z^{0}(\overline{\mathcal{X}}^{I-\mathrm{qtri}}_{\infty,x,\mathcal{R}})$

for $ww_{0}\geqslant w_{x,\mathcal{R}}$, where $m_{x}$ is the integer defined
by (8).

  3. 3)

There is an equality

$[\mathcal{M}_{\infty,x,\mathcal{R}}(L(ww_{0}\cdot\lambda))]=m_{x}\sum_{w^{\prime}\leqslant
w}a_{w,w^{\prime}}[\mathcal{Z}_{w}]\in
Z^{0}(\overline{\mathcal{X}}^{I-\mathrm{qtri}}_{\infty,x,\mathcal{R}})$

where the $a_{w,w^{\prime}}\in\mathbb{N}$ are the integers defined in [BHS19,
Thm. 2.4.7]. In particular $a_{w,w}=1$.

  4. 4)

For all $I\subset\Delta$, the sheaves

$\mathcal{M}_{\infty,x,{\mathcal{R}}}(M_{I}(w^{\rm min}\cdot\lambda))\
\text{and}\ \mathcal{M}_{\infty,x,{\mathcal{R}}}(M_{I}(w^{\rm
min}\cdot\lambda)^{\vee})$

are non zero if and only if $w^{\rm min}w_{0}\geqslant w_{x,\mathcal{R}}$.

  5. 5)

For all $I\subset\Delta$, the support of

$\mathcal{M}_{\infty,x,{\mathcal{R}}}(\widetilde{M}_{I}(w^{\rm
min}\cdot\lambda))\ \text{and}\
\mathcal{M}_{\infty,x,{\mathcal{R}}}(\widetilde{M}_{I}(w^{\rm
min}\cdot\lambda)^{\vee}),$

for $\overline{w}\in W_{I}\backslash W$, is
${\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},w^{\rm min}w_{0}}$ if
$w^{\rm min}w_{0}\geqslant w_{x,\mathcal{R}}$ and these sheaves are zero
otherwise.

  6. 6)

The module $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ is free of rank
$m_{x}$ over
$\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{cris},h}\subset\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},w_{0}}$.

  7. 7)

For any $I\subset\Delta$ and any $w\in W$, the sheaves

$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w^{\rm
min}\cdot\lambda))\ \text{and}\
\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w^{\rm
min}\cdot\lambda)^{\vee})$

are generically free of rank $m_{x}$ over their support.

###### Proof.

We first prove point 1). As $\mathcal{X}^{p}$ is smooth at $\rho^{p}$ (as
recalled in the begining of this section), the formal completion
$\widehat{\mathcal{X}^{p}}_{\rho^{p}}$ is formally smooth. As
$\widehat{\mathbb{U}^{g}}$ is also formally smooth, the claim follows from the
fact that

$\mathcal{X}_{r_{v},\mathcal{R}_{v}}^{I_{v}-\mathrm{qtri},\Box}\longrightarrow\mathcal{X}_{r_{v},\mathcal{R}_{v}}^{I_{v}-\mathrm{qtri}}\
\text{and}\
\mathcal{X}_{r_{v},\mathcal{R}_{v}}^{I_{v}-\mathrm{qtri},\Box}\longrightarrow\widehat{X_{I,x_{\mathrm{pdR}}}}$

are formally smooth and that $X_{I,w,x_{\mathrm{pdR}}}$ is an irreducible
component of $\widehat{X_{I,x_{\mathrm{pdR}}}}$.

By Theorem 6.16, the schematic support of the Cohen-Macaulay sheaves

$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w\cdot\lambda))\
\text{and}\
\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w\cdot\lambda)^{\vee})$

is contained in $\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},w}$ which
is irreducible. By Proposition 6.5, as the sheaves are Cohen–Macaulay of
dimension
$t+\dim\mathfrak{z}_{I}=\dim\mathcal{X}^{I-\mathrm{qtri}}_{\infty,x,\mathcal{R}}$
(e.g. [BHS19, equation (5.8)] and Proposition 3.20), we deduce that, if non
empty, their schematic support is all
$\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},w}$.

By Remark 6.3 we deduce also that

${\rm
supp}\big{(}\mathcal{M}_{\infty,x,\mathcal{R}}(M_{I}(w\cdot\lambda))\big{)}\subset\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},w^{\rm
min}w_{0}}$

for $w\in{}^{I}W$. Note that the Jordan–Hölder factors of
$M_{I}(w\cdot\lambda)$ are among the the $L(w^{\prime}\cdot\lambda)$ with
$w^{\prime}\geqslant w$ and that $L(w\cdot\lambda)$ is the cosocle of
$M_{I}(w\cdot\lambda)$. Therefore
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w\cdot\lambda))\neq 0$
if and only if $\mathcal{M}_{\infty,x,\mathcal{R}}(M_{I}(w\cdot\lambda))\neq
0$ if and only if $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w\cdot\lambda))\neq
0$. Therefore the non nullity assertions in 4 and 5 follow from the exactness
of $\mathcal{M}_{\infty,x,\mathcal{R}}$ (Proposition 6.5) and from [BHS19,
Thm. 5.3.3]. This proves 4 and 5

We prove point 6. By [BHS19, Remark 4.3.1 and Proof of Theorem 5.3.3, Step 7],
the schematic support of $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ is
contained in the crystalline locus
$\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{cris},h}\subset\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$,
which is smooth and irreducible of the same dimension as the support of
$\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$. Thus these coincide and
$\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))$ is free of rank $m_{x}$ over
the crystalline locus.

No we prove point 2. The first assertion has already been proved with 4 and 5,
therefore it remains to prove the assertion on the cycle. It follows from the
proof [BHS19, Thm. 5.3.3] that
$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda))$ is generically free of
rank $m_{x}$ for $ww_{0}\geqslant w_{x,\mathcal{R}}$. As
${\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},ww_{0}}}$ is
Cohen–Macaulay, the result is a consequence of point 5 and of Lemma 7.1
applied with

$M=\mathcal{O}_{\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},ww_{0}}}^{m_{x}}\
\text{and}\ M^{\prime}=\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda))$

and to a regular sequence generating the maximal ideal of
$U(\mathfrak{t})_{\mathfrak{m}}$. This sequence is $M^{\prime}$-regular by
Proposition 6.5.

We deduce 3 from 2 together with formulas (5.23) and (5.24) of [BHS19] and the
fact that the Verma modules form a basis of the Grothendieck group of the
category $\mathcal{O}_{\chi_{\lambda}}$.

We prove point 7. As
$\mathcal{X}^{I-\mathrm{qtri},w^{\prime}}_{\infty,x,\mathcal{R}}$ is
generically smooth for any $w^{\prime}$, the module
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is generically free, say of rank $r$,
over its support where

$M\in\\{\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w^{\rm
min}\cdot\lambda)),\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w^{\rm
min}\cdot\lambda)^{\vee})\\}.$

Now we claim that there exists an open an subset $U$ in the regular locus of
$\operatorname{\mathrm{Spec}}(R_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},w^{\rm
min}w_{0}})$ such that $U$ intersects the support of
$\mathcal{M}_{\infty,x,\mathcal{R}}(L(w^{\rm min}\cdot\lambda))$. The claim
then implies $r=m_{x}$. Indeed, the restriction of
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w^{\rm
min}\cdot\lambda))$ to $U$ is locally free since $U$ is regular. Therefore
$\mathcal{M}_{\infty,x,\mathcal{R}}(M_{I}(w^{\rm min}\cdot\lambda))$ is
locally free of rank $r$ over its support intersected with $U$. It follows
from the point 3 that
$\mathcal{M}_{\infty,x,\mathcal{R}}(L(w^{\prime}\cdot\lambda))$ is not
supported at the generic point of $\mathcal{Z}_{w^{\rm min}w_{0}}$ for
$w^{\prime}>w^{\rm min}$ and that $\mathcal{M}_{\infty,x,\mathcal{R}}(L(w^{\rm
min}\cdot\lambda))$ has length $m_{x}$ at the generic point of
$\mathcal{Z}_{w^{\rm min}w_{0}}$. As $L(w^{\rm min}\cdot\lambda)$ appears with
multiplicity one in $M_{I}(w^{\rm min}\cdot\lambda)$ and all other subquotient
are of the form $L(w^{\prime}\cdot\lambda)$ with $w^{\prime}>w^{\rm min}$, we
have $r=m_{x}$. We now construct an open subset $U$ with the claimed
properties. We set

$U=g(f^{-1}(V_{w^{\rm min}w_{0}}\cap\widehat{X}_{I,w^{\rm
min}w_{0},x_{\mathrm{dR}}}))\times\widehat{\mathcal{X}}^{p}_{\rho^{p}}\times\widehat{\mathbb{U}}^{g},$

where $f$ and $g$ are the maps of Theorem 4.7 and $V_{w^{\rm min}w_{0}}$ is
the preimage of the Schubert cell $\underline{G}_{L}(1,w^{\rm
min}w_{0})\subset\underline{G}_{L}/\underline{B}\times\underline{G}_{L}/\underline{B}$
in $X_{I,w^{\rm min}w_{0}}$. This is an open and smooth subset of $X_{I,w^{\rm
min}w_{0}}$: indeed, the maps $f$ and $g$ are formally smooth, the formal
scheme
$\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}\longrightarrow\mathcal{X}_{\rho_{p},\mathcal{R}}^{\mathrm{qtri}}$
is formally smooth and the point as $\rho^{p}$ lies in the smooth locus of
$\mathcal{X}^{p}$. ∎

######  Proposition 7.3.

Assume that $x_{\mathrm{pdR}}$ is a smooth point of $X_{w^{\rm min}w_{0}}$.
Then

$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w^{\rm min}\cdot\lambda))\ \text{and}\
\mathcal{M}_{\infty,x,\mathcal{R}}(M(w^{\rm min}\cdot\lambda)^{\vee})$

are finite free
$\mathcal{O}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},w^{min}w_{0}}}$-modules.

###### Proof.

We write $w^{\rm min}=w$ to simplify the notations. By Remark 6.3, the two
$U(\mathfrak{t})$-module structures on
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}(w\cdot\lambda))$ coming from
the $U(\mathfrak{t})$-action on $\widetilde{M}(w\cdot\lambda)$ and the one
coming from the derivative of the locally analytic action, coincide. Thus we
have the equality between
$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda))$ and the localisation

$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda))\simeq
i_{*}i^{*}\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}(w\cdot\lambda)),$

where $i:\widehat{T}^{\rm sm}\longrightarrow\widehat{T}$ denotes the inclusion
of the closed subspace of smooth characters. A similar remark applies to the
dual Verma module. In particular, it is enough to show that the
$\mathcal{O}_{{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},ww_{0}}}$-modules

$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}(w\cdot\lambda))\ \text{and}\
\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}^{\vee}(w\cdot\lambda))$

are finite free. But these modules are Cohen-Macaulay with support the
localization at $x$ of $\mathcal{X}_{\infty,x,\mathcal{R}}^{{\rm
qtri},ww_{0}}$, which is smooth. ∎

###  7.2 Recollection on Bezrukavnikov’s functor

The aim if this section (or even of the paper) is to identify the patching
functor that takes objects in $\mathcal{O}_{\rm alg}$ (or more generally in
$\mathcal{O}_{\rm alg}^{\infty}$) to Cohen-Macaulay modules on certain Galois
deformation rings with a functor constructed by Bezrukavnikov in geometric
representation theory (more precisely: with the pullback from our local models
to the Galois deformation rings). Before doing so, we will need to recall the
result of Bezrukavnikov.

Recall that
$X=\widetilde{\mathfrak{g}}\times_{\mathfrak{g}}\widetilde{\mathfrak{g}}$
where $\mathfrak{g}$ is the Lie algebra of
$\underline{G}_{L}=\prod_{v\in\Sigma}(L\times_{\mathbb{Q}_{p}}\operatorname{\mathrm{Res}}_{F_{v}/\mathbb{Q}_{p}}\operatorname{\mathrm{GL}}_{n})$
as in section 4.1 and denote by $X^{\wedge}$ the completion of $X$ along the
preimage of $\\{(0,0)\\}\in\mathfrak{t}\times_{\mathfrak{t}/W}\mathfrak{t}$ in
$X$. Moreover, we write $\overline{X}=X\times_{\mathfrak{t}}\\{0\\}$, where
the fiber product is taken with respect to the map
$\kappa_{1}:X\longrightarrow\mathfrak{t}$ of 4.1 that maps
$(g\underline{B},h\underline{B},N)$ to
$\mathrm{ad}(g^{-1})(N)\pmod{\mathfrak{n}}\in\mathfrak{t}$. As in the
preceding sections we fix the shift

$\delta^{\prime}_{\underline{G}}={\rm
det}^{\tfrac{1-n}{2}}\delta_{\underline{G}}\in X^{\ast}(\underline{T})$

of the half sum of the positive roots $\delta_{\underline{G}}$.

######  Theorem 7.4 (Bezrukavnikov).

Let $\lambda\in X^{*}(\underline{T})$ be a dominant character. There exists an
exact functor

$\mathcal{B}:{\mathcal{O}_{\chi_{\lambda}}}\longrightarrow\operatorname{\mathrm{Coh}}^{\underline{G}_{L}}(X^{\wedge}),$

such that

  1. 1)

for all $M\in\mathcal{O}_{\chi_{\lambda}}$ the sheaf $\mathcal{B}(M)$ is a
Cohen-Macaulay sheaf,

  2. 2)

for all $w\in W$ there is an isomorphism
$\mathcal{B}(M(ww_{0}\cdot\lambda)^{\vee})\simeq\mathcal{O}_{\overline{X_{w}}}$,

  3. 3)

for all $w\in W$ there is an isomorphism
$\mathcal{B}(M(ww_{0}\cdot\lambda))\simeq\omega_{\overline{X_{w}}}$,

  4. 4)

the image $\mathcal{B}(P(w_{0}\cdot\lambda))$ of the anti-dominant projective
$P(w_{0}\cdot\lambda)$ is the structure sheaf $\mathcal{O}_{\overline{X}}$,

  5. 5)

the image $\mathcal{B}(L(\lambda))$ of the algebraic representation
$L(\lambda)$ is the line bundle
$\mathcal{O}(-\delta^{\prime}_{\underline{G}})\boxtimes\mathcal{O}(-\delta^{\prime}_{\underline{G}})$
on $\underline{G}_{L}/\underline{B}\times\underline{G}_{L}/\underline{B}$
which is viewed as a closed subscheme of $X^{\wedge}$ via

$(g\underline{B},h\underline{B})\mapsto(g\underline{B},h\underline{B},0).$

This result is (a small part of a result) due to Bezrukavnikov and his
collaborators whose proof is spread out through the papers [Bez16, BR12, BL23,
BR22]). For the convenience of the reader, we explain how to get the result in
the previous form.

###### Proof.

By the main result of [Bez16], there are reverse equivalence of categories

$\Psi:D_{I^{0},I^{0}}\leftrightarrow
D^{b}(\operatorname{\mathrm{Coh}}(\widetilde{\mathfrak{g}}\times_{\mathfrak{g}}\widetilde{\mathfrak{g}})):\Phi_{I^{0},I^{0}},$

which we can then localize on $X^{\wedge}\subset X$. Up to use translation
functors, we can focus on the case $\lambda=0$. By [Bez16, Corollary 42 ] the
functor $\Psi$ in fact takes values in ($\underline{G}$-equivariant) coherent
sheaves on $X$, when restricted to perverse sheaves
$F\in\operatorname{Perv}_{\underline{N}}(\underline{G}/\underline{B})$.
Moreover, the Beillinson–Bernstein localization theorem, more precisely by
[BG99] Localization Theorem 2.2, provides an exact fully faithfull embedding
of categories

$\mathcal{O}_{\chi_{0}}\longrightarrow\operatorname{Perv}_{\underline{N}}(\underline{G}/\underline{N}).$

Composing the Beillinson–Bernstein equivalence with Bezrukavnikov’s functor
(noting that the blocks $\mathcal{O}_{\chi_{0}}$ and
$\mathcal{O}_{\chi_{\lambda}}$ are equivalent) we get the exact functor
$\mathcal{B}$.

Denote $\mu=w_{0}\cdot\lambda$ denote the antidominant weight in the dot-orbit
of $\lambda$. Now the proof of [BL23, Proposition 5.8] implies that
$\mathcal{B}({M}(s\cdot\mu)^{\vee})=\mathcal{O}_{\overline{X_{s}}}$ for all
simple reflection $s$ and $\mathcal{B}(P(\mu))=\mathcal{O}_{\overline{X}}$.
Bezrukavnikov’s main result [Bez16, Theorem 1] implies that $\Psi$ (hence
$\mathcal{B}$) intertwines the convolutions on both sides. Here the
convolution on the category
$\mathcal{O}_{\chi_{\lambda}}\simeq\mathcal{O}_{\chi_{0}}$ is inherited from
the convolution in
$\operatorname{Perv}_{\underline{N}}(\underline{G}/\underline{B})$ defined as
in [BR22, 7.]. We write $w=s_{1}\dots s_{r}$ and compute convolutions on both
sides. By [BR12, Theorem 2.2.1] we have

$
\mathcal{O}_{\overline{X_{w}}}=\mathcal{O}_{\overline{X_{s_{1}}}}\star\dots\star\mathcal{O}_{\overline{X_{s_{r}}}}.$

By [BR22, Lemma 7.7] we have
${M}(w\cdot\mu)^{\vee}={M}(s_{1}\cdot\mu)^{\vee}\star\dots\star{M}(s_{r}\cdot\mu)^{\vee}$
and hence $\mathcal{B}({M}(w\cdot\mu)^{\vee})=\mathcal{O}_{\overline{X_{w}}}$.
Moreover, by [BR12, Theorem 2.2.1] again, the dualizing sheaf of
$\overline{X_{w}}$ is given by the convolution

$
\omega_{\overline{X_{w}}}=\omega_{\overline{X_{s_{1}}}}\star\dots\star\omega_{\overline{X_{s_{r}}}}.$

But [BR12, Proposition 1.10.3] implies that the inverse of
$\mathcal{O}_{\overline{X_{s}}}$ for the convolution is
$\omega_{\overline{X_{s}}}$, and as $\mathcal{B}$ is compatible with
convolution, and as the inverse of ${M}(s\cdot\mu)^{\vee}$ is ${M}(s\cdot\mu)$
(again using [BR22, Lemma 7.7]for example), we deduce
$\omega_{\overline{X_{s}}}=\mathcal{B}(M(s\cdot\mu))$. Finally 5. is a
consequence of [BL23, Lemma 6.7] (with $P=\underline{G}$). ∎

Recall that we have fixed a point $x\in\mathcal{X}_{\infty}$ associated which
we have defined the positive integer $m_{x}$ in (8).

######  Corollary 7.5.

The functor $\mathcal{B}$ induces an exact functor

$\mathcal{B}_{x}:\mathcal{O}_{\chi_{\lambda}}\longrightarrow\operatorname{\mathrm{Coh}}(\mathcal{X}_{\infty,x,\mathcal{R}}^{\rm
qtri})$

such that, for all $M\in\mathcal{O}_{\chi_{\lambda}}$ the sheaf
$\mathcal{B}_{x}(M)$ is a Cohen-Macaulay sheaf and such that

$ [\mathcal{M}_{\infty,x,\mathcal{R}}(M)]=m_{x}[\mathcal{B}_{x}(M)]\in
Z^{0}(\overline{\mathcal{X}}^{I-\mathrm{qtri}}_{\infty,x,\mathcal{R}}).$

###### Proof.

Let $\underline{G}_{1}$ be the completion of $\underline{G}$ at the unit
element. As the representations $(\rho_{v})_{v|p}$ defined by the point $x$
are crystalline and hence de Rham we may choose a basis $\alpha$ of
$W(x)=\prod_{v\in\Sigma}W_{\rm dR}(D_{\rm rig}(\rho_{x,v})[1/t])$ and define a
point $x_{\rm pdR}$ associated to $x$ (or rather to the representations
$(\rho_{v})_{v|p}$) as in (6). For all $M\in\mathcal{O}_{\chi_{\lambda}}$, the
sheaf $\mathcal{B}(M)$ is a $\underline{G}_{L}$-equivariant sheaf on
$X^{\wedge}$ and hence gives rise to a $\underline{G}_{1}$-equivariant sheaf
on $\widehat{X}_{x_{\mathrm{pdR}}}$. Now by [BHS19, Theorem 3.4.4. and
Corollary 3.5.8], see also Theorem 4.7 above, we have a diagram

${\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},\Box}}$${\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}}$${X^{\wedge}_{x_{\mathrm{pdR}}}.}$$\scriptstyle{\pi}$$\scriptstyle{W}$

More precisely, the map $\pi$ forgets the deformation of the fixed basis
$\alpha$, and hence it is a $\underline{G}_{1}$-torsor. Moreover, $W$ formally
smooth and $\underline{G}_{1}$-equivariant for the natural left actions
$g\cdot\widetilde{\alpha}:=\widetilde{\alpha}\circ g^{-1}$ on the source
(acting only on the deformation of the isomorphisms
$\alpha_{v}:L\otimes_{\mathbb{Q}_{p}}F_{v}\overset{\sim}{\longrightarrow}W_{v}$)
and $g\cdot(kB,hB,N)=(gkB,ghB,g^{-1}Ng)$ on the target of $W$.

It follows that the pullback of $\mathcal{B}(M)^{\wedge}_{x_{\mathrm{pdR}}}$
at $\widehat{X}_{x_{\mathrm{pdR}}}$ along $W$ is a
$\underline{G}_{1}$-equivariant sheaf and hence descends to a coherent sheaf

$\mathcal{B}_{x}(M)\in{\rm
Coh}(\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}).$

It follows from the construction that $M\mapsto\mathcal{B}_{x}(M)$ and that
$\mathcal{B}_{x}(M)$ is Cohen-Macaulay, as $\mathcal{B}(M)$ is. Moreover,
$\mathcal{B}_{x}$ is exact, as $W$ is formally smooth and hence flat.

It remains to check the assertion on cycles. But as taking cycles is additive
and $\mathcal{B}_{x}$ is exact, we only need to check this equality on a
generating set of the Grothendieck group of $\mathcal{O}_{\chi_{\lambda}}$,
such as the Verma modules $M(w\cdot\mu)$. Hence the desired equality follows
from the previous result on Bezrukavnikov’s functor together with Proposition
7.2. ∎

###  7.3 A detail study of local models when $n=3$

From now on we assume $n=3$, so that the group $\underline{G}_{L}$ is

$\underline{G}_{L}\simeq(\operatorname{\mathrm{Res}}_{F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}/\mathbb{Q}_{p}}\operatorname{\mathrm{GL}}_{3})\times_{\mathbb{Q}_{p}}
L\simeq\prod_{v\in
S_{p}}(L\times_{\mathbb{Q}_{p}}\operatorname{\mathrm{Res}}_{F_{v}/\mathbb{Q}_{p}}\operatorname{\mathrm{GL}}_{3,F_{v}})\simeq\prod_{\tau\in\Sigma_{F}}\operatorname{\mathrm{GL}}_{3,L}.$

We identify the previous local Weyl group $W$ with $\prod_{\tau}W_{\tau}$ and
each $W_{\tau}$ with
$W_{\operatorname{\mathrm{GL}}_{3}}\simeq\mathfrak{S}_{3}$ and denote
$s_{1,\tau},s_{2,\tau}$ the two simple reflection corresponding to the choice
of the upper Borel, and $w_{0,\tau}=s_{1,\tau}s_{2,\tau}s_{1,\tau}$ the
longuest element in $W_{\tau}$. If $\tau$ is understood, we often omit it from
the notation.

As in section 4.1 we denote by $X$ the Steinberg variety for the group

$\underline{G}=\operatorname{\mathrm{Res}}_{F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}/\mathbb{Q}_{p}}\operatorname{\mathrm{GL}}_{3},$

over $L$. As $L$ is assumed to contain all Galois conjugates of $F$ we have
$X\simeq\prod_{\tau\in\Sigma_{F}}X_{3}$ (see Remark 4.6 for the notation
$X_{3}$). The Steinberg variety $X$ (resp. $X_{3}$) has dimension
$9^{\lvert\Sigma_{F}\rvert}$ (resp. $9$) and $6^{\lvert\Sigma_{F}\rvert}$
(resp. $6$) irreducible components $X_{w},w\in W$ (resp.
$X_{3,w},w\in\mathfrak{S}_{3}$), see e.g. [BHS19, Proposition 2.2.5].

######  Proposition 7.6.

For $w=(w_{\tau})_{\tau\in\Sigma_{F}}$, let $s=\lvert\\{\tau\in\Sigma_{F}\mid
w_{\tau}=w_{0}\\}\rvert$. Then the component $X_{w}$ is smooth if and only if
$s=0$. Moreover, if $s\neq 0$, then the component $X_{w}$ is Cohen–Macaulay
but not Gorenstein. More precisel, let

$
x_{\mathrm{pdR}}=(g\underline{B},h\underline{B},N)=(g_{\tau}\underline{B}_{\tau},N_{\tau},h_{\tau}\underline{B}_{\tau})\in
X_{w}(L)=\prod_{\tau\in\Sigma_{F}}X_{3,w_{\tau}}(L),$

and assume that $N_{\tau}=0$ when $w_{\tau}=w_{0}$. Then

$\dim_{L}\omega_{X_{w}}\otimes k(x_{\mathrm{pdR}})=2^{r},$

where $r\coloneqq\lvert\\{\tau\mid w_{\tau}=w_{0},\text{ and
}g_{\tau}\underline{B}_{\tau}=h_{\tau}\underline{B}_{\tau}\\}\rvert$.

###### Proof.

The smoothness is a consequence of Proposition 4.1. As
$X=\prod_{\tau\in\Sigma_{F}}X_{3}$, it is enough to prove the analogous result
for $X_{3}$ only. Indeed, by base change and composition of upper shriek
functors, the dualizing sheaf of $X$ is a derived tensor product
$\bigotimes^{\mathbb{L}}_{\tau}p_{\tau}^{*}\omega_{X_{3}}$, where
$p_{\tau}:X\longrightarrow X_{3}$ is projection to the $\tau$-component. But
as the product $X=\prod_{\tau}X_{3}$ is a product over a field, we find

$\omega_{X}=\bigotimes_{\tau}p_{\tau}^{*}\omega_{X_{3}}.$

Thus from now on we denote $X_{3}$ simply by $X$.

It is thus enough to prove that the fiber of $\omega_{X_{w_{0}}}$, is
2-dimensional at a point of the form $(g\underline{B},0,g\underline{B})$. Let
$q:\widetilde{\mathfrak{g}}\longrightarrow\mathfrak{g}$ denote the
Grothendieck resolution, then
$X\simeq\underline{G}_{L}\times^{\underline{B}}q^{-1}(\mathfrak{b})$.
Moreover, $Y\coloneqq q^{-1}(\mathfrak{b})$ decomposes into irreducible
components $Y=\bigcup_{w\in W}Y_{w}$ such that
$X_{w}\simeq\underline{G}_{L}\times^{\underline{B}}Y_{w}$. Hence it is enough
to prove that $\omega_{Y_{w_{0}}}$ has fiber dimension $2$ at the point
$y_{\rm pdR}=(\underline{B},0)$. As $X_{w_{0}}$ is Cohen-Macaulay and flat
over $\mathfrak{t}$ (cf [BHS19, Proposition 2.2.3]), we have the base change
formula
$\omega_{X_{w_{0}}}\otimes_{X}\overline{X}\simeq\omega_{\overline{X_{w_{0}}}}$.
We are thus reduced to compute the dualizing sheaf
$\omega_{\overline{Y_{w_{0}}}}$ of the irreducible component

$\overline{Y_{w_{0}}}=Y_{w_{0}}\times_{\mathfrak{t}}\\{0\\}$

of $\overline{Y}=q^{-1}(\mathfrak{n})$. This scheme now has dimension $3$ and
we can use explicit computations.

A point of $\overline{Y}(L)$ is of the form
$(g\underline{B},N)\in(\underline{G}/\underline{B}\times\mathfrak{g})(L)$. We
use the embedding
$\underline{G}/\underline{B}\hookrightarrow\mathbb{P}_{L}^{2}\times(\mathbb{P}_{L}^{2})^{\vee}$
that sents a full flag $(0\subset\mathcal{L}\subset\mathcal{P}\subset k^{3})$
to $(\mathcal{L}\subset k^{3},\mathcal{P}\subset k^{3})$. In homogeneous
coordinates $([x_{0}:x_{1}:x_{2}],[y_{0}:y_{1}:y_{2}])$ the condition
$\mathcal{L}\subset\mathcal{P}$ is given by
$x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}=0$. Let
$\overline{Y}^{0}\subset\overline{Y}$ denote the open subset defined by the
condition $x_{0}=y_{2}=1$. It is enough to compute on this open subset, as
this is a neighborhood of the point $y_{\rm
pdR}=(\underline{B},0)=([1:0:0],[0:0:1])$. On $\overline{Y}^{0}$ we can thus
remove $y_{0}$ from our equations. Let us write

$N=\left(\begin{array}[]{ccc}0&u_{12}&u_{13}\\\ &0&u_{23}\\\
&&0\end{array}\right)$

for the universal matrix over $\overline{Y}^{0}$. The ideal defining

$\overline{Y}^{0}_{w_{0}}\subset
Z\coloneqq\operatorname{\mathrm{Spec}}(k[x_{1},x_{2},y_{1},u_{12},u_{23},u_{13}])$

is then given by

$I_{w_{0}}=(u_{23}x_{2},u_{12}(x_{2}+x_{1}y_{1}),u_{12}x_{1}+u_{13}x_{2},u_{23}y_{1}-u_{13}(x_{2}+x_{1}y_{1})).$

We remark that we can replace $u_{12}(x_{2}+x_{1}y_{1})$ by
$u_{12}x_{2}-x_{13}x_{2}y_{1}$ using the third equation, and that
automatically $y_{1}u_{12}u_{23}=0$ using our new equation and
$u_{23}y_{1}-u_{13}(x_{2}+x_{1}y_{1})=0$. We then check (e.g. using Macaulay2)
that

$0\longrightarrow
\mathcal{O}_{Z}^{2}\overset{A^{\prime}}{\longrightarrow}\mathcal{O}_{Z}^{6}\overset{A}{\longrightarrow}\mathcal{O}_{Z}^{5}\overset{A^{\prime\prime}}{\longrightarrow}\mathcal{O}_{Z}$

is a resolution of $\mathcal{O}_{Z}/I_{w_{0}}$, where

$A^{\prime}=\left(\begin{array}[]{cc}y_{1}&y_{1}u_{13}-u_{12}\\\ -x_{2}&0\\\
x_{1}&u_{23}\\\ 0&-u_{12}u_{23}\\\ 0&-x_{2}u_{23}\\\
0&x_{1}u_{12}+x_{2}u_{12}\end{array}\right)\ ,\
A^{\prime\prime}=\left(\begin{array}[]{c}x_{1}u_{12}+x_{2}u_{13}\\\
x_{2}u_{23}\\\ y_{1}u_{12}u_{23}\\\
x_{1}y_{1}u_{13}-y_{1}u_{23}+x_{2}u_{13}\\\
x_{2}y_{1}u_{13}-x_{2}u_{12}\end{array}\right)^{t}$

$A=\begin{pmatrix}-x_{2}u_{23}&-y_{1}u_{23}&0&x_{2}&-y_{1}u_{13}&0\\\
x_{1}u_{12}+x_{2}u_{13}&y_{1}u_{13}&-y_{1}u_{12}&-y_{1}&0&-y_{1}u_{13}+u_{12}\\\
0&x_{1}&x_{2}&0&1&0\\\ 0&0&0&{-x_{2}}&u_{12}&0\\\
0&0&0&x_{1}&u_{13}&u_{23}\end{pmatrix}.$

Let $i:\overline{Y}^{0}_{w_{0}}\hookrightarrow Z$ denote the canonical closed
embedding. Then the dualizing sheaf can be computed as
$\omega_{\overline{Y}^{0}_{w_{0}}}=i^{*}{\rm
Ext}^{3}_{\mathcal{O}_{Z}}(\mathcal{O}_{\overline{Y^{0}_{w_{0}}}},\mathcal{O}_{Z})$
which is given by

$\omega_{\overline{Y}^{0}_{w_{0}}}\simeq\mathcal{O}_{Z}^{2}/<(y_{1},y_{1}u_{13}-u_{12}),(x_{2},0),(x_{1},u_{12}),(0,u_{12}u_{23})>,$

as $x_{2}u_{23}=x_{2}u_{12}+x_{2}u_{12}=0$ on $\overline{Y}^{0}_{w_{0}}$. It
follows that the fiber of $\omega_{\overline{Y}^{0}_{w_{0}}}$ at
$y_{\mathrm{pdR}}$ is $2$-dimensional. ∎

######  Lemma 7.7.

Let $J\subset\Delta_{\operatorname{\mathrm{GL}}_{3}}$.

  1. 1.

For $w\in W(\operatorname{\mathrm{GL}}_{3})\simeq\mathfrak{S}_{3}$ the
component $X_{3,w}$ is smooth if $w\neq w_{0}$.

  2. 2.

If $x_{\mathrm{pdR}}=(g\underline{B}_{3},h\underline{B}_{3},0)\in
X_{3,w_{0}}(L)$, with $g\underline{B}_{3}\neq h\underline{B}_{3}$, then
$x_{\mathrm{pdR}}$ is a smooth point of $X_{3,w_{0}}$.

  3. 3.

For $\emptyset\neq
J\subset\Delta_{\operatorname{\mathrm{GL}}_{3}}=\\{s_{1},s_{2}\\}$ the
component $X_{3,J,\overline{w}}$ is smooth for any $\overline{w}\in
W_{J}\backslash W_{\operatorname{\mathrm{GL}}_{3}}$.

###### Proof.

Point 1 is Proposition 7.6. For the point 2, denote $w^{\prime}$ the index of
the Schubert stratum in which $x_{pdR}$ lies. By [BHS19, Proposition
2.5.3(ii)] it is thus enough (as
$\overline{U_{w_{0}}}=\operatorname{\mathrm{GL}}_{3}/\underline{B}_{3}\times\operatorname{\mathrm{GL}}_{3}/\underline{B}_{3}$
is smooth) to prove that
$\operatorname{\mathrm{codim}}_{\mathfrak{t}}(\mathfrak{t}^{w_{0}w{{}^{\prime}-1}})=\lg(w_{0})-\lg(w^{\prime})$.
But this codimension is what we have denoted $\ell(w_{0}w^{{}^{\prime}-1})$ in
the proof of Proposition 4.1. As $w^{\prime}\neq 1$ and $n=3$,
$w_{0}w^{{}^{\prime}-1}$ is a product of distinct simple reflections thus
$\ell(w_{0}w^{{}^{\prime}-1})=\lg(w_{0}w^{{}^{\prime}-1})=\lg(w_{0})-\lg(w^{\prime})$.
For point 3, as $n=3$ we have that $J= \\{s_{1}\\},\\{s_{2}\\}$ or
$J=\\{s_{1},s_{2}\\}$. Denote $\underline{P}=\underline{P}_{J}$. In the case
$J=\\{s_{1},s_{2}\\}$, then $\underline{P}_{J}=\operatorname{\mathrm{GL}}_{3}$
and $X_{3,J}=\widetilde{\mathfrak{g}}$ is smooth. It is sufficient to prove
the case of $J=\\{s_{1}\\}$ (the other case is exactly the same), where an
explicite computation gives the smoothness (alternatively, when $w^{min}$ has
length $\leqslant 1$, [BD, Corollary 5.3.4] also implies smoothness). ∎

######  Corollary 7.8.

Let $w=(w_{\tau})_{\tau}\in W$ and let
$I=\coprod_{\tau}I_{\tau}\subset\Delta$. Let
$x_{\mathrm{pdR}}=(x_{\mathrm{pdR},\tau})_{\tau}=(g_{\tau}\underline{B}_{\tau},h_{\tau}\underline{B}_{\tau},N_{\tau})$
be a point such that $N_{\tau}=0$ whenever $I_{\tau}=\emptyset,w_{\tau}=1$. If

$\mathcal{M}_{\infty,x,\mathcal{R}}(M_{I}(w^{\rm
min}\cdot\lambda))\quad\text{(resp.
}\mathcal{M}_{\infty,x,\mathcal{R}}(M_{I}(w^{\rm min}\cdot\lambda)^{\vee})),$

is not a finite free over
${\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},w^{\rm
min}w_{0}}}$-module, then there exists an embedding $\tau$ such that
$I_{\tau}=\emptyset$, $w_{\tau}=1$ and $w_{x,\mathcal{R},\tau}=1$.

###### Proof.

Assume that there is no $\tau$ such that $I_{\tau}=\emptyset$ and
$w_{\tau}=w_{x,\mathcal{R},\tau}=1$. Lemma 7.7, then shows that the local
model $X_{I}$ is smooth at $x_{\mathrm{pdR}}$. By 7.2 the support

$\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},ww_{0}}={\rm supp}\
\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w\cdot\lambda))$

is smooth. Thus
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w\cdot\lambda))$ is a
free of rank $m_{x}$ over
${\mathcal{X}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},ww_{0}}}$. By Remark 6.3
its follows that $\mathcal{M}_{\infty,x,\mathcal{R}}({M}_{I}(w\cdot\lambda))$
is a free of rank $m_{x}$ over
${\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{I-\mathrm{qtri},ww_{0}}}$.

The same argument also applies to
$\mathcal{M}_{\infty,x,\mathcal{R}}(\widetilde{M}_{I}(w\cdot\lambda))$. ∎

######  Proposition 7.9.

For all $w\in W$ the sheaf $\mathcal{B}_{x}(L(w\cdot\lambda))$ is cyclic.
Moreover, for all $w\in W$ such that $ww_{0}\geqslant w_{x,\mathcal{R}}$ the
sheaf $\mathcal{M}_{\infty}(L(w\cdot\lambda))$ is free of rank $m_{x}$ over
its support.

###### Proof.

Recall that, for $w\in W$, $Z_{w}$ is the closure in
$\widetilde{\mathcal{N}}\times_{\mathcal{N}}\widetilde{\mathcal{N}}$ of the
preimage $V_{w}$ of the Bruhat Cell
$U_{w}=\underline{G}_{L}(1,w)\subset\underline{G}_{L}/\underline{B}\times\underline{G}_{L}/\underline{B}$.
By [CG10, Prop. 3.3.4], $V_{w}$ can be identified with the conormal bundle of
$U_{w}$ in $\widetilde{\mathcal{N}}\times\widetilde{\mathcal{N}}\simeq
T^{*}(\underline{G}_{L}/\underline{B}\times\underline{G}_{L}/\underline{B})$.
As $\mathfrak{g}$ is isomorphic to direct sum of copies of
$\mathfrak{gl}_{3}$, the closure $\overline{U_{w}}$ of $U_{w}$ in
$\underline{G}_{L}/\underline{B}\times\underline{G}_{L}/\underline{B}$ is
smooth, hence a local complete intersection. This proves that the conormal
bundle of $\overline{U_{w}}$ is a closed smooth subscheme of
$\widetilde{\mathcal{N}}\times\widetilde{\mathcal{N}}$ containing $V_{w}$ as
an open dense subset so that it coincides with $Z_{w}$ and $Z_{w}$ is smooth.
This implies that $\mathfrak{Z}_{w}$ is a smooth. As
$\mathcal{M}_{\infty,x,\mathcal{R}}(L(ww_{0}\cdot\lambda))$ is Cohen–Macaulay,
it follows from Proposition 7.2 3 and from the fact that $a_{w,w^{\prime}}=0$
for $w\neq w^{\prime}$ (see [BHS19, Rk. 2.4.5]) that the sheaf
$\mathcal{M}_{\infty,x,\mathcal{R}}(L(ww_{0}\cdot\lambda))$ is locally free
over its support. ∎

###  7.4 The case of dual Vermas

For later use, let us recall the following Lemma.

######  Lemma 7.10.

Let $R$ be a commutative local ring and let $I\subset J$ two ideals of $R$.
Let $m\geqslant 1$ and $\pi:(R/I)^{m}\longrightarrow(R/J)^{m}$ a surjective
$R$-linear map. Then there exist isomorphisms

$\varphi:(R/J)^{m}\longrightarrow(R/J)^{m},\quad\psi:(R/I)^{m}\longrightarrow(R/I)^{m}$

such that $\varphi\circ\pi=\pi\circ\psi=\mathrm{can}^{\oplus m}$ where
$\mathrm{can}:R/I\longrightarrow R/J$ is the quotient map.

###### Proof.

Let $(e_{1},\dots,e_{m})$ be the standard basis of $(R/I)^{m}$ as an
$(R/I)$-module and $(f_{1},\dots,f_{m})$ the standard basis of $(R/J)^{m}$.
Then $(\pi(e_{1}),\dots,\pi(e_{m}))$ is a generating family of $(R/J)^{m}$. As
a surjective endomorphism of a module is bijective, we see that
$(\pi(e_{1}),\dots,\pi(e_{m}))$ is also a basis of $(R/I)^{m}$. Therefore we
can define $\varphi$ by the formula $\varphi(\pi(e_{i}))=f_{i}$. Now, for any
$1\leqslant i\leqslant m$, let $f^{\prime}_{i}\in(R/I)^{m}$ such that
$\pi(f_{i}^{\prime})=f_{i}$. By Nakayama Lemma the family
$(f^{\prime}_{1},\dots,f^{\prime}_{m})$ generates $(R/I)^{m}$ and so is a
basis of $(R/I)^{m}$. We can therefore define $\psi$ by the formula
$\psi(e_{i})=f_{i}^{\prime}$. ∎

We will use the previous Corollary 7.8 to start a devissage which will be
assured by the following two Lemmas.

######  Lemma 7.11.

Let $M$ be an object of $\mathcal{O}_{\chi_{\lambda}}$ and let
$Q_{1},\dots,Q_{r}$ be quotients of $M$. Let $Q$ be the smallest quotient of
$M$ dominating all the $Q_{i}$, i.e. $Q=M/(M_{1}\cap\cdots\cap M_{r})$ where
$M_{i}=\operatorname{\mathrm{Ker}}(M\rightarrow Q_{i})$ for $1\leqslant
i\leqslant r$. We assume that

  1. (i)

for any $1\leqslant i\leqslant r$, the sheaf
$\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{i})$ is free of rank $m_{x}$ over it
support;

  2. (ii)

for any $1\leqslant i\leqslant r$, the sheaf $\mathcal{B}_{x}(Q_{i})$ is
cyclic (generated by one element);

  3. (iii)

for any $1\leqslant i\leqslant r$,
$\operatorname{\mathrm{Supp}}\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{i})=\operatorname{\mathrm{Supp}}\mathcal{B}_{x}(Q_{i})$
;

  4. (iv)

the sheaf $\mathcal{B}_{x}(Q)$ is cyclic.

Then the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is free of rank $m_{x}$
over its support and

$\operatorname{\mathrm{Supp}}(\mathcal{M}_{x,\infty,\mathcal{R}}(Q)=\operatorname{\mathrm{Supp}}(\mathcal{B}_{x}(Q)).$

###### Proof.

To ease notation we note $m=m_{x}$. Let’s prove the result when $r=2$. Let
$A=\overline{R}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$ be the ring of global
sections of $\overline{\mathcal{X}}^{\mathrm{qtri}}_{\infty,x,\mathcal{R}}$
and let $I_{i}=\mathrm{Ann}(\mathcal{B}_{x}(Q_{i}))$ for $i\in\\{1,2\\}$.
Define $Q_{0}$ the largest common quotient of $Q_{1}$ and $Q_{2}$, i.e.
$Q_{0}=M/(M_{1}+M_{2})$. Then we have a short exact sequence

$0\longrightarrow Q\longrightarrow Q_{1}\oplus Q_{2}\longrightarrow
Q_{0}\longrightarrow 0.$

By exactness of $\mathcal{M}_{\infty,x,\mathcal{R}}$, we have a short exact
sequence

$0\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(Q)\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{1})\oplus\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{2})\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{0})\longrightarrow
0$

where the map $Q_{1}\oplus Q_{2}\rightarrow Q_{0}$ is given by $(x,y)\mapsto
x-y$.

We fix isomorphisms
$(A/I_{i})^{m}\xrightarrow{\sim}\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{i})$ for
$i\in\\{1,2\\}$. As $Q_{0}$ is a quotient of both $Q_{1}$ and $Q_{2}$, we have
surjective maps

$(A/I_{i})^{m}\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{i})\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{0}),$

which factor through $(A/(I_{1}+I_{2}))^{m}$. Using Lemma 7.10 we can choose
the previous isomorphisms such that the following diagram commutes

${(A/I_{1})^{m}\oplus(A/I_{2})^{m}}$${A/(I_{1}+I_{2})^{m}}$${0}$${\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{1})\oplus\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{2})}$${\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{0})}$${0.}$$\scriptstyle{(x{,}y)\mapsto
x-y}$$\scriptstyle{\simeq}$ (9)

As the kernel of the upper horizontal map is isomorphic to $(A/(I_{1}\cap
I_{2}))^{m}$, we obtain a commutative diagram

${0}$${(A/(I_{1}\cap
I_{2}))^{m}}$${(A/I_{1})^{m}\oplus(A/I_{2})^{m}}$${A/(I_{1}+I_{2})^{m}}$${0}$${0}$${\mathcal{M}_{\infty,x,\mathcal{R}}(Q)}$${\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{1})\oplus\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{2})}$${\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{0})}$${0.}$$\scriptstyle{\simeq}$
(10)

As $\mathrm{Ann}(\mathcal{B}_{x}(Q))=I_{1}\cap I_{2}$ and $\mathcal{B}_{x}(Q)$
is cyclic, there exists an isomorphism $\mathcal{B}_{x}(Q)\simeq A/(I_{1}\cap
I_{2})$. Moreover, by hypothesis, we have
$\operatorname{\mathrm{Supp}}(\mathcal{B}_{x}(Q_{i}))=\operatorname{\mathrm{Spec}}(A/I_{i})$
so that the maps $A/(I_{1}\cap
I_{2})\simeq\mathcal{B}_{x}(Q)\twoheadrightarrow\mathcal{B}_{x}(Q_{i})$
factors through isomorphisms $A/I_{i}\simeq\mathcal{B}_{x}(Q_{i})$. Therefore,
by exactness of $\mathcal{B}_{x}$, we also have a commutatif diagram

${0}$${(A/(I_{1}\cap
I_{2}))}$${(A/I_{1})\oplus(A/I_{2})}$${0}$${\mathcal{B}_{x}(Q)}$${\mathcal{B}_{x}(Q_{1})\oplus\mathcal{B}_{x}(Q_{2})}$${\mathcal{B}_{x}(Q_{0})}$${0.}$$\scriptstyle{x\mapsto(x{,}x)}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

This implies that we have an isomorphism
$A/(I_{1}+I_{2})\simeq\mathcal{B}_{x}(Q_{0})$. As $\mathcal{B}_{x}(Q_{0})$ is
Cohen–Macaulay, so is $A/(I_{1}+I_{2})$. As the ring $A/(I_{1}+I_{2})$ is
Cohen–Macaulay, the vertical right arrow of diagram (9) is a surjective map
$(A/(I_{1}+I_{2}))^{m}\twoheadrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{0})$
between two Cohen–Macaulay modules with the same cycle by Corollary 7.5. It is
therefore an isomorphism and the Snake Lemma allows us to conclude that the
left vertical arrow in (10) is an isomorphism.

Assume that the result is proved for some integer $r\geqslant 2$. Let
$Q_{1},\dots,Q_{r+1}$ be quotients of $M$ satisfying the hypotheses of the
Lemma. Let $Q^{\prime}$ be the smallest quotient of $M$ dominating all the
$Q_{i}$ for $1\leqslant i\leqslant r$. Note that $\mathcal{B}_{x}(Q^{\prime})$
is a quotient of $\mathcal{B}_{x}(Q)$ and is therefore cyclic. By induction,
$\mathcal{M}_{\infty,x,\mathcal{R}}(Q^{\prime})$ is free of rank $m$ over its
support and
$\operatorname{\mathrm{Supp}}\mathcal{M}_{\infty,x,\mathcal{R}}(Q^{\prime})=\operatorname{\mathrm{Supp}}\mathcal{B}_{x}(Q^{\prime})$.
The quotient $Q$ is now the smallest quotient of $M$ dominating $Q^{\prime}$
and $Q_{r+1}$. Therefore the case $r=2$ implies that
$\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is free of rank $m$ over its support
and
$\operatorname{\mathrm{Supp}}\mathcal{M}_{\infty,x,\mathcal{R}}(Q)=\operatorname{\mathrm{Supp}}\mathcal{B}_{x}(Q)$,
which concludes the induction. ∎

######  Lemma 7.12.

Let $M$ be an object of the category $\mathcal{O}_{\chi_{\lambda}}$. Assume
that $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is generated by $m_{x}$ elements
and $\mathcal{B}_{x}(M)$ is cyclic. Then
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is locally free of rank $m_{x}$ over
its support, its support is Cohen–Macaulay and
$\operatorname{\mathrm{Supp}}\mathcal{M}_{\infty,x,\mathcal{R}}(M)=\operatorname{\mathrm{Supp}}\mathcal{B}_{x}(M)$.

###### Proof.

We prove the result by induction on the length of $M$. If $M$ is simple this
is done in Proposition 7.9. Thus we can assume that we have a short exact
sequence

$0\longrightarrow L\longrightarrow M\longrightarrow Q\longrightarrow 0$

with $L$ simple such that $\mathcal{M}_{\infty,x,\mathcal{R}}(L)\neq 0$ and
that the result is true for $Q$. Let
$I=\mathrm{Ann}(\mathcal{M}_{\infty,x,\mathcal{R}}(M))$,
$I_{B}=\mathrm{Ann}(\mathcal{B}_{x}(M))$, $J=\mathrm{Ann}(\mathcal{B}_{x}(Q))$
and $K=\mathrm{Ann}(\mathcal{B}_{x}(L))$. Then we have two short exact
sequences

$\displaystyle
0\longrightarrow\mathcal{B}_{x}(L)\longrightarrow\mathcal{B}_{x}(M)\longrightarrow\mathcal{B}_{x}(Q)\longrightarrow
0$ $\displaystyle
0\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(L)\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(M)\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(Q)\longrightarrow
0.$

The first exact sequence shows that $\widehat{R}_{\infty,x}/K\simeq J/I_{B}$
so that $I_{B}=JK$. The second exact sequence shows that $I_{B}\subset I$.
Therefore, as $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is generated by $m_{x}$
elements, we have a surjective map

$\mathcal{B}_{x}(M)^{m_{x}}\simeq(\widehat{R}_{\infty,x}/I_{B})^{m_{x}}\twoheadrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(M).$

These modules are both Cohen–Macaulay of the same dimension with identical
associated maximal cycle by Corollary 7.5, therefore this map is an
isomorphism and $I_{B}=I$. Moreover as $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$
is Cohen–Macaulay, so is its support. ∎

######  Theorem 7.13.

For any $w\in W$ such that $ww_{0}\geqslant w_{x,\mathcal{R}}$, the coherent
sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda)^{\vee})$ is locally
free of rank $m_{x}$ over its support.

###### Proof.

As $M(w\cdot\lambda)^{\vee}$ is a quotient of $M(\lambda)^{\vee}$ for any
$w\in W$, Lemma 7.12 implies that it is sufficient to prove the result for
$w=1$.

Recall that $W=\prod_{\tau:F\hookrightarrow L}W_{\tau}$ and write
$w_{x,\mathcal{R}}=(w_{x,\tau})$. Let
$J\subset\operatorname{\mathrm{Hom}}(F,L)$ be the set embeddings such that
$w_{x,\tau}=1$. Let $E$ be the set of elements $w=(w_{v})\in W$ such that
$w_{\tau}\in\\{s_{1},s_{2}\\}$ if $\tau\in J$ and $w_{\tau}=1$ if $\tau\notin
J$. By Corollary 7.8 and Lemma 7.12, for $w\in E$, the module
$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda)^{\vee})$ is free of rank
$m_{x}$ over its support and
$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda)^{\vee})=\mathcal{B}_{x}(M(w\cdot\lambda)^{\vee})^{m_{x}}$.
Let $Q$ be the smallest quotient of $M(\lambda)^{\vee}$ dominating all the
$M(w\cdot\lambda)^{\vee}$ for $w\in E$. Lemma 7.11 implies that
$\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is free of rank $m_{x}$ over its
support and
$\mathcal{M}_{\infty,x,\mathcal{R}}(Q)=\mathcal{B}_{x}(Q)^{m_{x}}$. Let $N$ be
the kernel of the map $M\twoheadrightarrow Q$.

Let $I$ of the form $\coprod_{\tau\in J}\\{s_{i_{\tau}}\\}$ where
$i_{\tau}\in\\{1,2\\}$. Then the image of the map
$M_{I}(\lambda)^{\vee}\hookrightarrow M(\lambda)^{\vee}\twoheadrightarrow Q$
is
$Q_{I}\coloneqq\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
J}L(s_{3-i_{\tau}}\cdot\lambda_{\tau})\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\notin
J}M(\lambda_{\tau})^{\vee}$. By Corollary 7.8, the module
$\mathcal{M}_{\infty,x,\mathcal{R}}(M_{I}(\lambda)^{\vee})$ is free of rank
$m_{x}$ over its support. Thus $\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{I})$ is
generated by $m_{x}$ elements, and its quotient

$L_{I}:=\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}{\tau\in
J}L(s_{3-i_{\tau}}\cdot\lambda_{\tau})\boxtimes\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\notin
J}M(w_{x,\tau}w_{0}\cdot\lambda_{\tau})^{\vee},$

satisfies

$\mathcal{M}_{\infty,x,\mathcal{R}}(L_{I})=\mathcal{M}_{\infty,x,\mathcal{R}}\big{(}\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\in
J}L(s_{3-i_{\tau}}\cdot\lambda_{\tau})\boxtimes\operatorname*{\mathchoice{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\displaystyle\boxtimes$}}}}}{\raisebox{-10.00012pt}{\leavevmode\resizebox{14.81496pt}{10.00012pt}{\hbox{\raisebox{0.0pt}{$\textstyle\boxtimes$}}}}}{\raisebox{-7.00009pt}{\leavevmode\resizebox{11.35814pt}{7.00009pt}{\hbox{\raisebox{0.0pt}{$\scriptstyle\boxtimes$}}}}}{\raisebox{-5.00006pt}{\leavevmode\resizebox{9.05354pt}{5.00006pt}{\hbox{\raisebox{0.0pt}{$\scriptscriptstyle\boxtimes$}}}}}}_{\tau\notin
J}L(w_{x,\tau}w_{0}\cdot\lambda_{\tau})\big{)}.$

by Proposition 7.2. Moreover, by Proposition 7.9, this module is free of rank
$m_{x}$ over its support so that its fiber at $x$ has dimension $m_{x}$. This
implies that the following surjective maps are all isomorphisms

$\displaystyle
k(x)^{m_{x}}\simeq\mathcal{M}_{\infty,x,\mathcal{R}}(M_{I}(\lambda)^{\vee})\otimes
k(x)$
$\displaystyle\xrightarrow{\sim}\mathcal{M}_{\infty,x,\mathcal{R}}(Q_{I})\otimes
k(x)$
$\displaystyle\xrightarrow{\sim}\mathcal{M}_{\infty,x,\mathcal{R}}(L_{I})\otimes
k(x)\simeq k(x)^{m_{x}}.$

As moreover $\operatorname{\mathrm{Ker}}(M_{I}(\lambda)^{\vee}\rightarrow
Q_{I})=N\cap M_{I}(\lambda)^{\vee}$, we see that the map

$\mathcal{M}_{\infty,x,\mathcal{R}}(N\cap M_{I}(\lambda)^{\vee})\otimes
k(x)\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda)^{\vee})\otimes
k(x)$

is zero. As $M(\lambda)^{\vee}$ is multiplicity-free, we have
$N=\sum_{I}(N\cap M_{I}(\lambda)^{\vee})$ and we conclude that the map

$\mathcal{M}_{\infty,x,\mathcal{R}}(N)\otimes
k(x)\longrightarrow\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda)^{\vee})\otimes
k(x)$

is zero. Therefore
$\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda)^{\vee})\otimes
k(x)\simeq\mathcal{M}_{\infty,x,\mathcal{R}}(Q)\otimes k(x)\simeq
k(x)^{m_{x}}$ and we conclude with Lemma 7.12 since
$\mathcal{B}_{x}(M(\lambda)^{\vee})$ is cyclic. ∎

###  7.5 The case of the antidominant projective

######  Theorem 7.14.

The coherent sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(P(w_{0}\cdot\lambda))$
is free of rank $m_{x}$ over its support.

###### Proof.

Recall that $A=U(\mathfrak{t})_{\mathfrak{m}}$ and set $D\coloneqq
L\otimes_{A^{W}}A$. By Proposition 2.17, the action of $Z(\mathfrak{g})$ on
$P(w_{0}\cdot\lambda)$ induces a structure of $D$-module on
$P(w_{0}\cdot\lambda)$. As $M(\lambda)^{\vee}$ is an injective object, it
follows from [Soe90, Prop. 6], that $M(\lambda)^{\vee}\simeq
P(w_{0}\cdot\lambda)\otimes_{D}(D/\mathfrak{m}_{D})$, where $\mathfrak{m}_{D}$
is the maximal ideal of $D$. We have also a local map of local algebras
$\alpha:D\rightarrow\mathcal{O}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}}$
defined in section 6.3. It follows from Corollary 6.14 that these define the
same action of $D$ on
$\mathcal{M}_{\infty,x,\mathcal{R}}(P(w_{0}\cdot\lambda))$. As moreover the
functor $\mathcal{M}_{\infty,x,\mathcal{R}}$ is exact, we have an isomorphism
$\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda)^{\vee})\simeq\mathcal{M}_{\infty,x,\mathcal{R}}(P(w_{0}\cdot\lambda))\otimes_{D}(D/\mathfrak{m}_{D})$.
As moreover the map
$A\otimes_{A^{W}}A\rightarrow\mathcal{O}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}}$
is a local map of local rings, we have an isomorphism
$\mathcal{M}_{\infty,x,\mathcal{R}}(P(w_{0}\cdot\lambda))\otimes
k(x)\xrightarrow{\sim}\mathcal{M}_{\infty,x,\mathcal{R}}(M(\lambda)^{\vee})\otimes
k(x)$ and thus
$\dim_{L}\mathcal{M}_{\infty,x,\mathcal{R}}(P(w_{0}\cdot\lambda))\otimes
k(x)=m_{x}$ by Theorem 7.13. We conclude by Lemma 7.12. ∎

######  Corollary 7.15.

Let $Q$ be a quotient of the anti-dominant projective $P(w_{0}\cdot\lambda)$
in the category $\mathcal{O}_{\chi_{\lambda}}$. If
$\mathcal{M}_{\infty,x,\mathcal{R}}(Q)\neq 0$, then it is finite free of rank
$m_{x}$ over its support and its support is Cohen–Macaulay.

###### Proof.

As $\mathcal{M}_{\infty,x,\mathcal{R}}(P(w_{0}\cdot\lambda))$ (resp.
$\mathcal{B}_{x}(P(w_{0}\cdot\lambda))$) is free of rank $m_{x}$ (resp. 1)
over its support by Theorems 7.14 and 7.4, we have that
$\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ (resp. $\mathcal{B}_{x}(Q)$) is
generated by at most $m_{x}$ elements (resp. cyclic). It follows from Lemma
7.12, $\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is free of rank $m_{x}$ over its
support and that its support is Cohen–Macaulay. ∎

######  Corollary 7.16.

For all $w\in W$, the coherent sheaf

$\mathcal{M}_{\infty,x,\mathcal{R}}(P(w\cdot\lambda)^{\vee}),$

is free of rank $m_{x}$ over its support.

###### Proof.

By Corollary 7.15, it is sufficient to prove that
$\mathcal{M}_{\infty,x,\mathcal{R}}(P(w\cdot\lambda)^{\vee})$ is non zero and
that there exists a surjective map

$P(w_{0}\cdot\lambda)\longrightarrow P(w\cdot\lambda)^{\vee}.$

As $P(w_{0}\cdot\lambda)$ is the projective envelope of
$L(w_{0}\cdot\lambda)$, this is equivalent to showing that the socle of
$P(w\cdot\lambda)$ is isomorphic to $L(w_{0}\cdot\lambda)$. By [Str03, Thm.
8.1], the socle of $P(w\cdot\lambda)$ is isomorphic to
$L(w_{0}\cdot\lambda)^{m}$ with
$m=[P(w\cdot\lambda):M(\lambda)]=[M(\lambda):L(w\cdot\lambda)]$ by [Hum08,
Thm. 3.9]. As $\mathfrak{g}$ is isomorphic to a direct sum of copies of
$\mathfrak{gl}_{3,L}$, we have $[M(\lambda):L(w\cdot\lambda)]=1$ for any $w\in
W$.

Moreover, as $[M(\lambda):L(\lambda)]=1$, we have

$[P(w\cdot\lambda)^{\vee}:L(\lambda)]=[P(w\cdot\lambda):L(\lambda)]=1.$

As $\mathcal{M}_{\infty,x,\mathcal{R}}(L(\lambda))\neq 0$, we have
$\mathcal{M}_{\infty,x,\mathcal{R}}(P(w\cdot\lambda)^{\vee})\neq 0$. ∎

###  7.6 Duality

For a Cohen–Macaulay sheaf $\mathcal{F}$ on
$\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$ of dimension
$\dim\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$, we write
$\omega^{\bullet}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}}$
for the dualizing complex and set

$\mathcal{F}^{\vee}:=R\operatorname{\mathrm{Hom}}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}}(\mathcal{F},\omega^{\bullet}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}})[-\dim\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}].$

This complex $\mathcal{F}^{\vee}$ is a coherent sheaf concentrated in degree
$0$ to which we refer to $\mathcal{F}^{\vee}$ as the _shifted_ Serre dual of
$\mathcal{F}$.

######  Lemma 7.17.

Let $\mathcal{F}$ be a maximal Cohen–Macaulay coherent sheaf over
$\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$. Then
$[\mathcal{F}^{\vee}]=[\mathcal{F}]$. As a consequence if
$\mathcal{Y}\subset\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$
is a maximal Cohen–Macaulay closed subscheme, we have
$[\omega_{\mathcal{Y}}]=[\mathcal{Y}]$.

###### Proof.

Let $R$ be local complete regular ring such that
$\mathcal{O}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}}$
is isomorphic to a quotient of $R$. Then we can compute $\mathcal{F}^{\vee}$
by the formula
$\mathcal{F}^{\vee}=\operatorname{\mathrm{Ext}}^{d}_{R}(\mathcal{F},R)$ where
$d$ is the codimension of
$\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$ in
$\operatorname{\mathrm{Spec}}(R)$. By definition, we have
$[\mathcal{F}]=\sum_{z}a(z)z$ where the sum is over all maximal points in
$\operatorname{\mathrm{Supp}}(\mathcal{F})$ and $a(z)$ is the length of the
finite length $R_{z}$-module $\mathcal{F}_{z}$. Let
$z\in\operatorname{\mathrm{Spec}}(R)$ be a maximal point of the support of
$\mathcal{F}$. The localization $R_{z}$ of $R$ at $z$ is a local regular ring
and we have
$\mathcal{F}^{\vee}_{z}\simeq\operatorname{\mathrm{Ext}}^{d}_{R_{z}}(\mathcal{F}_{z},R_{z})$.
As $\operatorname{\mathrm{Ext}}^{d}_{R_{z}}(-,R_{z})$ is a an exact functor on
the subcategory of finite length $R_{z}$-modules and
$\dim_{k(z)}\operatorname{\mathrm{Ext}}_{R_{z}}^{d}(k(z),R_{z})=1$, the length
of the $R_{z}$-module
$\operatorname{\mathrm{Ext}}^{d}_{R_{z}}(\mathcal{F}_{z},R_{z})$ is $a(z)$. So
we have the proved the claim. ∎

######  Proposition 7.18.

Let $M$ be a subobject of the anti-dominant projective $P(w_{0}\cdot\lambda)$.
Assume that $\mathcal{M}_{\infty,x,\mathcal{R}}(M)\neq 0$ and let
$\mathcal{Y}$ be the support of $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$. Then
$\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ is isomorphic to
$\omega_{\mathcal{Y}}^{m_{x}}$ and $\mathcal{Y}$ is Cohen–Macaulay.

###### Proof.

Let $Q$ be the quotient of $P(w_{0}\cdot\lambda)$ by $M$. If
$\mathcal{M}_{\infty,x,\mathcal{R}}(Q)=0$, then Theorem 7.14 implies the
result. So we can assume that $\mathcal{M}_{\infty,x,\mathcal{R}}(M)\neq 0$
and $\mathcal{M}_{\infty,x,\mathcal{R}}(Q)\neq 0$. By Corollary 7.15,
$\mathcal{M}_{\infty,x,\mathcal{R}}(Q)$ is isomorphic to
$\mathcal{O}_{\mathcal{Z}}^{m_{x}}$ for
$\mathcal{Z}\subset\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}$
maximal Cohen–Macaulay. Using Lemma 7.10, we can construct a commutative
diagram

${0}$${\mathcal{M}_{\infty,x,\mathcal{R}}(M)}$${\mathcal{M}_{\infty,x,\mathcal{R}}(P(w_{0}\cdot\lambda))}$${\mathcal{M}_{\infty,x,\mathcal{R}}(Q)}$${0}$${0}$${\operatorname{\mathrm{Ker}}}$${\mathcal{O}_{\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}}^{m_{x}}}$${\mathcal{O}_{\mathcal{Z}}^{m_{x}}}$${0.}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{can}^{m_{x}}}$

Let $I$ be the ideal defining $\mathcal{Z}$. As
$\overline{\mathcal{X}}^{\mathrm{qtri}}_{\infty,x,\mathcal{R}}$ and
$\mathcal{Z}$ are Cohen–Macaulay of the same dimension, the involutivity of
the duality implies that we have $I\simeq\omega_{\mathcal{O}_{\mathcal{Y}}}$
where $\mathcal{Y}=\operatorname{\mathrm{Supp}}(I)$ so that we have the
result. ∎

######  Theorem 7.19.

For all $w\in W$, with $ww_{0}\geqslant w_{x,\mathcal{R}}$, the sheaf
$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda))$ is isomorphic to

$\big{(}\omega_{\overline{\mathcal{X}}^{\mathrm{qtri},ww_{0}}_{\infty,x,\mathcal{R}}}\big{)}^{\oplus
m_{x}}.$

###### Proof.

It follows from Propositions 7.18 and 7.3 that
$\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda))\simeq\omega_{\mathcal{Y}}^{m_{x}}$
where
$\mathcal{Y}=\operatorname{\mathrm{Supp}}\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda))$
is Cohen–Macaulay. However it follows from Theorem 6.16 that
$\mathcal{Y}\subset\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},ww_{0}}$.
By Lemma 7.17, we have an equality $[\omega_{\mathcal{Y}}]=[\mathcal{Y}]$ and
it follows from Corollary 7.5 that
$[\mathcal{M}_{\infty,x,\mathcal{R}}(M(w\cdot\lambda))]=m_{x}[\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},ww_{0}}]$.
Therefore we have
$[\mathcal{Y}]=[\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},ww_{0}}]$
and thus
$\mathcal{Y}=\overline{\mathcal{X}}_{\infty,x,\mathcal{R}}^{\mathrm{qtri},ww_{0}}$.
∎

We choose for all $\lambda$ dominant weight, and all $w\in W$ a surjective map
$\pi_{w}:P(w_{0}\cdot\lambda)\longrightarrow P(w\cdot\lambda)^{\vee}$ (see
proof of Corollary 7.16).

######  Lemma 7.20.

For all map $f_{w,w^{\prime}}:P(w\cdot\lambda)^{\vee}\longrightarrow
P(w^{\prime}\cdot\lambda)^{\vee}$ there exists a map
$\widetilde{f}_{w,w^{\prime}}:P(w_{0} \cdot\lambda)\longrightarrow
P(w_{0}\cdot\lambda)$ such that the following diagram commutes

${P(w_{0}\cdot\lambda)}$${P(w_{0}\cdot\lambda)}$${P(w\cdot\lambda)^{\vee}}$${P(w^{\prime}\cdot\lambda)^{\vee}}$$\scriptstyle{\widetilde{f}_{w,w^{\prime}}}$$\scriptstyle{\pi_{w}}$$\scriptstyle{\pi_{w^{\prime}}}$$\scriptstyle{f_{w,w^{\prime}}}$
(11)

###### Proof.

As $\pi_{w^{\prime}}:P(w_{0} \cdot\lambda)\longrightarrow
P(w^{\prime}\cdot\lambda)^{\vee}$ is surjective and $P(w_{0}\cdot\lambda)$ is
projective, the map
$\operatorname{\mathrm{Hom}}(P(w_{0}\cdot\lambda,P(w_{0}\cdot\lambda))\longrightarrow\operatorname{\mathrm{Hom}}(P(w_{0}\cdot\lambda),P(w^{\prime}\cdot\lambda)^{\vee})$
is surjective, thus there exists $\widetilde{f}_{w,w^{\prime}}$ mapping to
$f_{w,w^{\prime}}\circ\pi_{w}$. This proves the claim. ∎

######  Lemma 7.21.

Let $\mathcal{F}$ be either $\mathcal{B},\mathcal{B}_{x}$ or
$\mathcal{M}_{\infty,x,\mathcal{R}}$. There exists a family of isomorphisms
indexed by $w\in W$

$\Psi_{w}:\mathcal{F}(P(w\cdot\lambda)^{\vee})\overset{\sim}{\longrightarrow}\mathcal{F}(P(w\cdot\lambda))^{\vee}.$

such that for any $w,w^{\prime}\in W$ and any if
$f_{w,w^{\prime}}:P(w\cdot\lambda)^{\vee}\longrightarrow
P(w^{\prime}\cdot\lambda)^{\vee}$, the following diagram commutes

${\mathcal{F}(P(w\cdot\lambda)^{\vee})}$${\mathcal{F}(P(w^{\prime}\cdot\lambda)^{\vee})}$${\mathcal{F}(P(w\cdot\lambda))^{\vee}}$${\mathcal{F}(P(w^{\prime}\cdot\lambda))^{\vee}}$$\scriptstyle{\mathcal{F}(f_{w,w^{\prime}})}$$\scriptstyle{\Psi_{w}}$$\scriptstyle{\Psi_{w^{\prime}}}$$\scriptstyle{\mathcal{F}(f_{w,w^{\prime}}^{\vee})^{\vee}}$
(12)

where we denote by the same symbol $(\cdot)^{\vee}$ the duality in
$\mathcal{O}$ and Serre duality on coherent sheaves.

###### Proof.

Let $w\in W$. The sheaves $\mathcal{F}(P(w\cdot\lambda)^{\vee})$ and
$\mathcal{F}(P(w\cdot\lambda))^{\vee}$ are isomorphic to the same quotient of
$\mathcal{F}(P(w_{0}\cdot\lambda))$ by Theorem 7.4 for
$\mathcal{B},\mathcal{B}_{x}$ and Corollary 7.16 and Proposition 7.18 for
$\mathcal{M}_{\infty,x,\mathcal{R}}$. This implies that there exists an
isomorphism
$\Psi_{w}:\mathcal{F}(P(w\cdot\lambda)^{\vee})\xrightarrow{\sim}\mathcal{F}(P(w\cdot\lambda))^{\vee}$
such that the following diagram commutes

${\mathcal{F}(P(w_{0}\cdot\lambda))}$${\mathcal{F}(P(w_{0}\cdot\lambda))^{\vee}}$${\mathcal{F}(P(w\cdot\lambda)^{\vee})}$${\mathcal{F}(P(w\cdot\lambda))^{\vee}}$$\scriptstyle{\Psi_{w_{0}}}$$\scriptstyle{\mathcal{F}(\pi_{w})}$$\scriptstyle{\mathcal{F}(\pi_{w}^{\vee})^{\vee}}$$\scriptstyle{\Psi_{w}}$
(13)

Fix $w,w^{\prime}$ and let’s show that the diagram (12) is commutative. Let
$f_{w,w^{\prime}}\in\operatorname{\mathrm{Hom}}(P(w\cdot\lambda)^{\vee},P(w^{\prime}\cdot\lambda)^{\vee})$.
By Lemma 7.20, there exists a map
$\widetilde{f}_{w,w^{\prime}}\in\operatorname{\mathrm{End}}(P(w_{0}\cdot\lambda))$
such that the diagram (11) is commutative. We first consider the following
diagram

${\mathcal{F}(P(w_{0}\cdot\lambda))}$${\mathcal{F}(P(w_{0}\cdot\lambda))^{\vee}}$${\mathcal{F}(P(w_{0}\cdot\lambda)^{\vee})}$${\mathcal{F}(P(w_{0}\cdot\lambda))^{\vee}}$$\scriptstyle{\Psi_{w_{0}}}$$\scriptstyle{\mathcal{F}(\widetilde{f}_{w,w^{\prime}})}$$\scriptstyle{\mathcal{F}(\widetilde{f}_{w,w^{\prime}}^{\vee})^{\vee}}$$\scriptstyle{\Psi_{w_{0}}}$
(14)

But as
$\widetilde{f}_{w,w^{\prime}}\in\operatorname{\mathrm{End}}_{\mathcal{O}}(P(w_{0}\cdot\lambda),P(w_{0}\cdot\lambda))\simeq
D=L\otimes_{A^{W}}A$, it follows from Corollary 6.14 for
$\mathcal{F}=\mathcal{M}_{\infty,x,\mathcal{R}}$ and [Bez16, Prop. 23] for
$\mathcal{F}=\mathcal{B}_{x}$, and the fact that $\Psi_{w_{0}}$ is
$\mathcal{O}_{\mathcal{X}^{\mathrm{qtri}}_{\infty,x,\mathcal{R}}}$-linear,
that this diagram commutes. Now consider the diagram

${\mathcal{F}(P(w_{0}\cdot\lambda))}$${\mathcal{F}(P(w^{\prime}\cdot\lambda))^{\vee}}$${\mathcal{F}(P(w_{0}\cdot\lambda))}$${\mathcal{F}(P(w^{\prime}\cdot\lambda)^{\vee})}$${\mathcal{F}(P(w_{0}\cdot\lambda))}$${\mathcal{F}(P(w\cdot\lambda))^{\vee}}$${\mathcal{F}(P(w_{0}\cdot\lambda))}$${\mathcal{F}(P(w\cdot\lambda)^{\vee})}$$\scriptstyle{\mathcal{F}(\pi_{w^{\prime}}^{\vee})^{\vee}}$$\scriptstyle{\mathcal{F}(\widetilde{f}_{w,w^{\prime}}^{\vee})^{\vee}}$$\scriptstyle{\mathcal{F}(\pi_{w^{\prime}})}$$\scriptstyle{\Psi_{w_{0}}}$$\scriptstyle{\Psi_{w^{\prime}}}$$\scriptstyle{\mathcal{F}(\pi_{w}^{\vee})^{\vee}}$$\scriptstyle{\mathcal{F}({f}_{w,w^{\prime}}^{\vee})^{\vee}}$$\scriptstyle{\mathcal{F}(\pi_{w})}$$\scriptstyle{\Psi_{w_{0}}}$$\scriptstyle{\mathcal{F}(\widetilde{f}_{w,w^{\prime}})}$$\scriptstyle{\Psi_{w}}$$\scriptstyle{\mathcal{F}({f}_{w,w^{\prime}})}$

All faces, except maybe the right hand one (which is the one of the
statement), of this cube are commutative diagrams by functoriality and
diagrams (11), (13), (14). Moreover $\mathcal{F}(\pi_{w})$,
$\mathcal{F}(\pi_{w}^{\vee})^{\vee}$, $\mathcal{F}(\pi_{w^{\prime}})$,
$\mathcal{F}(\pi_{w^{\prime}}^{\vee})^{\vee}$ are surjective, thus the last
right hand face also commutes. ∎

######  Corollary 7.22.

For any $M\in\mathcal{O}_{\mathrm{alg}}$, there is a compatible choice of
isomorphisms

$ \Psi_{M}:\mathcal{F}(M^{\vee}) \overset{\sim}{\longrightarrow}
\mathcal{F}(M)^{\vee},$

where $\mathcal{F}$ is either the functor $\mathcal{B},\mathcal{B}_{x}$ or
$\mathcal{M}_{\infty,x,\mathcal{R}}$. In particular, $\mathcal{F}$ is
compatible with duality.

###### Proof.

Choose a resolution

$
\bigoplus_{i}P(\mu_{i})\longrightarrow\bigoplus_{j}P(\lambda_{j})\longrightarrow
M\longrightarrow 0.$ (15)

Then we have two exact sequences

$0\longrightarrow\mathcal{F}(M^{\vee})\longrightarrow\bigoplus_{j}\mathcal{F}(P(\lambda_{j})^{\vee})\longrightarrow
\bigoplus_{i}\mathcal{F}(P(\mu_{i})^{\vee}),$

and

$0\longrightarrow\mathcal{F}(M)^{\vee}\longrightarrow\bigoplus_{j}\mathcal{F}(P(\lambda_{j}))^{\vee}\longrightarrow
\bigoplus_{i}\mathcal{F}(P(\mu_{i}))^{\vee}.$

For the second one, recall that if $K$ denote the Kernel in equation (15) so
that

$ 0\longrightarrow K\longrightarrow\bigoplus_{j}P(\lambda_{j})\longrightarrow
M\longrightarrow 0,$

then, as $\mathcal{F}(K)$ is CM of the same dimension as the other modules, we
have

$
0\longrightarrow\mathcal{F}(M)^{\vee}\longrightarrow\bigoplus_{j}\mathcal{F}(P(\lambda_{j}))^{\vee}\longrightarrow\mathcal{F}(K)^{\vee}\longrightarrow
0,$

which is exact. Moreover, by the previous Lemma 7.21 we have a commutative
diagram with vertical isomorphisms

${0}$${\mathcal{F}(M^{\vee})}$${\bigoplus_{j}\mathcal{F}(P(\lambda_{j})^{\vee})}$${\bigoplus_{i}\mathcal{F}(P(\mu_{i})^{\vee})}$${0}$${\mathcal{F}(M)^{\vee}}$${\bigoplus_{j}\mathcal{F}(P(\lambda_{j}))^{\vee}}$${\bigoplus_{i}\mathcal{F}(P(\mu_{i}))^{\vee}}$$\bigoplus_{j}\Psi_{\lambda_{j}}$$\bigoplus_{i}\Psi_{\mu_{i}}$

which induces an isomorphism
$\Psi_{M}:\mathcal{F}(M^{\vee})\longrightarrow\mathcal{F}(M)^{\vee}$. ∎

######  Corollary 7.23.

There exists an isomorphism of functors
$\mathcal{B}_{x}^{m_{x}}\simeq\mathcal{M}_{\infty,x,\mathcal{R}}$.

###### Proof.

By a similar argument to the proof of Lemma 7.21, we can construct a family
indexed by $w\in W$ of isomorphisms

$\Phi_{w}:\mathcal{B}_{x}(P(w\cdot\lambda)^{\vee})^{m_{x}}\overset{\sim}{\longrightarrow}\mathcal{M}_{\infty,x,\mathcal{R}}(P(w\cdot\lambda)^{\vee})$

such that, for any $w,w^{\prime}\in W$ and any
$f_{w,w^{\prime}}\in\operatorname{\mathrm{Hom}}(P(w\cdot\lambda)^{\vee},P(w^{\prime}\cdot\lambda)^{\vee}$,
the following diagram commutes

${\mathcal{B}_{x}(P(w\cdot)^{\vee})^{m_{x}}}$${\mathcal{B}_{x}(P(w^{\prime}\cdot\lambda)^{\vee})}$${\mathcal{M}_{\infty,x,\mathcal{R}}(P(w\cdot\lambda)^{\vee})}$${\mathcal{M}_{\infty,x,\mathcal{R}}(P(w^{\prime}\cdot\lambda)^{\vee}).}$$\scriptstyle{\mathcal{B}_{x}(f_{w,w^{\prime}})}$$\scriptstyle{\Phi_{w}}$$\scriptstyle{\Phi_{w^{\prime}}}$$\scriptstyle{\mathcal{M}_{\infty,x,\mathcal{R}}(f_{w,w^{\prime}})}$

Such a family of isomorphisms provides an isomorphism of functors between
$\mathcal{B}_{x}^{m_{x}}$ and $\mathcal{M}_{\infty,x,\mathcal{R}}$ restricted
to the full subcategory of $\mathcal{O}_{\lambda}$ of injective objects. As
the category $\mathcal{O}_{\lambda}$ has enough injectives, this isomorphism
extends to all of $\mathcal{O}_{\lambda}$. ∎

###  7.7 Consequences

In this section we keep the setting introduced in subsection 7.3. In
particular $n=3$.

######  Lemma 7.24.

Let $\rho,\lambda,\mathcal{R}$ be as above and let
$x\in\mathcal{X}_{\infty}(L)$ the point corresponding to $\rho$. Then for all
$M\in\mathcal{O}_{\chi_{\lambda}}$,

$ \mathcal{M}_{\infty,x,\mathcal{R}}(M)\otimes
k(x)\simeq\left(\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M,\Pi^{\rm
la}[\mathfrak{m}_{\rho}])^{N^{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}]\right)^{\prime}.$

###### Proof.

By construction (see Remark 6.2), we have

$
\mathcal{M}_{\infty,x,\mathcal{R}}(M)\simeq\left(\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M,\Pi^{\mathrm{la}}_{\infty}[\mathfrak{m}_{x}^{\infty}])^{N_{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}^{\infty}]\right)^{\prime}.$

By Corollary 6.11, the $\mathcal{X}_{\infty}\times\widehat{T}$-structure on
the sheaf $\mathcal{M}_{\infty,x,\mathcal{R}}(M)$ factors through
$\mathcal{X}_{\infty,x,\mathcal{R}}^{\mathrm{qtri}}
\longrightarrow\mathcal{X}_{\infty}\times\widehat{T}$. Thus,

$\mathcal{M}_{\infty,x,\mathcal{R}}(M)\otimes
k(x)\simeq\left(\operatorname{\mathrm{Hom}}_{U(\mathfrak{g})}(M,\Pi^{\mathrm{la}}_{\infty}[\mathfrak{m}_{x}])^{N_{0}}[\mathfrak{m}_{\delta_{\mathcal{R}}}]\right)^{\prime}.\qed$

######  Corollary 7.25.

Let $\delta:T\rightarrow L^{\times}$ be a continuous character and let
$\chi^{S}:\mathbb{T}^{S}\rightarrow L$ be a character such that there exists
$f\in S^{\dagger}(K^{p})[\chi^{S}\otimes\delta]$ an overconvergent $p$-adic
eigenform on the group $U(3)$. Assume that the Galois representation $\rho$
associated to $f$ is crystalline strictly dominant and $\varphi$-generic at
$p$ satisfying (1). Let