$w\kappa(t)=\kappa(w^{-1}tw)$ for $\kappa\in X^{\star}(T)$ and $w\in W$. Let
$W_{M}$ be the Weyl group of $M$. The quotient $W_{M}\backslash W$ has a set
of coset representatives of minimal length (the Kostant representatives)
called ${}^{M}W$. This is the subset of $W$ of elements $w$ that satisfy
$\Phi_{M}^{+}\subseteq w\Phi^{+}$.

Let $FL=P\backslash G\rightarrow\mathrm{Spec}~{}\mathcal{O}_{F}$ be the flag
variety associated with $P$. We let $d=\\#\Phi^{+,M}$ be the (relative)
dimension of $FL$. The group $G$ acts on the right on $FL$.

For any $w\in{}^{M}W$, we let $C_{w}=P\backslash PwB$ be the Bruhat cell
corresponding to $w$. We have the decomposition into $B$-orbits
$FL=\coprod_{w\in{}^{M}W}C_{w}$. We can also consider the opposite Bruhat
cell: $C^{w}=P\backslash Pw\overline{B}$ for the opposite Borel
$\overline{B}$.

We let $X_{w}$ be the Schubert variety equal to the Zariski closure of $C_{w}$
in $FL$. We also let $X^{w}$ be the opposite Schubert variety, equal to the
Zariski closure of $C^{w}$ in $FL$. There is a partial order $\leq$ on
${}^{M}W$ for which $X_{w}=\cup_{w^{\prime}\leq w}C_{w^{\prime}}$ and
$X^{w}=\cup_{w^{\prime}\geq w}C^{w^{\prime}}$. For the length function
$\ell:{}^{M}W\rightarrow[0,\mathrm{dim}FL]$, we have $\ell(w)=\dim C_{w}$
(here dimensions are relative dimensions over
$\mathrm{Spec}~{}\mathcal{O}_{F}$).

We also define $Y_{w}=\cup_{w^{\prime}\geq w}C_{w}$ for all $w\in{}^{M}W$.
This is an open subscheme of $FL$ containing $C_{w}$.

######  Lemma 3.1.2.

We have an inclusion $X^{w}\hookrightarrow Y_{w}$.

###### Proof.

By [BL03], I, lemma 1, we know that $X^{w}\cap
X_{v}\neq\emptyset\Leftrightarrow v\geq w$. We deduce that $X^{w}\cap
C_{v}\neq\emptyset\Rightarrow v\geq w$, so that $X^{w}\subseteq\cup_{v\geq
w}C_{v}$. ∎

The following lemma gives a description of the Bruhat cells. For all
$\alpha\in\Phi$, we let $U_{\alpha}$ be the one parameter subgroup
corresponding to $\alpha$.

######  Lemma 3.1.3.

The product map (for any ordering of the factors)

$\displaystyle\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{+}}U_{\alpha}$
$\displaystyle\rightarrow$ $\displaystyle C_{w}$ $\displaystyle(x_{\alpha})$
$\displaystyle\mapsto$ $\displaystyle w\prod_{\alpha}x_{\alpha}$

is an isomorphism of schemes.

###### Proof.

We need to prove that the map
$\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{+}}U_{\alpha}\rightarrow(B\cap
w^{-1}Pw)\backslash B$ is an isomorphism. This follows easily from the
following facts:

  * •

$B=T\times\prod_{\alpha\in\Phi^{+}}U_{\alpha}$ (in any order),

  * •

$\Phi^{+}=(w^{-1}\Phi^{-,M})\cap\Phi^{+}\coprod(w^{-1}\Phi^{+})\cap\Phi^{+}\coprod(w^{-1}\Phi^{-}_{M})\cap\Phi^{+}$,

  * •

$B\cap
w^{-1}Pw=T\times\prod_{\alpha\in(w^{-1}\Phi^{+})\cap\Phi^{+}\coprod(w^{-1}\Phi^{-}_{M})\cap\Phi^{+}}U_{\alpha}$.

∎

We also introduce the translated open cell $U_{w}:=C_{w_{0}}\cdot
w_{0}^{-1}w$, so that by the lemma we have an isomorphism (for any ordering of
the roots):

$\displaystyle\prod_{\alpha\in w^{-1}\Phi^{-,M}}U_{\alpha}$
$\displaystyle\rightarrow$ $\displaystyle U_{w}$ $\displaystyle(x_{\alpha})$
$\displaystyle\mapsto$ $\displaystyle w\prod_{\alpha}x_{\alpha}.$

We see that $U_{w}$ contains the Bruhat cell $C_{w}$ as a closed subset.

###  3.2. Interlude: the cohomology of the flag variety

In this section, which is independent of the rest of the paper, we discuss the
coherent cohomology of the flag variety following [Kem78], section 12. We
assume here that $P=B$ is a Borel subgroup. We consider the stratification
$Z_{0}=FL\supseteq Z_{1}\supseteq\cdots\supseteq Z_{d}\supseteq
Z_{d+1}=\emptyset$ where $d=\mathrm{dim}FL$ and $Z_{i}=\bigcup_{w\in
W,\ell(w)=d-i}X_{w}$. Let $\kappa\in X^{\star}(T)$. We associate to $\kappa$ a
$G$-equivariant line bundle $\mathcal{L}_{\kappa}$ on $FL$ as follows: if
$\pi:G\rightarrow FL$ is the projection map, then for any open
$U\hookrightarrow FL$,

$\mathcal{L}_{\kappa}(U)=\\{f:\pi^{-1}(U)\rightarrow\mathbb{A}^{1}\mid
f(bu)=(w_{0}\kappa)(b)f(u),~{}\forall(b,u)\in B\times\pi^{-1}(U)\\}.$

We consider the spectral sequence associated to the filtration (section 2.3)

$E_{1}^{p,q}=\mathrm{H}^{p+q}_{Z_{p}/Z_{p+1}}(FL,\mathcal{L}_{\kappa})\Rightarrow\mathrm{H}^{p+q}(FL,\mathcal{L}_{\kappa}).$

The abutment of this spectral sequence carries an action of $G$. The terms on
the left do not carry an action of $G$, but then do carry an action of $B$, as
well as the Lie algebra $\mathfrak{g}$ (see [Kem78], lemma 12.8), and the
spectral sequence is equivariant for these actions.

In order to study this spectral sequence, we need the following basic result:

######  Lemma 3.2.1.

Let $m\geq n$. Let
$\mathbb{A}^{m}=\mathrm{Spec}~{}\mathcal{O}_{F}[x_{1},\cdots,x_{m}]$ and
$\mathbb{A}^{n}=\mathrm{Spec}~{}\mathcal{O}_{F}[x_{1},\cdots,x_{n}]$. Let
$\mathbb{A}^{n}\hookrightarrow\mathbb{A}^{m}$ be the closed immersion given by
$x_{n+1}=\cdots=x_{m}=0$. We have
$\mathrm{H}^{i}_{\mathbb{A}^{n}}(\mathbb{A}^{m})=0$ if $i\neq m-n$, and

$\mathrm{H}^{m-n}_{\mathbb{A}^{n}}(\mathbb{A}^{m})=\bigoplus_{k_{1},\cdots,k_{n}\geq
0,\\\
~{}k_{n+1},\cdots,k_{m}<0}\mathcal{O}_{F}\cdot\prod_{i=1}^{m}x_{i}^{k_{i}}$

###### Proof.

One can use a Koszul complex. See [Gro05], exposé II, proposition 5. ∎

Let $\rho=\frac{1}{2}\sum_{\alpha\in\Phi^{+}}\alpha$. We define the dotted
action of $W$ on $X^{\star}(T)$ by $w\cdot\kappa=w(\kappa+\rho)-\rho$. We also
denote
$\mathrm{H}^{\star}_{w}(FL,\mathcal{L}_{\kappa})=\mathrm{H}^{\star}_{C_{w}}(U_{w},\mathcal{L}_{\kappa})$.

######  Lemma 3.2.2.

We have a decomposition

$\mathrm{H}^{p+q}_{Z_{p}/Z_{p+1}}(FL,\mathcal{L}_{\kappa})=\bigoplus_{w\in
W,\ell(w)=d-p}\mathrm{H}^{p+q}_{w}(FL,\mathcal{L}_{\kappa})$

and these groups vanish when $q\not=0$. Moreover, $T$ acts on
$\mathrm{H}^{d-\ell(w)}_{w}(FL,\mathcal{L}_{\kappa})$ with character:

$\frac{[(w^{-1}w_{0})\cdot\kappa]}{\prod_{\alpha\in\Phi^{-}}([1]-[\alpha])}$

Here the character is viewed as a function $X^{\star}(T)\to\mathbb{Z}$ and for
$\lambda\in X^{\star}(T)$, we write $[\lambda]$ for the function which is 1 on
$\lambda$ and 0 otherwise. The multiplication is convolution. We note that
this says that the character of
$\mathrm{H}^{d-\ell(w)}_{w}(FL,\mathcal{L}_{\kappa})$ is the same as that of
the Verma module with highest weight $(w^{-1}w_{0})\cdot\kappa$.

###### Proof.

Since $Z_{p}\setminus Z_{p+1}=\coprod_{w\in W,\ell(w)=d-p}C_{w}$ and $U_{w}$
is a neighborhood of $C_{w}$ in $FL$, we deduce from lemma 2.1.1 that

$\mathrm{H}^{p+q}_{Z_{p}/Z_{p+1}}(FL,\mathcal{L}_{\kappa})=\oplus_{w\in
W,\ell(w)=d-p}\mathrm{H}^{p+q}_{C_{w}}(U_{w},\mathcal{L}_{\kappa}).$

By lemma 3.1.3, after choosing an ordering of the roots, we have
$U_{w}\simeq\mathbb{A}^{d}$ with coordinates $x_{\alpha}$ for $\alpha\in
w^{-1}\Phi^{-}$, and $C_{w}\simeq\mathbb{A}^{\ell(w)}$ with coordinates
$x_{\alpha}$ for $\alpha\in w^{-1}\Phi^{-}\cap\Phi^{+}$. Moreover the
inclusion $C_{w}\subseteq U_{w}$ is given by the vanishing of $x_{\alpha}$ for
$\alpha\in w^{-1}\Phi^{-}\cap\Phi^{-}$. The torus $T$ acts on $U_{w}$ and
$C_{w}$, and the action on the coordinates is given by $t\cdot
x_{\alpha}=(-\alpha)(t)x_{\alpha}$.

Over $U_{w}$, the line bundle $\mathcal{L}_{\kappa}$ has a non-vanishing
section $s$ given by the function $bw_{0}uw_{0}^{-1}w\mapsto(w_{0}\kappa)(b)$
on $\pi^{-1}(U_{w})=Bw_{0}Uw_{0}^{-1}w$. We compute that the action of $T$ on
$s$ is given by $t\cdot s=(w^{-1}w_{0}\kappa)(t)s$.

We now deduce from lemma 3.2.1 that
$\mathrm{H}^{p+q}_{C_{w}}(U_{w},\mathcal{L}_{\kappa})$ is concentrated in
degree $p$, and the cohomology is isomorphic to the free
$\mathcal{O}_{F}$-module:

$\bigoplus_{k_{\alpha}\geq 0~{}\forall\alpha\in
w^{-1}\Phi^{-}\cap\Phi^{+},~{}\\\ k_{\alpha}<0~{}\forall\alpha\in
w^{-1}\Phi^{-}\cap\Phi^{-}}\mathcal{O}_{F}\cdot\prod x_{\alpha}^{k_{\alpha}}s$

from which we can read off the character of the $T$-action:

$\displaystyle\mathrm{ch}\big{(}\mathrm{H}^{p}_{C_{w}}(U_{w},\mathcal{L}_{\kappa})\big{)}$
$\displaystyle=$ $\displaystyle\sum_{k_{\alpha}\geq 0~{}\forall\alpha\in
w^{-1}\Phi^{+}\cap\Phi^{-},k_{\alpha}>0~{}\forall\alpha\in
w^{-1}\Phi^{-}\cap\Phi^{-}}[w^{-1}w_{0}\kappa+\sum k_{\alpha}\alpha]$
$\displaystyle=$ $\displaystyle[w^{-1}w_{0}\kappa+\sum_{\alpha\in
w^{-1}\Phi^{-}\cap\Phi^{-}}\alpha]\frac{1}{\prod_{\alpha\in\Phi^{-}}[1]-[\alpha]}$

Finally we note

$(w^{-1}w_{0})\cdot\kappa=w^{-1}w_{0}(\kappa+\rho)-\rho=w^{-1}w_{0}\kappa-w^{-1}\rho-\rho=w^{-1}w_{0}\kappa+\sum_{\alpha\in
w^{-1}\Phi^{-}\cap\Phi^{-}}\alpha.$

∎

It follows from the lemma that the complex $E_{1}^{\bullet,0}$ from the
spectral sequence (the Grothendieck-Cousin complex):

$\mathcal{C}ous(\kappa):0\rightarrow\mathrm{H}^{0}_{Z_{0}/Z_{1}}(FL,\mathcal{L}_{\kappa})\rightarrow\cdots\rightarrow\mathrm{H}^{d}_{Z_{d}/Z_{d+1}}(FL,\mathcal{L}_{\kappa})\rightarrow
0$

computes $\mathrm{R}\Gamma(FL,\mathcal{L}_{\kappa})$. Each group decomposes

$\mathrm{H}^{p}_{Z_{p}/Z_{p+1}}(FL,\mathcal{L}_{\kappa})=\bigoplus_{w\in
W,\ell(w)=d-p}\mathrm{H}^{p}_{w}(FL,\mathcal{L}_{\kappa})$

and we have established a precise formula for the weights of $T$ on each
module.

There is a partial order on $X^{\star}(T)$ where $\lambda\geq 0$ if and only
if $\lambda$ is a sum of positive roots. Lemma 3.2.2 tells us that the weights
occurring in $\mathrm{H}^{d-\ell(w)}_{w}(FL,\mathcal{L}_{\kappa})$ are exactly
those which are $\leq(w^{-1}w_{0})\cdot\kappa$. In particular certain “big
weights” will occur in as few of the terms of the Cousin complex as possible.
We begin with the following lemma:

######  Lemma 3.2.3.

Let $\nu\in X^{\star}(T)$ be such that $\nu+\rho$ is dominant. Then the
following conditions on a weight $\lambda\in X^{\star}(T)$ are equivalent:

  1. (1)

$\lambda\not\leq w\cdot\nu$ for all $w\in W$ with $w\cdot\nu\not=\nu$.

  2. (2)

$\lambda\not\leq s_{\alpha}\cdot\nu$ for all $\alpha\in\Delta$ with
$s_{\alpha}\cdot\nu\not=\nu$.

Moreover if we additionally assume that $\lambda\leq\nu$ then we have the
further equivalent condition:

  1. (3)

$\lambda=\nu-\sum_{\alpha\in\Delta}n_{\alpha}\alpha$ with
$n_{\alpha}<\langle\alpha^{\vee},\nu\rangle+1$ for all $\alpha\in\Delta$ with
$\langle\alpha^{\vee},\nu\rangle+1>0$.

###### Proof.

Clearly the first condition implies the second. For the converse, writing $w$
as a reduced product of simple reflections, there must be at least one factor
$s_{\alpha}$ with $s_{\alpha}\cdot\nu\not=\nu$. Then $w\geq s_{\alpha}$, and
so $w\cdot\nu\leq s_{\alpha}\cdot\nu$ (see lemma 5.11.4) and hence
$\lambda\not\leq s_{\alpha}\cdot\nu$ implies $\lambda\not\leq w\cdot\nu$.

The equivalence of the second and third points follows from the formula
$s_{\alpha}\cdot\nu=\nu-(\langle\alpha^{\vee},\nu\rangle+1)\alpha$. ∎

We say that a weight $\lambda$ satisfying the conditions of the proposition
has big weight (with respect to $\nu$) and for a $T$-module $M$ which is a
direct sum of its weight spaces we denote by $M^{bw(\nu)}$ the direct sum of
its weight spaces corresponding to big weights. We note that if $\nu+\rho$ is
regular the last condition may be expressed as
$\lambda>\nu-\sum_{\alpha\in\Delta}\langle(\alpha^{\vee},\nu\rangle+1)\alpha$.

The set $X^{\star}(T)_{\mathbb{Q}}^{+}-\rho$ is a fundamental domain for the
dotted action of $W$ on $X^{\star}(T)_{\mathbb{Q}}$, and so there is a unique
$\nu\in X^{\star}(T)_{\mathbb{Q}}^{+}-\rho$ with $\nu\in W\cdot\kappa$. We let
$C(\kappa)=\\{w\in W\mid(w^{-1}w_{0})\cdot\kappa=\nu\\}$. This set is a right
torsor under $W_{\nu}=\\{w\in W\mid w\cdot\nu=\nu\\}$, and hence it is always
nonempty and consists of a single element if and only if $\kappa+\rho$ is
regular. For example, if $\kappa$ is dominant, $C(\kappa)=\\{w_{0}\\}$.

######  Proposition 3.2.4.

The cohomology complex $\mathrm{R}\Gamma(FL,\mathcal{L}_{\kappa})^{bw(\nu)}$
is a perfect complex of $\mathcal{O}_{F}$-modules of amplitude
$[\mathrm{min}_{w\in C(\kappa)}(d-\ell(w)),\mathrm{max}_{w\in
C(\kappa)}(d-\ell(w))]$.

###### Proof.

We will show that for $w\in W\setminus C(\kappa)$, the big weight part of the
term in the Grothendieck-Cousin complex corresponding to $w$ vanishes. Indeed,
by lemma 3.2.2, all the weights of
$\mathrm{H}^{d-\ell(w)}_{w}(FL,\mathcal{L}_{\kappa})$ are
$\leq(w^{-1}w_{0})\cdot\kappa$, but
$(w^{-1}w_{0})\cdot\kappa\not=w^{\prime}\cdot\nu\not=\nu$ and so
$\mathrm{H}^{d-\ell(w)}_{w}(FL,\mathcal{L}_{\kappa})^{bw(\nu)}=0$. ∎

######  Remark 3.2.5.

In particular when $\kappa+\rho$ is regular so that $C(\kappa)=\\{w\\}$
consists of a single element, we have that
$\mathrm{H}^{\star}(FL,\mathcal{L}_{\kappa})^{bw(\nu)}=\mathrm{H}_{w}^{\star}(FL,\mathcal{L}_{\kappa})^{bw(\nu)}$.
We view this statement as a sort of analog of Coleman’s classicality theorem,
where the algebraic local cohomology groups $\mathrm{H}_{w}^{\star}$ play the
role of the overconvergent cohomology groups introduced in this paper, and the
big weight condition plays the role of the small slope condition.

We emphasize that the vanishing of Proposition 3.2.4 is characteristic
independent. Of course in characteristic zero, the classical Borel-Weil-Bott
theorem gives a precise description of
$\mathrm{H}^{\star}(FL,\mathcal{L}_{\kappa})$. For the sake of completeness,
we explain how the Borel-Weil-Bott theorem may be deduced from the computation
of $\mathrm{H}^{\star}(FL,\mathcal{L}_{\kappa})$ via the Cousin complex and
basic properties of the BGG category $\mathcal{O}$.

######  Theorem 3.2.6 ([Jan03], 5.5, corollary).

Let $\kappa\in X^{\star}(T)$.

  1. (1)

If there exists no $w\in W$ such that $w\cdot\kappa$ is dominant then
$\mathrm{H}^{i}(FL,\mathcal{L}_{\kappa})\otimes F=0$ for all $i$.

  2. (2)

If there exists $w\in W$ such that $w\cdot\kappa$ is dominant, then there is a
unique such $w$, and $\mathrm{H}^{i}(FL,\mathcal{L}_{\kappa})\otimes F=0$ if
$\ell(w)\neq i$, while $\mathrm{H}^{\ell(w)}(FL,\mathcal{L}_{\kappa})\otimes
F$ is a highest weight $w\cdot\kappa$ representation.

###### Proof.

It suffices to prove the theorem after tensoring with $\bar{F}$, an algebraic
closure of $F$. For simplicity, we slightly change our notation and we assume
that $F$ is algebraically closed for this proof. There is a famous sub-
category of the category of $\mathcal{U}(\mathfrak{g})$-modules, called the
category $\mathcal{O}$. See [Hum08], chapter 1 for the definition and
properties of the category $\mathcal{O}$. We recall a number of basic results
concerning the category $\mathcal{O}$ that will be used in the argument. The
category $\mathcal{O}$ is abelian, Artinian and Noetherian. The simple objects
are indexed by weights $\lambda\in X^{\star}(T)\otimes F$ and denoted by
$L_{\lambda}$. If the module $L_{\lambda}$ is finite dimensional then
$\lambda\in X^{\star}(T)^{+}$ is dominant, and moreover $L_{\lambda}$ arises
from the highest weight $\lambda$ representation of $G$. For all $\lambda\in
X^{\star}(T)\otimes F$ we also denote by
$M_{\lambda}=\mathcal{U}(\mathfrak{g})\otimes_{\mathcal{U}(\mathfrak{b})}F(\lambda)$
the Verma module of weight $\lambda$. The simple module $L_{\lambda}$ is the
unique simple quotient of $M_{\lambda}$. The Grothendieck group of
$\mathcal{O}$, denoted by $K(\mathcal{O})$, is the free module on the
$[L_{\lambda}]$. We denote by $M\mapsto[M]$ the semi-simplification map from
$\mathcal{O}$ to $K(\mathcal{O})$. In $K(\mathcal{O})$ we have
$[M_{\lambda}]=\oplus_{w\cdot\lambda\leq\lambda}a(w\cdot\lambda,\lambda)[L_{w\cdot\lambda}]$
with $a(\lambda,\lambda)=1$. Since any element $M\in\mathcal{O}$ has
diagonalizable $\mathfrak{t}$-action, we can associate to $M$ its formal
character $\mathrm{ch}M$ which is an element of the group $\mathcal{X}$ of
functions $X^{\star}(T)\otimes F\rightarrow\mathbb{Z}$. The character is
additive on short exact sequence and we get a map
$K(\mathcal{O})\rightarrow\mathcal{X}$, $[M]\mapsto\mathrm{ch}[M]$. Moreover,
this last map is a group injection. We denote by $\mathcal{X}_{\mathcal{O}}$
its image. Finally, any $\mathcal{U}(\mathfrak{g})$-module with diagonalizable
$\mathfrak{t}$-action, and whose formal character belongs to
$\mathcal{X}_{\mathcal{O}}$ is an object of $\mathcal{O}$.

It follows from lemma 3.2.2 that
$\mathrm{ch}(\mathrm{H}^{p}_{Z_{p}/Z_{p+1}}(FL,\mathcal{L}_{\kappa})\otimes_{\mathcal{O}_{F}}F)=\oplus_{w,\ell(w)=p}\mathrm{ch}(M_{w\cdot\kappa})$.
The Grothendieck-Cousin complex

$0\rightarrow\mathrm{H}^{0}_{Z_{0}/Z_{1}}(FL,\mathcal{L}_{\kappa})\otimes
F\rightarrow\cdots\rightarrow\mathrm{H}^{d}_{Z_{d}/Z_{d+1}}(FL,\mathcal{L}_{\kappa})\otimes
F\rightarrow 0$

carries an action $\mathcal{U}(\mathfrak{g})$ by [Kem78], lemma 12.8, and is
therefore a complex in the category $\mathcal{O}$. At this stage, we see that
if none of the elements in the set $\\{w\cdot\kappa\\}$ is dominant, then none
of the
$\mathrm{H}^{p}_{Z_{p}/Z_{p+1}}(FL,\mathcal{L}_{\kappa})\otimes_{\mathcal{O}_{F}}F$
contains a finite dimensional subquotient. Otherwise there is a unique $w$
such that $w\cdot\kappa$ is dominant. We see that for $p=\ell(w)$,
$[\mathrm{H}^{p}_{Z_{p}/Z_{p+1}}(FL,\mathcal{L}_{\kappa})\otimes_{\mathcal{O}_{F}}F]$
has a unique finite dimensional constituent (with multiplicity one) equal to
$[L_{w\cdot\kappa}]$. On the other hand
$[\mathrm{H}^{i}_{Z_{i}/Z_{i+1}}(FL,\mathcal{L}_{\kappa})\otimes_{\mathcal{O}_{F}}F]$
has no finite dimensional constituent for $i\neq\ell(w)$. The cohomology
groups $\mathrm{H}^{i}(FL,\mathcal{L}_{\kappa})\otimes_{\mathcal{O}_{F}}F$ are
finite dimensional because $FL$ is proper. If none of the elements in the set
$\\{w\cdot\kappa\\}$ is dominant, the cohomology is therefore trivial. If
there is a unique $w$ such that $w\cdot\kappa$ is dominant, we see that
$\mathrm{H}^{i}(FL,\mathcal{L}_{\kappa})\otimes_{\mathcal{O}_{F}}F=0$ if
$i\neq\ell(w)$, and
$\mathrm{H}^{\ell(w)}(FL,\mathcal{L}_{\kappa})\otimes_{\mathcal{O}_{F}}F=L_{w\cdot\kappa}$.
∎

###  3.3. Analytic geometry

If $S\rightarrow\mathrm{Spec}~{}\mathcal{O}_{F}$ is a finite type morphism of
schemes, we let
$\mathcal{S}=S\times_{\mathrm{Spec}~{}\mathcal{O}_{F}}\mathrm{Spa}(F,\mathcal{O}_{F})$
be the associated analytic adic space and
${S}_{k}=S\times_{\mathrm{Spec}~{}\mathcal{O}_{F}}\mathrm{Spec}~{}k$ be the
special fiber. One can also consider $\mathcal{S}^{an}$, the analytification
of the scheme $S\times_{\mathrm{Spec}~{}\mathcal{O}_{F}}\mathrm{Spec}~{}F$,
and there is a map $\mathcal{S}\rightarrow\mathcal{S}^{an}$ which is an
isomorphism when $S$ is proper (see [Hub94], section 4). There is a continuous
specialization map $\mathrm{sp}_{\mathcal{S}}:\mathcal{S}\rightarrow{S}_{k}$
and the preimage of a subset $U\subset{S}_{k}$ is denoted by
$\mathrm{sp}_{\mathcal{S}}^{-1}(U)$. If $U$ is a locally closed subset of
$S_{k}$, we let $]U[_{\mathcal{S}}$ be the interior of $\mathrm{sp}^{-1}(U)$.
This is an adic space, called the tube of $U$ (see [Ber91]). The difference
between $\mathrm{sp}_{\mathcal{S}}^{-1}(U)$ and $]U[_{\mathcal{S}}$ consists
only of certain higher rank points. The tube $]U[_{\mathcal{S}}$ is the adic
space associated to a “classical” rigid space, while
$\mathrm{sp}_{\mathcal{S}}^{-1}(U)$ is not in general.

####  3.3.1. The Iwahori decomposition

Let $\mathcal{G}$ be the quasi-compact adic space associated to $G$ and let
$\mathrm{Iw}=]{B}_{k}[_{\mathcal{G}}$ be the Iwahori subgroup of
$\mathcal{G}$.

For any root $\alpha\in\Phi$, we have an algebraic root space
$U_{\alpha}\rightarrow\mathrm{Spec}~{}\mathcal{O}_{F}$. We let
$\mathcal{U}_{\alpha}$ be the corresponding quasi-compact adic space
(isomorphic to a unit ball) and we let
$\mathcal{U}_{\alpha}^{o}=]\\{1\\}[_{\mathcal{U}_{\alpha}}$ be the tube of the
identity element (isomorphic to an increasing union of balls of radii $r<1$).
We also let $\mathcal{U}_{\alpha}^{an}$ be the analytification of $U_{\alpha}$
(isomorphic to the affine line).

The following result gives a strong form of the Iwahori decomposition.

######  Proposition 3.3.2.

Let $\alpha_{1},\cdots,\alpha_{n}$ be an enumeration of the roots in $\Phi$.
The product map

$\mathcal{T}\times\prod\mathcal{U}_{\alpha_{i}}^{\star_{i}}\rightarrow\mathrm{Iw}$

is an isomorphism of analytic adic spaces, where
$\mathcal{U}_{\alpha_{i}}^{\star_{i}}=\mathcal{U}_{\alpha_{i}}$ if
$\alpha_{i}\in\Phi^{+}$ and
$\mathcal{U}_{\alpha_{i}}^{\star_{i}}=\mathcal{U}_{\alpha_{i}}^{o}$ if
$\alpha_{i}\in\Phi^{-}$.

######  Remark 3.3.3.

The existence of a product decomposition
$\mathcal{T}(K)\times\prod\mathcal{U}_{\alpha_{i}}^{\star_{i}}(K)\rightarrow\mathrm{Iw}(K)$
for $K$ a discretely valued field is a consequence of Bruhat-Tits theory
[Tit79], sect. 3.1.1.

###### Proof.

We let $\mathfrak{Iw}$ be the formal group scheme obtained by completing the
group $G$ along the closed subscheme $B_{k}$. For each root $\alpha$, we let
$\mathfrak{U}_{\alpha}=\mathrm{Spf}(\mathcal{O}_{F}\langle T\rangle)$ be the
formal completion of $U_{\alpha}$ along its special fiber, and we let
$\mathfrak{U}^{o}_{\alpha}=\mathrm{Spf}(\mathcal{O}_{F}[[T]])$ be the formal
completion of $\mathfrak{U}_{\alpha}$ at the identity. We also let
$\mathfrak{T}$ be the formal completion of $T$ along its special fiber.

We consider the map of formal schemes:
$\mathfrak{T}\times\prod\mathfrak{U}_{\alpha_{i}}^{\star_{i}}\rightarrow\mathfrak{Iw}$
which yields the map of the proposition after passing to the generic fiber. We
will show that this map of formal schemes is an isomorphism. Passing to
underlying reduced schemes, we obtain the isomorphism
$T_{k}\times\prod_{\alpha\in\Phi^{+}}U_{\alpha,k}\rightarrow B_{k}$. Hence it
suffices to show that it is formally étale.

We claim that this map is formally étale. Since it is an isomorphism on the
associated reduced schemes because
$T_{k}\times\prod_{\alpha\in\Phi^{+}}U_{\alpha,k}\rightarrow B_{k}$ is an
isomorphism, we deduce that the map is an isomorphism. The associated map on
the generic fiber (which is the map of the lemma) is an isomorphism. We are
left to prove that the map is formally étale. Let $k^{\prime}$ be a finite
field extension of $k$. Let
$(t,u_{\alpha_{i}})\in\mathfrak{T}\times\prod\mathfrak{U}_{\alpha_{i}}^{\star_{i}}(k^{\prime})$.
Note that $u_{\alpha_{i}}=1$ if $\alpha_{i}\in\Phi^{-}$. The map on Zariski
tangent spaces is given by:

$(t(1+\epsilon T),u_{\alpha_{i}}(1+\epsilon U_{\alpha_{i}})_{1\leq i\leq
n})\mapsto t(1+\epsilon T)\prod_{i}u_{\alpha_{i}}(1+\epsilon U_{\alpha_{i}})$

where $(T,(U_{\alpha_{i}})_{1\leq i\leq
n})\in\mathrm{Lie}(T_{k})\otimes_{k}k^{\prime}\oplus_{i}\mathrm{Lie}(U_{\alpha_{i}})\otimes_{k}k^{\prime}$,
and there is an equality:

$t(1+\epsilon T)\prod_{i}u_{\alpha_{i}}(1+\epsilon
U_{\alpha_{i}})=t\prod_{i}u_{\alpha_{i}}(1+\epsilon\mathrm{Ad}((\prod_{i}u_{\alpha_{i}})^{-1})(T)+\sum_{i=2}^{n+1}\mathrm{Ad}((\prod_{k=i}^{n}u_{\alpha_{k}})^{-1})U_{\alpha_{i-1}}).$

Therefore, if we identify the tangent space of
$\mathfrak{T}\times\prod\mathfrak{U}_{\alpha_{i}}^{\star_{i}}$ at
$(t,u_{\alpha_{i}})$ and the tangent space of $\mathfrak{Iw}$ at $t\prod
u_{\alpha_{i}}$ with

$\mathrm{Lie}(G_{k})\otimes
k^{\prime}=\mathrm{Lie}(T_{k})\otimes_{k}k^{\prime}\oplus_{i}\mathrm{Lie}(U_{\alpha_{i}})\otimes_{k}k^{\prime},$

the map on tangent spaces is given by the endomorphism:

$(T,(U_{\alpha_{i}})_{1\leq i\leq
n})\mapsto\mathrm{Ad}((\prod_{i}u_{\alpha_{k}})^{-1})(T)+\sum_{i=2}^{n+1}\mathrm{Ad}((\prod_{k=i}^{n}u_{\alpha_{k}})^{-1})(U_{\alpha_{i-1}}).$

Observe that $(\prod_{k=i}^{n}u_{\alpha_{k}})^{-1}\in U(k^{\prime})$ for all
$1\leq i\leq n$. By Chevalley’s commutativity relations ([Ste16], chapter 6),
we know that for any $u\in U(k^{\prime})$, any $\alpha\in\Phi\cup\\{0\\}$ and
any $v\in\mathrm{Lie}(G_{k})_{\alpha}\otimes k^{\prime}$, $Ad(u).v=v+w$ where
$w\in\bigoplus_{\alpha^{\prime}>\alpha}\mathrm{Lie}(G_{k})_{\alpha^{\prime}}\otimes
k^{\prime}$. A simple inductive argument proves that the map on tangent spaces
is an isomorphism. ∎

####  3.3.4. Tubes of Bruhat cells

For any $w\in{}^{M}W$, we now consider the tube of the Bruhat cell
$]C_{w,k}[_{\mathcal{FL}}=\mathcal{P}\backslash\mathcal{P}w\mathrm{Iw}$, as
well as the tubes $]X_{w,k}[_{\mathcal{FL}}$ and $]Y_{w,k}[_{\mathcal{FL}}$.
It follows from lemma 3.1.2 that
$\mathcal{X}^{w}\hookrightarrow]Y_{w,k}[_{\mathcal{FL}}$.

We can now give a very precise description of the tubes of Bruhat cells.

######  Corollary 3.3.5.

For any $w\in{}^{M}W$, and for any ordering of the roots in $\Phi$, we have an
isomorphism of analytic spaces:

$\displaystyle\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{+}}\mathcal{U}_{\alpha}\times\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{-}}\mathcal{U}_{\alpha}^{o}$
$\displaystyle\rightarrow$ $\displaystyle]C_{w,k}[_{\mathcal{FL}}$
$\displaystyle(u_{\alpha})_{\alpha\in w^{-1}\Phi^{-,M}}$
$\displaystyle\mapsto$ $\displaystyle w\prod_{\alpha}u_{\alpha}$

where the product $\prod_{\alpha}u_{\alpha}$ is taken according to our fixed
ordering.

###### Proof.

This follows easily from proposition 3.3.2. ∎

####  3.3.6. Analytic subspaces of $\mathcal{U}_{w}^{an}$

We now consider $\mathcal{U}_{w}^{an}$, the analytification of $U_{w}$. This
open subset of $\mathcal{FL}$ contains $]C_{w,k}[_{\mathcal{FL}}$, and we have
an isomorphism of analytic spaces (for any ordering of the roots):

$\displaystyle\prod_{\alpha\in(w^{-1}\Phi^{-,M})}\mathcal{U}^{an}_{\alpha}$
$\displaystyle\rightarrow$ $\displaystyle\mathcal{U}_{w}^{an}$
$\displaystyle(u_{\alpha})_{\alpha\in w^{-1}\Phi^{-,M}}$
$\displaystyle\mapsto$ $\displaystyle w\prod_{\alpha}u_{\alpha}$

We now introduce certain quasi-Stein subspaces of $\mathcal{U}_{w}^{an}$ as
well as some partial closures of them. In order to do this we introduce a
little bit of language.

If $T$ is a topological space and $S\subseteq T$ is a subset, we let as usual
$\overline{S}$ be the closure of $S$ in $T$. We will use repeatedly the
property that in a spectral space, the closure of a pro-constructible set is
the set of all its specializations (see [Sch17], lemma 2.4).

Let $\mathcal{X}$ be an adic space. Using the Yoneda lemma, we identify
$\mathcal{X}$ with a functor on complete Huber pairs $(A,A^{+})$. Let $T$ be a
subset of $\mathcal{X}$. In general $T$ is not an adic space but using the
Yoneda point of view, we can attach to $T$ a subfunctor of $\mathcal{X}$: we
let $T((A,A^{+}))$ be the subset of $\mathcal{X}((A,A^{+}))$ of morphisms
$\mathrm{Spa}(A,A^{+})\rightarrow\mathcal{X}$ whose image factors through $T$.
We observe that this functor is completely determined by its value on fields
$(K,K^{+})$. In general, we say that a subfunctor $F$ of $\mathcal{X}$ is a
subset of $\mathcal{X}$ if it is the functor attached to a subset of
$\mathcal{X}$.

######  Example 3.3.7.

Let $\mathbb{B}(0,1)$ and $\mathbb{B}^{o}(0,1)$ be respectively the quasi-
compact unit ball and the open unit ball of radius $1$ over
$\mathrm{Spa}(F,\mathcal{O}_{F})$. These are open subsets of the affine line
$(\mathbb{A}^{1})^{an}$. For any $(F,\mathcal{O}_{F})$-complete Huber pair
$(A,A^{+})$, we have $(\mathbb{A}^{1})^{an}((A,A^{+}))=A$,
$\mathbb{B}(0,1)((A,A^{+}))=A^{+}$, $\mathbb{B}^{o}(0,1)((A,A^{+}))=\\{a\in
A^{+},~{}\exists n\in\mathbb{Z}_{\geq 1},a^{n}\in pA^{+}\\}$. The closed
subset $\overline{\mathbb{B}(0,1)}$ is the union of $\mathbb{B}(0,1)$ together
with a rank two point whose maximal generalization is the Gauss point and
which points towards infinity. This is still an adic space equal to
$\mathrm{Spa}(F\langle
X\rangle,\mathcal{O}_{F}+\mathfrak{m}_{\mathcal{O}_{F}}X+\mathfrak{m}_{\mathcal{O}_{F}}X^{2}+\cdots)$.
We have $\overline{\mathbb{B}(0,1)}((A,A^{+}))=\\{a\in A,\forall
n\in\mathbb{Z}_{\geq 1},pa^{n}\in A^{+}\\}$. The closed subset
$\overline{\mathbb{B}^{o}(0,1)}$ is the union of $\mathbb{B}^{o}(0,1)$
together with a rank two point whose maximal generalization is the Gauss point
and which points towards the center of the disc. This is not adic space. We
have $\overline{\mathbb{B}^{o}(0,1)}((K,K^{+}))=K^{++}=\\{a\in K,|a|<1\\}$.

We identify each $\mathcal{U}_{\alpha}$ with the unit ball of center $0$ and
coordinate $u_{\alpha}$ with its additive group law (the coordinate
$u_{\alpha}$ is well defined up to multiplication by a unit). For all
$m\in\mathbb{Q}\cup\\{-\infty\\}$ and all $\alpha\in\Phi$, we let
$\mathcal{U}_{\alpha,m}=\\{|.|\in\mathcal{U}^{an}_{\alpha},~{}|u_{\alpha}|\leq|p^{m}|\\}$
and
$\mathcal{U}^{o}_{\alpha,m}=\cup_{m^{\prime}>m}\mathcal{U}_{\alpha,m^{\prime}}$.
For all $m,n\in\mathbb{Q}\cup\\{-\infty\\}$, we let $]C_{w,k}[_{m,n}$ be the
image of

$w\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{-}}\mathcal{U}_{\alpha,m}^{o}\times\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{+}}\mathcal{U}_{\alpha,n}\rightarrow\mathcal{U}_{w}^{an},$

For all $m,n\in\mathbb{Q}$, we let $]C_{w,k}[_{\overline{m},n}$ be the image
of

$w\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{-}}\overline{\mathcal{U}_{\alpha,m}^{o}}\times\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{+}}\mathcal{U}_{\alpha,n}\rightarrow\mathcal{U}_{w}^{an},$

we let $]C_{w,k}[_{m,\overline{n}}$ be the image of

$w\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{-}}\mathcal{U}_{\alpha,m}^{o}\times\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{+}}\overline{\mathcal{U}_{\alpha,n}}\rightarrow\mathcal{U}_{w}^{an},$

and we let $]C_{w,k}[_{\overline{m},\overline{n}}$ be the image of

$w\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{-}}\overline{\mathcal{U}^{o}_{\alpha,m}}\times\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{+}}\overline{\mathcal{U}_{\alpha,n}}\rightarrow\mathcal{U}_{w}^{an}.$

Clearly, $]C_{w,k}[=]C_{w,k}[_{0,0}$, $]C_{w,k}[_{m,n}\subseteq]C_{w,k}[$ if
$m,n\geq 0$, and moreover then we have
$]C_{w,k}[_{m,n}=]C_{w,k}[_{m,0}\cap]C_{w,k}[_{0,n}$,
$]C_{w,k}[_{\overline{m},n}=]C_{w,k}[_{\overline{m},0}\cap]C_{w,k}[_{0,n}$ and
$]C_{w,k}[_{m,\overline{n}}=]C_{w,k}[_{m,0}\cap]C_{w,k}[_{0,\overline{n}}$.

######  Remark 3.3.8.

In the formulas defining the sets $]C_{w,k}[_{m,n}$ and their partial
closures, the image is taken in the sheaf theoretic sense. If the spaces at
hand are adic spaces, this is also the images of the corresponding morphisms
of adic spaces.

######  Remark 3.3.9.

In the above formulas, we make the product for any ordering of the roots in
$(w^{-1}\Phi^{-,M})\cap\Phi^{+}$ or $(w^{-1}\Phi^{-,M})\cap\Phi^{-}$. See
[Ste16], lemma 17 for a justification that the order doesn’t matter. We also
point out that in our applications to Shimura varieties the unipotent radical
of $P$ is abelian so that the root groups $U_{\alpha}$ for $\alpha\in
w^{-1}\Phi^{-,M}$ commute with each other.

####  3.3.10. Orbits of Cells

We can give a more group theoretic description of certain of the above sets.
We introduce some subgroups of $\mathcal{G}$.

For $m\in\mathbb{Q}_{\geq 0}$ we let $\mathcal{G}_{B,m}$ (resp.
$\mathcal{G}_{U,m}$, $\mathcal{G}_{\overline{B},m}$,
$\mathcal{G}_{\overline{U},m}$) for the affinoid subgroup of $\mathcal{G}$ of
elements reducing to $B$ (resp. $U$, $\overline{B}$, $\overline{U}$) mod
$p^{m}$. We also define
$\mathcal{G}_{B,m}^{\circ}=\cup_{m^{\prime}>m}\mathcal{G}_{B,m^{\prime}}$,
$\mathcal{G}_{U,m}^{\circ}=\cup_{m^{\prime}>m}\mathcal{G}_{U,m^{\prime}}$.

Then for all $m,n\in\mathbb{Q}_{\geq 0}$ we let
$\mathcal{G}_{m,n}=\mathcal{G}_{B,m}^{\circ}\cap\mathcal{G}_{\overline{B},n}$.
In particular we note that $\mathcal{G}_{0,0}=\mathrm{Iw}$. We also let
$\mathcal{G}^{1}_{m,n}=\mathcal{G}_{U,m}^{\circ}\cap\mathcal{G}_{\overline{U},n}$.

We would also like to introduce some “partial closures” of these groups. In
general these won’t be adic spaces, but simply certain subsets of
$\mathcal{G}^{an}$. For all $m,n\in\mathbb{Q}_{\geq 0}$ we let
$\mathcal{G}_{\overline{m},n}=\overline{\mathcal{G}_{B,m}^{\circ}}\cap\mathcal{G}_{\overline{B},n}$
and
$\mathcal{G}^{1}_{\overline{m},n}=\overline{\mathcal{G}_{U,m}^{\circ}}\cap\mathcal{G}_{\overline{U},n}$,
where the closures are taken inside $\mathcal{G}$. We note that
$\mathcal{G}_{\overline{0},0}=\mathrm{sp}^{-1}(B_{k})$, the closure of
$\mathrm{Iw}$ in $\mathcal{G}$. For all $m,n\in\mathbb{Q}_{\geq 0}$, we let
$\mathcal{G}_{m,\overline{n}}$ and $\mathcal{G}^{1}_{m,\overline{n}}$ be set
of all specializations of $\mathcal{G}_{m,n}$ and $\mathcal{G}^{1}_{m,n}$
respectively in $\mathcal{G}^{an}$. For all $m,n\in\mathbb{Q}_{\geq 0}$ with
$n>0$ we also observe that
$\mathcal{G}_{m,\overline{n}}=\mathcal{G}_{B,m}^{\circ}\cap\overline{\mathcal{G}_{\overline{B},n}}$
and
$\mathcal{G}^{1}_{m,\overline{n}}=\mathcal{G}_{U,m}^{\circ}\cap\overline{\mathcal{G}_{\overline{U},n}}$
where the closures are taken inside $\mathcal{G}$.

######  Lemma 3.3.11.

The subsets $\mathcal{G}_{m,n}$, $\mathcal{G}_{\overline{m},n}$ and
$\mathcal{G}_{m,\overline{n}}$, $\mathcal{G}^{1}_{m,n}$,
$\mathcal{G}^{1}_{\overline{m},n}$ and $\mathcal{G}^{1}_{m,\overline{n}}$ of
$\mathcal{G}^{an}$ are subgroups (in the sense that their functor of points is
a subgroup of $\mathcal{G}^{an}$).

###### Proof.

It is clear that $\mathcal{G}_{B,m}$, $\mathcal{G}_{U,m}$,
$\mathcal{G}_{\overline{B},n}$, $\mathcal{G}_{\overline{U},n}$ are groups.
Therefore, the $\mathcal{G}_{m,n}$ are groups. We deduce that the
$\mathcal{G}_{m,\overline{n}}$ are groups. Indeed,
$\mathcal{G}_{m,\overline{n}}((K,K^{+}))=\mathcal{G}_{m,{n}}(K,K^{0})$ for any
field (where $K^{0}$ is the ring of power bounded elements in $K$). We also
check that $\mathcal{G}_{\overline{m},0}$ are groups. Indeed,
$\mathcal{G}_{\overline{m},0}((K,K^{+}))=\\{g\in G(K^{+}),g\mod p^{m}K^{++}\in
B(K^{+}/p^{m}K^{++})\\}$. We finally deduce that
$\mathcal{G}_{\overline{m},n}$ is a subgroup. ∎

Note that the groups $\mathcal{G}_{m,n}$, $\mathcal{G}_{\overline{m},n}$, and
$\mathcal{G}_{m,\overline{n}}$ all have the same rank 1 points. Moreover they
all have Iwahori decompositions:

######  Proposition 3.3.12.

Let $w\in W$. Then for $m,n\in\mathbb{Q}_{\geq 0}$ and for
$\mathcal{G}^{\prime}$ one of $\mathcal{G}^{\star}_{m,n}$,
$\mathcal{G}^{\star}_{\overline{m},n}$, $\mathcal{G}^{\star}_{m,\overline{n}}$
with $\star\in\\{\emptyset,1\\}$ the product map

$\mathcal{T}\cap\mathcal{G}^{\prime}\times\prod_{\alpha_{i}\in
w^{-1}\Phi^{+}}\mathcal{U}_{\alpha_{i}}^{\star_{i}}\prod_{\alpha_{i}\in
w^{-1}\Phi^{-}}\mathcal{U}_{\alpha_{i}}^{\star_{i}}\rightarrow\mathcal{G}^{\prime}$

is an isomorphism, where if $\alpha_{i}\in\Phi^{+}$ then
$\mathcal{U}_{\alpha_{i}}^{\star_{i}}$ is $\mathcal{U}_{\alpha_{i},n}$ (resp.
$\overline{\mathcal{U}_{\alpha_{i},n}}$) if
$\mathcal{G}^{\prime}=\mathcal{G}_{m,n}^{\star},\mathcal{G}_{\overline{m},n}^{\star}$
(resp. $\mathcal{G}^{\prime}=\mathcal{G}_{m,\overline{n}}^{\star}$) while if
$\alpha_{i}\in\Phi^{-}$ then $\mathcal{U}_{\alpha_{i}}^{\star_{i}}$ is
$\mathcal{U}^{o}_{\alpha_{i},m}$ (resp.
$\overline{\mathcal{U}^{o}_{\alpha_{i},m}}$) if
$\mathcal{G}^{\prime}=\mathcal{G}_{m,n}^{\star},\mathcal{G}_{m,\overline{n}}^{\star}$.
(resp. $\mathcal{G}^{\prime}=\mathcal{G}_{\overline{m},n}^{\star}$.)

###### Proof.

We first give the argument for $\mathcal{G}_{\overline{0},0}$. Since
$\mathcal{G}_{\overline{0},0}=\mathrm{sp}^{-1}(B_{k})$, and
$B_{k}\hookrightarrow w^{-1}\overline{B_{k}}B_{k}w$, it follows that
$\mathcal{G}_{\overline{0},0}\hookrightarrow\mathcal{T}\times\prod_{\alpha\in
w^{-1}\Phi^{-}}\mathcal{U}_{\alpha}\prod_{\alpha\in
w^{-1}\Phi^{+}}\mathcal{U}_{\alpha}=\mathrm{sp}^{-1}(w^{-1}\overline{B_{k}}B_{k}w)$.
We now let $g\in\mathcal{G}_{\overline{0},0}(K,K^{+})$ for a field $K$ and
valuation ring $K^{+}\subseteq K$. Let $K^{++}$ be the maximal ideal of
$K^{+}$. By definition, $g\in G(K^{+})$ and its image $\overline{g}$ in
$G(K^{+}/K^{++})$ lies in $B$. Let $g=t\prod_{\alpha\in
w^{-1}\Phi^{+}}u_{\alpha}\prod_{\alpha\in w^{-1}}u_{\alpha}$. We have
$\overline{g}=\overline{t}\prod_{\alpha\in
w^{-1}\Phi^{+}}\overline{u_{\alpha}}\prod_{\alpha\in
w^{-1}}\overline{u_{\alpha}}$ and we find that $\overline{u_{\alpha}}=1$ if
$\alpha\in\Phi^{-}$ by unicity of the decomposition. The case of
$\mathcal{G}_{0,0}$ has already been treated in proposition 3.3.2. The case of
$\mathcal{G}_{0,\overline{0}}$ follows. The general case is left to the
reader. ∎

######  Remark 3.3.13.

For the group $\mathcal{G}_{0,0}=\mathrm{Iw}$, the decomposition holds for any
ordering of the roots $\alpha\in\Phi$ by proposition 3.3.2. We don’t know if
this property holds for $\mathcal{G}_{\overline{0},0}$ for example.

######  Corollary 3.3.14.

We have that for all $m,n\in\mathbb{Q}_{\geq 0}$,
$]C_{w,k}[_{m,n}=\mathcal{P}\backslash\mathcal{P}w\mathcal{G}_{m,n}=\mathcal{P}\backslash\mathcal{P}w\mathcal{G}_{m,n}^{1}$,
$]C_{w,k}[_{\overline{m},n}=\mathcal{P}\backslash\mathcal{P}w\mathcal{G}_{\overline{m},n}=\mathcal{P}\backslash\mathcal{P}w\mathcal{G}_{\overline{m},n}^{1}$
and ,
$]C_{w,k}[_{m,\overline{n}}=\mathcal{P}\backslash\mathcal{P}w\mathcal{G}_{m,\overline{n}}=\mathcal{P}\backslash\mathcal{P}w\mathcal{G}_{m,\overline{n}}^{1}$.

###### Proof.

We only prove the identity
$]C_{w,k}[_{m,n}=\mathcal{P}\backslash\mathcal{P}w\mathcal{G}_{m,n}$, the
other identities are proved similarly. By proposition 3.3.12, we have that

$\displaystyle\mathcal{P}\backslash\mathcal{P}w\mathcal{G}_{m,n}$
$\displaystyle=$ $\displaystyle
w.w^{-1}\mathcal{P}w\cap\mathcal{G}_{m,n}\backslash\mathcal{G}_{m,n}$
$\displaystyle=$ $\displaystyle w.\prod_{\alpha_{i}\in
w^{-1}\Phi^{-}_{M}}\mathcal{U}^{\star_{i}}_{\alpha_{i}}\prod_{\alpha_{i}\in
w^{-1}\Phi^{+}}\mathcal{U}_{\alpha_{i}}^{\star_{i}}\backslash\prod_{\alpha_{i}\in
w^{-1}\Phi^{+}}\mathcal{U}_{\alpha_{i}}^{\star_{i}}\prod_{\alpha_{i}\in
w^{-1}\Phi^{-}}\mathcal{U}_{\alpha_{i}}^{\star_{i}}$ $\displaystyle=$
$\displaystyle w\prod_{\alpha\in
w^{-1}\Phi^{-,M}}\mathcal{U}_{\alpha_{i}}^{\star_{i}}$

where if $\alpha_{i}\in\Phi^{+}$ then $\mathcal{U}_{\alpha_{i}}^{\star_{i}}$
is $\mathcal{U}_{\alpha_{i},n}$ and if $\alpha_{i}\in\Phi^{-}$ then
$\mathcal{U}_{\alpha_{i}}^{\star_{i}}$ is $\mathcal{U}^{o}_{\alpha_{i},m}$. ∎

######  Lemma 3.3.15.

For all $m\in\mathbb{Z}_{\geq 0}$ and $k\in\mathbb{Z}_{\geq 0}$, the groups
$\mathcal{G}^{1}_{m+k,k}$, $\mathcal{G}^{1}_{\overline{m+k},k}$ and
$\mathcal{G}^{1}_{m+k,\overline{k}}$ are normalized by $\mathcal{G}_{m,0}$.

###### Proof.

The case $k=0$ is trivial. We assume that $k\geq 1$. We can find a closed
embedding $G\hookrightarrow\mathrm{GL}_{r}$ and a Borel $B_{\mathrm{GL_{r}}}$
of $\mathrm{GL}_{r}$ with the property that $B=G\cap B_{GL_{n}}$. Indeed,
first consider a faithful representation of $G$ into $\mathrm{GL}_{r}$, then
consider the action of $B$ on the flag variety of Borels of $\mathrm{GL}_{r}$.
This action of a solvable group on a proper scheme must have a fixed point
$B_{GL_{r}}$, and then $B\subseteq G\cap B_{GL_{r}}$. But then we must have
$B=G\cap B_{GL_{r}}$ as the later is a solvable subgroup of $G$ containing
$B$. Therefore, the problem is reduced to the case of the group
$\mathrm{GL}_{r}$. We now consider certain sub-algebras of the algebra of
$r\times r$ matrices $\mathcal{M}^{an}_{r}$. For all $s\in\mathbb{Q}$, we let
$\mathbb{B}(0,s)$ the (quasi-compact) ball of center $0$ and radius $s$, and
$\mathbb{B}^{o}(0,s)=\cup_{s^{\prime}<s}\mathbb{B}(0,s^{\prime})$ be the
“open” ball of radius $s$ (which is a Stein space). For any $m\geq 0$, we let:

$\mathrm{Lie}_{m,0}(S,S^{+})=$
$\\{(a_{i,j})\in\mathcal{M}^{an}_{r}(S,S^{+}),~{}a_{i,j}\in\mathbb{B}^{o}(0,|p^{m}|)(S,S^{+})~{}\textrm{if}~{}i\geq
j,a_{i,j}\in{\mathbb{B}(0,1)}(S,S^{+})~{}\textrm{if}~{}i\leq j\\},$
$\mathrm{Lie}_{\overline{m},0}(S,S^{+})=$
$\\{(a_{i,j})\in\mathcal{M}^{an}_{r}(S,S^{+}),a_{i,j}\in\overline{\mathbb{B}^{o}(0,|p^{m}|)}(S,S^{+})~{}\textrm{if}~{}i\geq
j,a_{i,j}\in{\mathbb{B}(0,1)}(S,S^{+})~{}\textrm{if}~{}i\leq j\\},$
$\mathrm{Lie}_{{m},\overline{0}}(S,S^{+})=$
$\\{(a_{i,j})\in\mathcal{M}^{an}_{r}(S,S^{+}),~{}a_{i,j}\in\mathbb{B}^{o}(0,|p^{m}|)(S,S^{+})~{}\textrm{if}~{}i\geq
j\,a_{i,j}\in\overline{\mathbb{B}(0,1)}(S,S^{+})~{}\textrm{if}~{}i\leq j\\}.$

We claim that these algebras are stable under the adjoint action of
$\mathcal{G}_{m,0}$. Since $\mathrm{Lie}_{m,0}$ is the Lie algebra of
$\mathcal{G}^{1}_{m,0}$, and $\mathcal{G}_{m,0}$ normalizes
$\mathcal{G}^{1}_{m,0}$, this case follows easily. The other cases are
elementary to check by hand. A typical element of $\mathcal{G}^{1}_{m+k,k}$
(resp. $\mathcal{G}^{1}_{\overline{m+k},k}$, resp.
$\mathcal{G}^{1}_{{m+k},\overline{k}}$) writes $(1+p^{k}g)$ where
$g\in\mathrm{Lie}_{m,0}$ (resp. $\mathrm{Lie}_{\overline{m},0}$, resp.
$\mathrm{Lie}_{m,\overline{0}}$). For $h\in\mathcal{G}_{m,0}$, we have
$h(1+p^{k}g)h^{-1}=1+p^{k}\mathrm{Ad}(h).g$. ∎

######  Lemma 3.3.16.

Let $w\in{}^{M}W$. Let $m\geq 0$, and let $K_{p}\subset\mathcal{G}_{m,0}$ be a
profinite subgroup. For all $k\geq 0$, the sets

$]C_{w,k}[_{m+k,k}K_{p}$, $]C_{w,k}[_{\overline{m+k},k}K_{p}$ and
$]C_{w,k}[_{{m+k},\overline{k}}K_{p}$

are a finite disjoint union of translates of the form

$]C_{w,k}[_{m+k,k}h$, $]C_{w,k}[_{\overline{m+k},k}h$, and respectively
$]C_{w,k}[_{{m+k},\overline{k}}h$ for $h\in K_{p}$.

###### Proof.

We prove the first statement, the others are identical. For $h\in K_{p}$ we
have
$]C_{w,k}[_{m+k,k}h=\mathcal{P}\backslash\mathcal{P}w\mathcal{G}_{m+k,k}^{1}h=\mathcal{P}\backslash\mathcal{P}wh\mathcal{G}_{m+k,k}^{1}$
by Lemma 3.3.15. This proves that if for $h,h^{\prime}\in K_{p}$,
$]C_{w,k}[_{m+k,k}h\cap]C_{w,k}[_{m+k,k}h^{\prime}\neq\emptyset$, then
$]C_{w,k}[_{m+k,k}h=]C_{w,k}[_{m+k,k}h^{\prime}$. Therefore,
$]C_{w,k}[_{m+k,k}K_{p}$ is a disjoint union of translates. This disjoint
union is finite because $\mathcal{G}_{m+k,k}\cap K_{p}$ is of finite index in
$K_{p}$. ∎

We now consider certain intersections.

######  Lemma 3.3.17.

Let $w\in{}^{M}W$. Let $m,n\in\mathbb{Z}_{\geq 0}$ and let $K_{p}$ be a
subgroup of $\mathcal{G}_{m,0}$. Then we have:

$\displaystyle]C_{w,k}[_{m,0}K_{p}\cap]C_{w,k}[_{0,n}K_{p}$ $\displaystyle=$
$\displaystyle]C_{w,k}[_{m,n}K_{p},$
$\displaystyle]C_{w,k}[_{m,\overline{0}}K_{p}\cap]C_{w,k}[_{0,\overline{n}}K_{p}$
$\displaystyle=$ $\displaystyle]C_{w,k}[_{m,\overline{n}}K_{p},$
$\displaystyle]C_{w,k}[_{\overline{m},0}K_{p}\cap]C_{w,k}[_{\overline{0},n}K_{p}$
$\displaystyle=$ $\displaystyle]C_{w,k}[_{\overline{m},n}K_{p}.$

###### Proof.

We only check the first statement as the others are identical. As
$K_{p}\subset\mathcal{G}_{m,0}$, $]C_{w,k}[_{m,0}K_{p}=]C_{w,k}[_{m,0}$. Let
$x\in]C_{w,k}[_{m,0}K_{p}\cap]C_{w,k}[_{0,n}K_{p}$. Then, there exists $h\in
K_{p}$, $xh\in]C_{w,k}[_{m,0}\cap]C_{w,k}[_{0,n}=]C_{w,k}[_{m,n}$. The reverse
inclusion is trivial. ∎

####  3.3.18. Intersections and unions of cells

We prove a few more results concerning the intersections and unions of various
subsets of $\mathcal{FL}$ we have introduced so far.

######  Lemma 3.3.19.

For any integer $r$, we have:

$\displaystyle\mathcal{FL}$ $\displaystyle=$
$\displaystyle]\cup_{w,\ell(w)\geq
r}C_{w,k}[\coprod\overline{]\cup_{w,\ell(w)\leq r-1}C_{w,k}[}$
$\displaystyle=$ $\displaystyle\overline{]\cup_{w,\ell(w)\geq
r}C_{w,k}[}\coprod{]\cup_{w,\ell(w)\leq r-1}C_{w,k}[}$

###### Proof.

The first equality is obtained by applying $\mathrm{sp}^{-1}$ to the
decomposition of the special fiber:

$FL_{k}=\cup_{w,\ell(w)\geq r}C_{w,k}\coprod\cup_{w,\ell(w)\leq r-1}C_{w,k}$

For the second equality, first observe that both
$\overline{]\cup_{w,\ell(w)\geq r}C_{w,k}[}$ and ${]\cup_{w,\ell(w)\leq
r-1}C_{w,k}[}$ are stable under specialization and generalization. Since the
disjoint union contains all rank one points of $\mathcal{FL}$, it contains all
of them. Since the intersection contains no rank one points, it is empty. ∎

######  Lemma 3.3.20.

Let $w,w^{\prime}\in{}^{M}W$. Assume that $\ell(w)\leq\ell(w^{\prime})$ and
$w\neq w^{\prime}$. Then:

  1. (1)

$\overline{]X_{w,k}[}\cap]Y_{w^{\prime},k}[=\emptyset$,

  2. (2)

$]X_{w,k}[\cap\overline{]Y_{w^{\prime},k}[}=\emptyset$.

###### Proof.

We prove the first point. Note that
$\overline{]X_{w,k}[}=\mathrm{sp}^{-1}(X_{w,k})$ and
$]Y_{w^{\prime},k}[=\mathrm{sp}^{-1}(Y_{w^{\prime},k})$. Thus the first point
follows from the fact that $X_{w,k}\cap Y_{w^{\prime},k}=\emptyset$. We prove
the second point. Since $]Y_{w^{\prime},k}[$ is quasi-compact open,
$\overline{]Y_{w^{\prime},k}[}$ is the set of all its specializations.
Therefore, both $\overline{]Y_{w^{\prime},k}[}$ and $]X_{w,k}[$ are stable
under generalization. But they have no common rank one point, again as
$X_{w,k}\cap Y_{w^{\prime},k}=\emptyset$. ∎

######  Lemma 3.3.21.

  1. (1)

For all $m,n\in\mathbb{Q}$, we have
$\overline{]C_{w,k}[_{m,n}}=]C_{w,k}[_{\overline{m},\overline{n}}\subseteq\mathcal{U}_{w}^{an}$,

  2. (2)

We have $]C_{w,k}[_{0,-\infty}\subseteq]X_{w,k}[.$

  3. (3)

We have $]C_{w,k}[_{-\infty,0}\subseteq]Y_{w,k}[.$

  4. (4)

We have
$\overline{]Y_{w,k}[}\cap]X_{w,k}[=]C_{w,k}[_{0,\overline{0}}\subseteq\overline{]C_{w,k}[}$.

  5. (5)

We have
${]Y_{w,k}[}\cap\overline{]X_{w,k}[}=]C_{w,k}[_{\overline{0},{0}}\subseteq\overline{]C_{w,k}[}$.

###### Proof.

We begin with the first point. Let $Z=FL\setminus U_{w}$, a closed subscheme.
We let $\mathcal{Z}$ be the associated analytic space which is proper. We have
$\mathcal{FL}=\mathcal{U}_{w}^{an}\coprod\mathcal{Z}$. Since $\mathcal{Z}$ is
proper, we deduce that $\mathcal{U}_{w}^{an}$ is stable under specializations.
Let us consider $]C_{w,k}[_{m,n}$. We claim that
$\overline{]C_{w,k}[_{m,n}}\subseteq\mathcal{U}_{w}^{an}$. To prove this, we
can first find a quasi-compact open $\mathcal{U}$ such that
$]C_{w,k}[_{m,n}\subseteq\mathcal{U}\subseteq\mathcal{U}_{w}^{an}$ and we see
that $\overline{\mathcal{U}}$ is the set of all its specializations and is
therefore included in $\mathcal{U}_{w}^{an}$. Now it is clear that
$\overline{]C_{w,k}[_{m,n}}=]C_{w,k}[_{\overline{m},\overline{n}}$.

We now prove the second point. We observe that $w\prod_{\alpha\in
w^{-1}\Phi^{-,M}\cap\Phi^{+}}\mathcal{U}_{\alpha}^{an}=\mathcal{C}_{w}^{an}\subseteq\mathcal{X}_{w}\subseteq]X_{w,k}[$
and since $]X_{w,k}[$ is invariant under multiplication by the Iwahori
subgroup,

$w\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{+}}{\mathcal{U}^{an}_{\alpha}}\times\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{-}}{\mathcal{U}_{\alpha}^{o}}\subseteq]X_{w,k}[.$

We check the third point. We observe that $w\prod_{\alpha\in
w^{-1}\Phi^{-,M}\cap\Phi^{-}}\mathcal{U}_{\alpha}^{an}=\mathcal{C}^{w,an}\subseteq\mathcal{X}^{w}\subseteq]Y_{w,k}[$
and again the conclusion follows because $]Y_{w,k}[$ is invariant under
multiplication by the Iwahori subgroup.

We prove the fourth point. Since $]Y_{w,k}[$ is quasi-compact, it is
constructible, and therefore $\overline{]Y_{w,k}[}$ is the set of all
specializations of $]Y_{w,k}[$ in $\mathcal{FL}$. Let
$x\in\overline{]Y_{w,k}[}\cap]X_{w,k}[$ and let $y$ be the maximal
generalization of $x$ in $\mathcal{FL}$. Then
$y\in]Y_{w,k}[\cap]X_{w,k}[=]C_{w,k}[$. The subset of $\overline{]C_{w,k}[}$
consisting of points whose maximal generalization is in $]C_{w,k}[$ is exactly
$w\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{+}}\overline{\mathcal{U}_{\alpha}}\times\prod_{\alpha\in(w^{-1}\Phi^{-,M})\cap\Phi^{-}}{\mathcal{U}_{\alpha}^{o}}$.
This proves that $\overline{]Y_{w,k}[}\cap]X_{w,k}[$ is included in that set.
The converse inclusion follows easily from the second and third points. We
prove the last point. Since $\overline{]X_{w,k}[}=\mathrm{sp}^{-1}(X_{w,k})$
and $]Y_{w,k}[=\mathrm{sp}^{-1}(Y_{w,k})$, we deduce that
${]Y_{w,k}[}\cap\overline{]X_{w,k}[}=\mathrm{sp}^{-1}(C_{w,k})\subseteq\overline{]C_{w,k}[}$
has exactly the announced description. ∎

######  Lemma 3.3.22.

  1. (1)

We have
$]X_{w,k}[=]C_{w,k}[_{0,\overline{0}}\coprod\cup_{w^{\prime}<w}]X_{w^{\prime},k}[$.

  2. (2)

We have
$]Y_{w,k}[=]C_{w,k}[_{\overline{0},{0}}\coprod\cup_{w^{\prime}>w}]Y_{w^{\prime},k}[$.

  3. (3)

We have
$\overline{]X_{w,k}[}=]C_{w,k}[_{\overline{0},{0}}\coprod\cup_{w^{\prime}<w}\overline{]X_{w^{\prime},k}[}$.

  4. (4)

We have
$\overline{]Y_{w,k}[}=]C_{w,k}[_{{0},\overline{0}}\coprod\cup_{w^{\prime}>w}\overline{]Y_{w^{\prime},k}[}$.

###### Proof.

We prove the first point and the direct inclusion. We first observe that since
$X_{w^{\prime},k}$ is closed, $]X_{w^{\prime},k}[$ is a finite union of Stein
spaces. Therefore $X_{w^{\prime},k}$ is stable under specialization. Let
$x\in]X_{w,k}[$. If its maximal generalization is in $]X_{w^{\prime},k}[$ for
some $w^{\prime}<w$, then actually $x\in]X_{w^{\prime},k}[$. Otherwise,
$x\in\overline{]C_{w,k}[}\cap]X_{w,k}[=]C_{w,k}[_{0,\overline{0}}$ by lemma
3.3.21. The converse inclusion follows from the same lemma. The union is
disjoint by lemma 3.3.20. We prove the second point. We note that
$]Y_{w,k}[=\mathrm{sp}^{-1}(Y_{w,k})$ and therefore,

$]Y_{w,k}[=\mathrm{sp}^{-1}(C_{w,k})\cap]Y_{w,k}[\bigcup\cup_{w^{\prime}>w}]Y_{w^{\prime},k}[.$

By lemma 3.3.21
$\mathrm{sp}^{-1}(C_{w,k})\cap]Y_{w,k}[\subset]C_{w,k}[_{\overline{0},0}$. The
reverse inclusion follows also from the lemma. The union is disjoint by lemma
3.3.20. We prove the third point. We have
$X_{w,k}=C_{w,k}\coprod\cup_{w^{\prime}<w}X_{w^{\prime},k}$. Taking
$\mathrm{sp}^{-1}$ gives the identity. We prove the last point. All three
spaces $\overline{]Y_{w,k}[}$,
$]C_{w,k}[_{{0},\overline{0}}=\overline{]Y_{w,k}[}\cap\overline{]X_{w,k}[}$
and $\cup_{w^{\prime}>w}\overline{]Y_{w^{\prime},k}[}$ are stable under
specialization and generalization. It suffices therefore to prove the identity
on rank one points, and it is clear. ∎

###  3.4. Dynamics

Let $v:F\rightarrow\mathbb{R}\cup\\{+\infty\\}$ be the $p$-adic valuation
normalized by $v(p)=1$. We consider certain sub-semigroups of $T(F)$. We let
$T^{+}(F)=\\{t\in T(F),v(\alpha(t))\geq 0,~{}\forall\alpha\in\Phi^{+}\\}$,
$T^{++}(F)=\\{t\in T(F),v(\alpha(t))>0,~{}\forall\alpha\in\Phi^{+}\\}$,
$T^{-}(F)=\\{t\in T(F),v(\alpha(t))\leq 0,~{}\forall\alpha\in\Phi^{+}\\}$,
$T^{--}(F)=\\{t\in T(F),v(\alpha(t))<0,~{}\forall\alpha\in\Phi^{+}\\}$.

######  Lemma 3.4.1.

  1. (1)

If $t\in T^{+}(F)$, $]X_{w,k}[.t\subseteq]X_{w,k}[$ for all $w\in~{}{}^{M}W$.

  2. (2)

If $t\in T^{-}(F)$, $]Y_{w,k}[.t\subseteq]Y_{w,k}[$ for all $w\in~{}{}^{M}W$.

###### Proof.

By [Hub93], corollary 4.2, we reduce to check the inclusions on rank $1$
points. For the first point, it suffices therefore to prove that
$]C_{w,k}[.t\subseteq]X_{w,k}[$. Let $wbu\in]C_{w,k}[$ with $b\in\mathcal{B}$
and $u\in\prod_{\alpha\in\Phi^{-}}\mathcal{U}_{\alpha}^{o}$. We find that
$wbut=wt^{-1}btt^{-1}ut$. Now,
$\mathrm{Ad}(t^{-1})u\in\prod_{\alpha\in\Phi^{-}}\mathcal{U}_{\alpha}^{o}$,
while $\mathrm{Ad}(t^{-1})b\in\mathcal{B}^{an}$. In particular
$w\mathrm{Ad}(t^{-1})b\in\mathcal{X}_{w}\subseteq]X_{w,k}[_{\mathcal{FL}}$.
Therefore $w\mathrm{Ad}(t^{-1})b\mathrm{Ad}(t^{-1})u\in]X_{w,k}[$. For the
second point, it suffices to prove that $]C_{w,k}[.t\subseteq]Y_{w,k}[$. Let
$wub\in]C_{w,k}[$ with $b\in\mathcal{B}$ and $u\in\overline{\mathcal{U}}^{o}$.
We find that $wubt=wt^{-1}utt^{-1}bt$. Now,
$\mathrm{Ad}(t^{-1})b\in{\mathcal{B}}$, while
$\mathrm{Ad}(t^{-1})u\in\overline{\mathcal{B}}^{an}$. In particular,
$w\mathrm{Ad}(t^{-1})u\in\mathcal{X}^{w}\subseteq]Y_{w,k}[_{\mathcal{FL}}$.
Therefore $w\mathrm{Ad}(t^{-1})u\mathrm{Ad}(t^{-1})b\in]Y_{w,k}[$. ∎

For $t\in T^{++}(F)$, we define $\min(t)=\min_{\alpha\in\Phi^{+}}v(\alpha(t))$
and $\max(t)=\max_{\alpha\in\Phi^{+}}v(\alpha(t))$. For $t\in T^{--}(F)$, we
let $\min(t)=\min_{\alpha\in\Phi^{-}}v(\alpha(t))$ and
$\max(t)=\max_{\alpha\in\Phi^{-}}v(\alpha(t))$. We note that $\min(t)>0$ and
that $\min(t)=\min(t^{-1})$ and $\max(t)=\max(t^{-1})$ for $t\in T^{++}(F)$.

######  Lemma 3.4.2.

  1. (1)

Let $t\in T^{++}(F)$. For all $w\in{}^{M}W$, $m,n\in\mathbb{Q}$, we have

$\displaystyle]C_{w,k}[_{m+\max(t),n-\min(t)}$ $\displaystyle\subseteq$
$\displaystyle]C_{w,k}[_{m,n}.t\subseteq]C_{w,k}[_{m+\min(t),n-\max(t)}$
$\displaystyle]C_{w,k}[_{\overline{m+\max(t)},n-\min(t)}$
$\displaystyle\subseteq$
$\displaystyle]C_{w,k}[_{\overline{m},n}.t\subseteq]C_{w,k}[_{\overline{m+\min(t)},n-\max(t)},$
$\displaystyle]C_{w,k}[_{m+\max(t),\overline{n-\min(t)}}$
$\displaystyle\subseteq$
$\displaystyle]C_{w,k}[_{m,\overline{n}}.t\subseteq]C_{w,k}[_{m+\min(t),\overline{n-\max(t)}}.$

  2. (2)

Let $t\in T^{--}(F)$. For all $w\in{}^{M}W$ and $m,n\in\mathbb{Q}$, we have

$\displaystyle]C_{w,k}[_{m-\min(t),n+\max(t)}$ $\displaystyle\subseteq$
$\displaystyle]C_{w,k}[_{m,n}.t\subseteq]C_{w,k}[_{m-\max(t),n+\min(t)}$
$\displaystyle]C_{w,k}[_{\overline{m-\min(t)},n+\max(t)}$
$\displaystyle\subseteq$
$\displaystyle]C_{w,k}[_{\overline{m},n}.t\subseteq]C_{w,k}[_{\overline{m-\max(t)},n+\min(t)},$
$\displaystyle]C_{w,k}[_{m-\min(t),\overline{n+\max(t)}}$
$\displaystyle\subseteq$
$\displaystyle]C_{w,k}[_{m,\overline{n}}.t\subseteq]C_{w,k}[_{m-\max(t),\overline{n+\min(t)}}.$

###### Proof.

Easy and left to the reader. ∎

###  3.5. Dynamics of correspondences

####  3.5.1. Certain compact open subgroups

We now assume that the group
$G_{F}=G\times_{\mathrm{Spec}~{}\mathcal{O}_{F}}\mathrm{Spec}~{}F$ is defined
and quasi-split over $\mathbb{Q}_{p}$. Therefore, we have a reductive group
$G_{\mathbb{Q}_{p}}$ with Borel $B_{\mathbb{Q}_{p}}$ and moreover the group
$G_{\mathbb{Q}_{p}}$ splits over the extension $F$ of $\mathbb{Q}_{p}$. We
have a reductive model $G$ of $G_{F}$ over $\mathrm{Spec}~{}\mathcal{O}_{F}$.
The Borel $B_{\mathbb{Q}_{p}}$ base changes to a Borel $B_{F}$ of $G_{F}$
which extends to a Borel $B$ of $G$. Similarly, we have a maximal torus
$T_{\mathbb{Q}_{p}}\subseteq B_{\mathbb{Q}_{p}}$ of $G_{\mathbb{Q}_{p}}$, and
its base change $T_{F}$ extends to a maximal (split) torus of $G$. We will
often drop the subscripts $\mathbb{Q}_{p}$ or $F$ when the context is clear.

######  Remark 3.5.2.

Starting from the next section, we will consider a Shimura datum $(G,X)$,
where $G$ is a reductive group over $\mathbb{Q}$. The group
$G_{\mathbb{Q}_{p}}$ that we consider here will be the base change to
$\mathbb{Q}_{p}$ of the group $G$ which is part of the Shimura datum. We
apologize for this slightly inconsistent notation.

We let $T(\mathbb{Z}_{p})$ be the maximal compact subgroup of
$T(\mathbb{Q}_{p})$. Note that $T(\mathbb{Z}_{p})=T(\mathcal{O}_{F})\cap
T(\mathbb{Q}_{p})$. We have an exact sequence
$0\rightarrow\mathcal{O}_{F}^{\times}\rightarrow
F^{\times}\stackrel{{\scriptstyle v}}{{\rightarrow}}\mathbb{Q}$ and tensoring
with $X_{\star}(T)$ and taking Galois invariants, we obtain the sequence:
$0\rightarrow T(\mathbb{Z}_{p})\rightarrow
T(\mathbb{Q}_{p})\stackrel{{\scriptstyle
v}}{{\rightarrow}}X_{\star}(T^{d})\otimes\mathbb{Q}$ where $T^{d}$ stands for
the maximal split torus inside $T$ and $X_{\star}(T^{d})$ for its cocharacter
group. The image of $T(\mathbb{Q}_{p})$ in $X_{\star}(T^{d})\otimes\mathbb{Q}$
is easily seen to be a $\mathbb{Z}$-lattice.

We let $T^{+}=T(\mathbb{Q}_{p})\cap T^{+}(F)$, $T^{++}=T(\mathbb{Q}_{p})\cap
T^{++}(F)$, $T^{-}=T(\mathbb{Q}_{p})\cap T^{-}(F)$, and
$T^{--}=T(\mathbb{Q}_{p})\cap T^{--}(F)$. These are semigroups in
$T(\mathbb{Q}_{p})$ and one proves easily that they generate
$T(\mathbb{Q}_{p})$ (since there are regular elements in the maximal split
torus $T^{d}$).

We will now consider certain compact open subgroups of $G(\mathcal{O}_{F})$.
For all $m\in\mathbb{Z}_{\geq 0}$, we let $\tilde{K}_{p,m}\subseteq
G(\mathcal{O}_{F})$ be the preimage of $B(\mathcal{O}_{F}/p^{m})$ under the
map $G(\mathcal{O}_{F})\to G(\mathcal{O}_{F}/p^{m})$. We observe that
$\tilde{K}_{p,0}=G(\mathcal{O}_{F})$ and $\tilde{K}_{p,1}$ is the Iwahori
subgroup of $G(\mathcal{O}_{F})$ with respect to the Borel
$B(\mathcal{O}_{F})$. For $b\in\mathbb{Z}_{\geq 0}$ we let
$\tilde{K}_{p,b,b}\subseteq G(\mathcal{O}_{F})$ be the preimage of
$U(\mathcal{O}_{F}/p^{b})$ under the map $G(\mathcal{O}_{F})\to
G(\mathcal{O}_{F}/p^{b})$. Finally for $m\geq b\geq 0$ we let
$\tilde{K}_{p,m,b}=\tilde{K}_{p,m}\cap\tilde{K}_{p,b,b}$. In other words
$\tilde{K}_{p,m,b}$ is the subgroup of $G(\mathcal{O}_{F})$ of elements whose
reduction mod $p^{m}$ lies in $B$, and whose reduction mod $p^{b}$ lies in
$U$. We note that we have $\tilde{K}_{p,m,b}\subset\mathcal{G}_{m-1,0}$.

For $m\geq b\geq 0$ and $m>0$, the groups $\tilde{K}_{p,m,b}$ have an Iwahori
decomposition, in the sense that the product map

$\overline{\tilde{U}}_{m}\times\tilde{T}_{b}\times{U}(\mathcal{O}_{F})\to
K_{p,m,b}$

is a bijection, where $\tilde{T}_{b}=\ker(T(\mathcal{O}_{F})\to
T(\mathcal{O}_{F}/p^{b}))$ and
$\overline{\tilde{U}}_{m}=\ker(\overline{U}(\mathcal{O}_{F})\to\overline{U}(\mathcal{O}_{F}/p^{m}))$.

We now let $K_{p,m,b}=G(\mathbb{Q}_{p})\cap\tilde{K}_{p,m,b}$. This is a
compact open subgroup of $G(\mathbb{Q}_{p})$.

For $m\geq b\geq 0$ and $m>0$, the groups $K_{p,m,b}$ have an Iwahori
decomposition, in the sense that the product map

$\overline{U}_{m}\times T_{b}\times U(\mathbb{Z}_{p})\to K_{p,m,b}$

is a bijection, where $T_{b}=\tilde{T}_{b}\cap T(\mathbb{Q}_{p})$ and
$\overline{U}_{m}=\overline{\tilde{U}}_{m}\cap\overline{U}(\mathbb{Q}_{p})$,
$U(\mathbb{Z}_{p})=U(\mathcal{O}_{F})\cap U(\mathbb{Q}_{p})$.

######  Remark 3.5.3.

We note that the groups $K_{p,m,b}$ depend (even up to conjugation in
general), on the choice of the reductive model $G$ over $\mathcal{O}_{F}$.

######  Remark 3.5.4.

If the group $G_{\mathbb{Q}_{p}}$ is unramified, then we can choose a
reductive model $G_{\mathbb{Z}_{p}}$ of $G_{\mathbb{Q}_{p}}$ with Borel and
maximal torus $B_{\mathbb{Z}_{p}}$ and $T_{\mathbb{Z}_{p}}$. In this case, if
we take
$G_{\mathcal{O}_{F}}=G_{\mathbb{Z}_{p}}\times_{\mathbb{Z}_{p}}\mathrm{Spec}\mathcal{O}_{F}$,
then we have that $K_{p,m}$ is the preimage of $B(\mathbb{Z}_{p}/p^{m})$ under
$G(\mathbb{Z}_{p})\to G(\mathbb{Z}_{p}/p^{m})$, and $K_{p,b,b}$ is the
preimage of $U(\mathbb{Z}_{p}/p^{b})$ under $G(\mathbb{Z}_{p})\to
G(\mathbb{Z}_{p}/p^{b})$. In particular, $K_{p,1,0}$ is an Iwahori subgroup,
and $K_{p,1,1}$ is a pro-$p$ Iwahori subgroup.

In the ramified case, there is still a notion of an Iwahori subgroup of
$G(\mathbb{Q}_{p})$ defined using Bruhat-Tits theory. However, our subgroup
$K_{p,1,0}$ will usually not be an Iwahori subgroup. In the general case, we
found it convenient to work with the groups $K_{p,m,b}$ constructed as above,
and it makes no difference for the purposes of this paper whether $K_{p,1,0}$
is an Iwahori.

####  3.5.5. Change of group

In this paragraph we make a short digression that will be useful when we deal
with abelian type Shimura varieties. Assume that we have an epimorphism of
reductive groups $G_{\mathbb{Q}_{p}}\rightarrow G^{\prime}_{\mathbb{Q}_{p}}$
with central kernel. This implies that $G$ and $G^{\prime}$ have the same
adjoint group $G^{ad}$. We assume that these group split over $F$ and we fix
models over $\mathcal{O}_{F}$, that we denote $G$ and $G^{\prime}$ together
with a map $G\rightarrow G^{\prime}$ over $\mathrm{Spec}~{}\mathcal{O}_{F}$,
extending the map $G_{F}\rightarrow G^{\prime}_{F}$. We also assume that
$G_{\mathbb{Q}_{p}}$ and $G^{\prime}_{\mathbb{Q}_{p}}$ are quasi-split and
pick Borels $B$ and $B^{\prime}$ defined over $\mathbb{Q}_{p}$ such that
$B\rightarrow B^{\prime}$. We can therefore define compact open subgroups
$K_{p,m,b}\subseteq G(\mathbb{Q}_{p})$ and $K^{\prime}_{p,m,b}\subseteq
G^{\prime}(\mathbb{Q}_{p})$ as in section 3.5.1.

######  Lemma 3.5.6.

The groups $K_{p,m,b}$ and $K^{\prime}_{p,m,b}$ have the same image in
$G^{ad}(\mathbb{Q}_{p})$.

###### Proof.

This follows from the Iwahori decomposition of these groups. Since
$G_{\mathbb{Q}_{p}}\rightarrow G^{\prime}_{\mathbb{Q}_{p}}$ is an epimorphism
with central kernel, the induced map on root groups are isomorphisms. ∎

####  3.5.7. Hecke algebras

We let $\mathcal{H}_{p,m,b}$ be the Hecke algebras
$\mathbb{Z}[K_{p,m,b}\backslash G(\mathbb{Q}_{p})/K_{p,m,b}]$. We denote by
$\mathcal{H}_{p,m,b}^{+}$ the sub-algebra generated by the double cosets
$[K_{p,m,b}tK_{p,m,b}]$ with $t\in T^{+}$ and by $\mathcal{H}_{p,m,b}^{++}$
the ideal generated by $[K_{p,m,b}tK_{p,m,b}]$ with $t\in T^{++}$. We define
similarly $\mathcal{H}_{p,m,b}^{-}$ and $\mathcal{H}_{p,m,b}^{--}$.

######  Lemma 3.5.8.

For all $m\geq b\geq 0$ with $m>0$, the map $t\mapsto[K_{p,m,b}tK_{p,m,b}]$
induces isomorphisms
$\mathbb{Z}[T^{+}/T_{b}]\rightarrow\mathcal{H}_{p,m,b}^{+}$ and
$\mathbb{Z}[T^{-}/T_{b}]\rightarrow\mathcal{H}_{p,m,b}^{-}$.

###### Proof.

This is [Cas], lem. 4.1.5. (Alternatively it can be deduced from lemma 4.2.13
below.) ∎

####  3.5.9. Action of correspondences on the flag variety

Let $K_{p}\subseteq G_{0}(\mathbb{Q}_{p})$ be a compact open subgroup.

We consider the quotient space $\mathcal{FL}/K_{p}$. This space carries an
action by correspondences of double cosets $K_{p}gK_{p}$ for $g\in
G_{0}(\mathbb{Q}_{p})$. Namely, given $xK_{p}\in\mathcal{FL}/K_{p}$, we let
$xK_{p}.K_{p}gK_{p}=\\{x\tilde{u}gK_{p},~{}u\in K_{p}/(gK_{p}g^{-1}\cap
K_{p}),~{}\tilde{u}\in K_{p}~{}\textrm{lifts}~{}u\\}$.

We now consider a compact open $K_{p}=K_{p,m^{\prime},b}$ for $m^{\prime}\geq
b\geq 0$ and $m^{\prime}>0$.

######  Lemma 3.5.10.

Let $w\in{}^{M}W$.

  1. (1)

Let $t\in T^{+}$. The sequence $\\{]X_{w,k}[.(K_{p}tK_{p})^{m}\\}_{m\geq 0}$
is nested.

  2. (2)

Let $t\in T^{-}$. The sequence $\\{]Y_{w,k}[.(K_{p}tK_{p})^{n}\\}_{n\geq 0}$
is nested.

  3. (3)

Let $t\in T^{++}$ and $m\geq 0$. We have

$]C_{w,k}[_{m\max(t),\overline{-m\min(t)}}.K_{p}\subseteq$
$]X_{w,k}[.(K_{p}tK_{p})^{m}\subseteq]C_{w,k}[_{m\min(t),\overline{-m\max(t)}}.K_{p}\bigcup\cup_{w^{\prime}<w}]X_{w^{\prime},k}[.$

  4. (4)

Let $t\in T^{--}$ and $n\geq 0$. We have

$]C_{w,k}[_{\overline{-n\min(t)},n\max(t)}.K_{p}\subseteq$
$]Y_{w,k}[.(K_{p}tK_{p})^{n}\subseteq]C_{w,k}[_{\overline{-n\max(t)},n\min(t)}.K_{p}\bigcup\cup_{w^{\prime}>w}]Y_{w^{\prime},k}[.$

  5. (5)

Let $t\in T^{++}$. For all $m,n\in\mathbb{Z}_{\geq 0}$,

$]X_{w,k}[.(K_{p}tK_{p})^{m}\cap\overline{]Y_{w,k}[}.(K_{p}t^{-1}K_{p})^{n}\subseteq]C_{w,k}[_{m\min(t),\overline{0}}K_{p}\cap]C_{w,k}[_{0,\overline{n\min(t)}}K_{p}.$

  6. (6)

Let $t\in T^{--}$. For all $m,n\in\mathbb{Z}_{\geq 0}$,

$\overline{]X_{w,k}[}.(K_{p}t^{-1}K_{p})^{m}\cap{]Y_{w,k}[}.(K_{p}tK_{p})^{n}\subseteq]C_{w,k}[_{\overline{m\min(t)},{0}}K_{p}\cap]C_{w,k}[_{\overline{0},{n}\min(t)}K_{p}.$

  7. (7)

Let $t\in T^{--}$. For all $n\geq d$,
$\overline{]Y_{w,k}[}.(K_{p}tK_{p})^{n}\subseteq]Y_{w,k}[$.

###### Proof.

We observe that $]X_{w,k}[.(K_{p}tK_{p})^{m}=]X_{w,k}[t^{m}K_{p}$ and
$]Y_{w,k}[.(K_{p}tK_{p})^{m}=]Y_{w,k}[t^{m}K_{p}$, using the very definition
of the action, and also noting that $(K_{p}tK_{p})^{m}=K_{p}t^{m}K_{p}$ by
lemma 3.5.8. Therefore, the first and second points follow from lemma 3.4.1.

We note that
$]X_{w,k}[=]C_{w,k}[_{0,\overline{0}}\bigcup\cup_{w^{\prime}<w}]X_{w^{\prime},k}[$
and
$]Y_{w,k}[=]C_{w,k}[_{\overline{0},0}\bigcup\cup_{w^{\prime}>w}]Y_{w^{\prime},k}[$
by lemma 3.3.22. The third and fourth point follow from this, the first two
points, and lemma 3.4.2.

We know that $]X_{w,k}[\cap\overline{]Y_{w,k}[}=]C_{w,k}[_{0,\overline{0}}$ by
lemma 3.3.21. Moreover,
$]C_{w,k}[_{0,\overline{0}}\cap]X_{w^{\prime},k}[=\emptyset$ if
$w^{\prime}\neq w$ and
$]C_{w,k}[_{0,\overline{0}}\cap]Y_{w^{\prime},k}[=\emptyset$ if
$w^{\prime}\neq w$ (because $]C_{w,k}[_{0,\overline{0}}$, $]X_{w^{\prime},k}[$
and $]Y_{w,k}[$ are all stable under generalization and they have no common
rank one point). Therefore,

$]X_{w,k}[.(K_{p}tK_{p})^{m}\cap\overline{]Y_{w,k}[}.(K_{p}t^{-1}K_{p})^{n}\subseteq$
$]C_{w,k}[_{0,\overline{0}}\cap]C_{w,k}[_{m\min(t),\overline{-m\max(t)}}K_{p}\cap]C_{w,k}[_{\overline{-n\max(t^{-1})},\overline{n\min(t^{-1})}}.K_{p}\subseteq$
$]C_{w,k}[_{m\min(t),\overline{0}}K_{p}\cap]C_{w,k}[_{0,\overline{n\min(t)}}K_{p}.$

The proof of the sixth point is similar. We pass to the last point. By taking
closures, we find that
$\overline{]Y_{w,k}[}.(K_{p}tK_{p})\subseteq]C_{w,k}[_{\overline{-\max(t)},\overline{\min(t)}}.K_{p}\bigcup\cup_{w^{\prime}>w}\overline{]Y_{w^{\prime},k}[}$.
Since
$]C_{w,k}[_{\overline{-\max(t)},\overline{\min(t)}}.K_{p}\subseteq]Y_{w,k}[$,
we deduce that for any $x\in\overline{]Y_{w,k}[}$, and any $y\in
x.K_{p}tK_{p}$, then either $y\in]Y_{w,k}[$ or
$y\in\overline{]Y_{w^{\prime},k}[}$ for some $w^{\prime}>w$. We conclude by
induction on $d-\ell(w)$.

∎

##  4\. Shimura varieties

Let $(G,X)$ be a Shimura datum. Thus $X$ is a $G(\mathbb{R})$-conjugacy class
of homomorphism
$h:\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m}\rightarrow
G_{\mathbb{R}}$ satisfying a list of familiar axioms ([Del79], section 2.1):

  1. (1)

For all $h\in X$, the Hodge structure on $\mathfrak{g}_{\mathbb{R}}$ has
weight $(1,-1)$, $(0,0)$ and $(-1,1)$.

  2. (2)

The involution $\mathrm{Ad}(h(i))$ is a Cartan involution on
$G^{ad}_{\mathbb{R}}$.

  3. (3)

The group $G^{ad}$ has no simple $\mathbb{Q}$-factor whose real points are
compact.

######  Remark 4.0.1.

The condition $(3)$ is mostly irrelevant for this paper. This axiom is used in
order to apply the strong approximation theorem, which is important for the
description of connected components of Shimura varieties, and therefore the
theory of canonical models over the reflex field. This is important for
certain rationality questions, but will not play much of a role in this paper.
Groups $G$ whose real points are compact modulo the center give Shimura sets.
Automorphic forms on these groups are called algebraic automorphic forms, and
all our results are well known for Shimura sets: all the vanishing theorems
are trivial, whereas the theory of overconvergent modular forms has been
extensively developed in this setting ([Buz07], [Che04], [Loe11]). We keep
this third axiom for convenience, as it is used in many references, but
believe all our results hold without this assumption, with minor modifications
of the proofs.

Via base change to $\mathbb{C}$, we have
$(\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m})_{\mathbb{C}}=\mathbb{G}_{m}\times\mathbb{G}_{m}$
(given by $z\mapsto(z,\bar{z})$) and via projection to the first factor, $X$
induces a conjugacy class of cocharacters $\mathbb{G}_{m}\rightarrow
G_{\mathbb{C}}$. We fix one such cocharacter $\mu$. Associated to $\mu$ we
have two opposite parabolic subgroups $P^{std}_{\mu}=\\{g\in
G_{\mathbb{C}},\lim_{t\rightarrow\infty}\mathrm{Ad}(\mu(t))g~{}\textrm{exists}\\}$
and $P_{\mu}=\\{g\in G_{\mathbb{C}},\lim_{t\rightarrow
0}\mathrm{Ad}(\mu(t))g~{}\textrm{exists}\\}$. We also let $M_{\mu}$ be the
Levi quotient of $P_{\mu}$ and $P_{\mu}^{std}$. We let
$FL^{std}_{G,\mu}=G_{\mathbb{C}}/P^{std}_{\mu}$ and
$FL_{G,\mu}=P_{\mu}\backslash G_{\mathbb{C}}$ be the Flag varieties. Let $E$
be the reflex field, which is the field of definition of the conjugacy class
of $\mu$. The two flag varieties are defined over $\mathrm{Spec}~{}E$.

Let $K\subset G(\mathbb{A}_{f})$ be a neat compact open subgroup, and let
$S_{K}\to\mathrm{Spec}~{}E$ be the canonical model of the corresponding
Shimura variety [Mil90]. We have $S_{K}(\mathbb{C})=G(\mathbb{Q})\backslash
X\times G(\mathbb{A}_{f})/K$. In the rest of this paper, all the compact open
subgroups $K\subset G(\mathbb{A}_{f})$ are assumed to be neat, so we do not
always repeat this assumption.

The most fundamental Shimura data are the Siegel data
$(\mathrm{GSp}_{2g},\mathcal{H}_{g})$ for all $g\in\mathbb{Z}_{\geq 1}$, where
$\mathcal{H}_{g}$ is the Siegel (upper and lower) half space of matrices
$M\in\mathrm{M}_{g}(\mathbb{C})$ such that $~{}^{t}M=M$ and $\mathrm{Im}(M)$
is definite. The corresponding Shimura varieties parametrize abelian varieties
of dimension $g$, with a polarization and level structure prescribed by $K$. A
Shimura datum $(G,X)$ is of Hodge type if it admits an embedding in a Siegel
datum. All PEL Shimura data are of Hodge type. A Shimura datum $(G,X)$ is of
abelian type if there exists a datum $(G_{1},X_{1})$ of Hodge type and a
central isogeny $G_{1}^{der}\rightarrow G^{der}$ which induces an isomorphism
of connected Shimura data $(G^{ad},X^{\mathrm{+}})=(G_{1}^{ad},X_{1}^{+})$
where $X^{+}$ is a connected component of $X$ (and similarly $X_{1}^{+}$ is a
connected component of $X_{1}$).

######  Example 4.0.2.

Let $L$ be a totally real number field. The datum
$(\mathrm{Res}_{L/\mathbb{Q}}\mathrm{GSp}_{2g},\mathcal{H}_{g}^{[L:\mathbb{Q}]})$
is of abelian type. We call it a symplectic datum. In the case $g=1$, the
corresponding Shimura varieties are the Hilbert modular varieties.

We let $S^{\star}_{K}\rightarrow\mathrm{Spec}~{}E$ be the minimal
compactification of $S_{K}$. Depending on the auxiliary choice of a projective
cone decomposition $\Sigma$, we let
$S^{tor}_{K,\Sigma}\rightarrow\mathrm{Spec}~{}E$ be the toroidal
compactification of $S_{K}$ corresponding to $\Sigma$. The (reduced) boundary
$D_{K,\Sigma}=S^{tor}_{K,\Sigma}\setminus S_{K}$ is a Cartier divisor. The
cone decompositions $\Sigma$ are partially ordered by inclusion, and any two
cone decompositions admit a common refinement. The cone decompositions
$\Sigma$ which are such that $S^{tor}_{K,\Sigma}\to\mathrm{Spec}~{}E$ is
smooth and projective are cofinal among all cone decompositions, and we
usually choose them this way. We refer to [Pin90] for the construction of
these compactifications.

###  4.1. Automorphic vector bundles and their cohomology

####  4.1.1. Automorphic vector bundles

We choose a field extension $F$ of $E$ which splits $G$ and we work over $F$
in this section. In most of this paper, $F$ will be a finite extension of
$\mathbb{Q}_{p}$, but this is not necessary for the moment. We can choose
$\mu$ so that it is defined over $F$, and choose a maximal split torus $T$
with $\mu(\mathbb{G}_{m})\subseteq T\subseteq M_{\mu}$.

We let $Z_{s}(G)$ be the largest subtorus of the center $Z(G)$ which is
$\mathbb{R}$-split but contains no $\mathbb{Q}$-split subtorus. We let
$G^{c}=G/Z_{s}(G)$, and define $M_{\mu}^{c}$, $T^{c}$, $P_{\mu}^{c}$, and
$P_{\mu}^{c,std}$ similarly.

######  Remark 4.1.2.

In the Hodge type case, $Z_{s}(G)=\\{1\\}$. For the symplectic datum
$(\mathrm{Res}_{L/\mathbb{Q}}\mathrm{GSp}_{2g},\mathcal{H}_{g}^{[L:\mathbb{Q}]})$,
$Z_{s}(G)$ is the kernel of the norm
$\mathrm{Res}_{L/\mathbb{Q}}\mathbb{G}_{m}\rightarrow\mathbb{G}_{m}$.

Let $\mathrm{Rep}(M^{c}_{\mu})$ be the category of finite dimensional
algebraic representations of the reductive group $M^{c}_{\mu}$ on $F$-vector
spaces.

By [Mil90], thm. 5.1 and [Har89], thm. 4.2 we have a right
$M^{c}_{\mu}$-torsor $M_{dR}$ over $S^{tor}_{K,\Sigma}$, and it corresponds to
a functor:

$\displaystyle\mathrm{Rep}(M^{c}_{\mu})$ $\displaystyle\rightarrow$
$\displaystyle VB(S_{K,\Sigma}^{tor})$ $\displaystyle V$
$\displaystyle\mapsto$ $\displaystyle\mathcal{V}_{K,\Sigma}$

where $VB(S_{K,\Sigma}^{tor})$ is the category of locally free sheaves of
finite rank over $S_{K,\Sigma}^{tor}$. This functor is compatible in a natural
way with change of level $K$ and of cone decompositions $\Sigma$. The locally
free sheaves in the essential image of this functor are called (totally
decomposed) automorphic vector bundles. They carry an equivariant action of
$G(\mathbb{A}_{f})$.

######  Remark 4.1.3.

We recall the description of
$M_{dR}\times_{S^{tor}_{K,\Sigma}}S_{K}(\mathbb{C})$ as a complex analytic
space. First, we have the Borel embedding $\beta:X\hookrightarrow
FL_{G,\mu}^{std}(\mathbb{C})=G^{c}(\mathbb{C})/P_{\mu}^{c,std}(\mathbb{C})$
which sends $h\in X$ to the parabolic stabilizing the Hodge filtration. The
Borel embedding is equivariant for the left action of $G(\mathbb{R})$. We have
the canonical map $G^{c}(\mathbb{C})\rightarrow FL_{G,\mu}^{std}(\mathbb{C})$.
This map is a right $P_{\mu}^{c,std}(\mathbb{C})$-torsor and is
$G(\mathbb{C})$-equivariant for the left action. Similarly, the canonical map
$G^{c}(\mathbb{C})/U_{P_{\mu}^{c,std}}(\mathbb{C})\rightarrow
FL_{G,\mu}^{std}(\mathbb{C})$ is a right $M_{\mu}^{c}(\mathbb{C})$-torsor
(where $U_{P_{\mu}^{c,std}}$ is the unipotent radical of $P_{\mu}^{c,std}$).
Then we have:

$M_{dR}\times_{{S^{tor}_{K,\Sigma}}}S_{K}(\mathbb{C})=G(\mathbb{Q})\backslash\beta^{-1}(G^{c}(\mathbb{C})/U_{P_{\mu}^{c,std}}(\mathbb{C}))\times{G}(\mathbb{A}_{f})/K.$

######  Remark 4.1.4.

We give a description of $M_{dR}$ for the Siegel Shimura datum
$(\mathrm{GSp}_{2g},\mathcal{H}_{g})$. Let us first work over $S_{K}$. Let
$\omega_{A}$ be the conormal sheaf of the universal abelian scheme
$A\rightarrow{S}_{K}$ along its unit section and let $\mathrm{Lie}(A)$ be its
dual. The torsor ${M}_{dR}$ parametrizes trivializations
$\psi_{1}\oplus\psi_{2}:\mathscr{O}_{S_{K}}^{g}\oplus\mathscr{O}_{S_{K}}^{g}\rightarrow\mathrm{Lie}(A)\oplus\omega_{A^{t}}$,
such that under the isomorphism $\mathrm{Lie}(A)^{\vee}=\omega_{A^{t}}$ given
by the polarization, we have $\psi_{1}=c(\psi_{2}^{-1})^{t}$ for a unit
$c\in\mathscr{O}_{S_{K}}^{\times}$. The extension of $M_{dR}$ over
$S_{K,\Sigma}^{tor}$ has a similar description. Indeed, $A$ can be extended to
a semi-abelian scheme $A_{\Sigma}$ over ${S}_{K,\Sigma}^{tor}$, and we can let
${M}_{dR}$ be the torsor of trivializations of
$\mathrm{Lie}(A_{\Sigma})\oplus\omega_{A_{\Sigma}^{t}}$, compatible, up to a
unit, with the polarization.

We now make the construction of automorphic vector bundles explicit, and label
them using weights. We begin by making a choice of positive roots of $T$: we
first choose a set of compact positive roots $\Phi^{+}_{c}$ which lie in
$\mathfrak{m}_{\mu}$ the Lie algebra of $M_{\mu}$. We then choose the non-
compact positive roots $\Phi^{+}_{nc}$ to lie in
$\mathfrak{g}/\mathfrak{p}_{\mu}^{std}$ where $\mathfrak{p}_{\mu}^{std}$ is
the Lie algebra of $P_{\mu}^{std}$. We let
$\Phi^{+}=\Phi^{+}_{c}\coprod\Phi^{+}_{nc}$.

######  Remark 4.1.5.

This choice implies that the Borel corresponding to $\Phi^{+}$ is included in
$P_{\mu}$, or equivalently that the cocharacter $\mu$ of the Shimura datum is
dominant. In section 3 we fixed a parabolic $P$ of $G$ containing a Borel $B$.
In the applications to Shimura varieties, $P$ will be $P_{\mu}$. Therefore our
convention is also compatible with the choice made in section 3.

We let $X^{\star}(T)^{M_{\mu},+}$ be the cone of characters of $T$ which are
dominant for $\Phi^{+}_{c}$. We label irreducible representations of $M_{\mu}$
by their highest weight $\kappa\in X^{\star}(T)^{M_{\mu},+}$. An explicit
construction of the highest weight $\kappa$ representation $V_{\kappa}$ is as
follows. Let $w_{0,M}$ be the longest element of the Weyl group of $M_{\mu}$.
For any $\kappa\in X^{\star}(T)^{M_{\mu},+}$ we consider the space
$V_{\kappa}$ of functions $f:M_{\mu}\rightarrow\mathbb{A}^{1}$ such that
$f(mb)=(w_{0,M}\kappa)(b^{-1})f(m)$ for all $m\in M_{\mu}$ and $b\in B\cap
M_{\mu}$. The action of $M_{\mu}$ on itself via left translation induces a
left action on $V_{\kappa}$, i.e. $(m^{\prime}\cdot
f)(m)=f({m^{\prime}}^{-1}m)$. The irreducible representations of $M^{c}_{\mu}$
are the irreducible representations of $M_{\mu}$ labelled by dominant
characters $\kappa$ of $T^{c}$. We let $X^{\star}(T^{c})^{M_{\mu},+}$ be the
cone of these characters.

We denote by $\mathcal{V}_{\kappa,K,\Sigma}$ the locally free sheaf associated
to the irreducible representation of highest weight $\kappa$ of $M^{c}_{\mu}$.
Concretely, we consider the right torsor
$g:M_{dR}\rightarrow{S}^{tor}_{K,\Sigma}$ and we let
$\mathcal{V}_{\kappa,K,\Sigma}$ be the subsheaf of
$g_{\star}\mathscr{O}_{M_{dR}}$ of sections $f(m)$ such that
$f(mb)=(w_{0,M}\kappa)(b^{-1})f(m)$ for all $b\in B\cap M_{\mu}$. We will
often abbreviate $\mathcal{V}_{\kappa,K,\Sigma}$ to $\mathcal{V}_{\kappa}$.

We also introduce the cuspidal subsheaf
$\mathcal{V}_{\kappa,K,\Sigma}(-D_{K,\Sigma})$ of sections vanishing on the
boundary divisor $D_{K,\Sigma}\hookrightarrow S_{K,\Sigma}^{tor}$. Again, we
often abbreviate this sheaf to $\mathcal{V}_{\kappa}(-D)$.

####  4.1.6. The cohomology of automorphic vector bundles

We let $\pi_{K,\Sigma}:S_{K,\Sigma}^{tor}\rightarrow S_{K}^{\star}$ be the
projection from the toroidal to the minimal compactification.

######  Theorem 4.1.7.

We have
$\mathrm{R}^{i}(\pi_{K,\Sigma})_{\star}\mathcal{V}_{\kappa,K,\Sigma}(-nD_{K,\Sigma})=0$
for all $i>0$ and all $n\geq 1$.

###### Proof.

In the PEL case (and for $n=1$), this is [Lan17], thm. 8.6. We give an
argument which follows closely [AIP15] and which is also similar to _loc.
cit_. Let $x\in{S}^{\star}_{K}$. We write $\widehat{{S}^{tor}_{K,\Sigma}}^{x}$
for the formal completion of $S^{tor}_{K,\Sigma}$ along
$\pi_{K,\Sigma}^{-1}(x)$. By the theorem on formal functions ([Sta13], Tag
02O7) it suffices to prove that
$\mathrm{H}^{i}(\widehat{{S}^{tor}_{K,\Sigma}}^{x},\mathcal{V}_{\kappa,K,\Sigma}(-nD_{K,\Sigma}))=0$
for all $i>0$ and $n\geq 1$.

We may now use the description of $\widehat{{S}^{tor}_{K,\Sigma}}^{x}$ in
terms of the local charts, following the notations of [MP19]. The argument is
mostly about torus embedding, so we do not need to explain in detail the
structure of the toroidal compactification, but just recall what is strictly
necessary. Suppose that $x$ belongs to a boundary component indexed by a cusp
label representative $\Phi$. There is a tower of spaces:

$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\stackrel{{\scriptstyle
g}}{{\rightarrow}}\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})$

where
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})$
is a torsor under a torus $\mathbf{E}_{K}(\Phi)$. There is a locally free
sheaf $\mathcal{V}_{\kappa,K}$ over
$\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})$. There is a
twisted torus embedding
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$
which depends on the choice of $\Sigma$. There is an arithmetic group
$\Delta_{K}(\Phi)$ acting on $X^{\star}(\mathbf{E}_{K}(\Phi))$ and on
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})\hookrightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.
Let
$\overline{g}:\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))\rightarrow\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})$.
The arithmetic group also acts on
$\overline{g}^{\star}\mathcal{V}_{\kappa,K}$. We have a
$\Delta_{K}(\Phi)$-invariant closed subscheme
$\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))\hookrightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.
There is a finite morphism
$\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h})\rightarrow{S}^{\star}_{K}$ whose
image contains $x$. There is a series of morphisms:

$\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))\rightarrow\mathbf{S}_{K_{\Phi}}(\overline{Q}_{\Phi},\overline{D}_{\Phi})\rightarrow\mathbf{S}_{K_{\Phi}}(G_{\Phi,h},D_{\Phi,h}).$

We let $\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{x}$ be the
closed subspace equal to the inverse image of $x$. The main result on the
description of toroidal compactifications states that:

$\widehat{{S}^{tor}_{K,\Sigma}}^{x}\simeq\Delta_{K}(\Phi)\backslash\big{(}\widehat{\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}^{\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{x}}\big{)}$

Moreover, the sheaf $\overline{g}^{\star}\mathcal{V}_{\kappa,K}$ on
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$ descends to

$\Delta_{K}(\Phi)\backslash\big{(}\widehat{\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))}^{\mathbf{Z}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))_{x}}\big{)}$

and identifies with
$\mathcal{V}_{\kappa,K,\Sigma}|_{\widehat{{S}^{tor}_{K,\Sigma}}^{x}}$. We let
$D$ be the boundary divisor ${S}^{tor}_{K,\Sigma}\setminus{S}_{K}$. Under the
isomorphism above, it corresponds to the divisor
$D_{0}=\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))\setminus\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi})$.

There is a divisor $D^{\prime}$ on ${S}^{tor}_{K,\Sigma}$ which has exactly
the same support as $D$ and such that
$\mathscr{O}_{{S}^{tor}_{K,\Sigma}}(-D^{\prime})$ is ample relatively to the
minimal compactification. It corresponds to a divisor $D^{\prime}_{0}$ on
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$.

We will prove the following statement: for any $C\in\mathbb{Z}_{\geq 0}$,
there exists $s\geq C$ and a finite morphism

$\psi:\widehat{{S}^{tor}_{K,\Sigma}}^{x}\rightarrow\widehat{{S}^{tor}_{K,\Sigma}}^{x}$

such that
$\psi^{\star}\mathcal{V}_{\kappa,K,\Sigma}=\mathcal{V}_{\kappa,K,\Sigma}$ and
$\mathscr{O}(-nD)\rightarrow\psi_{\star}\mathscr{O}(-sD^{\prime})$ is a split
injection. This implies the theorem, as we deduce that for all $i\geq 0$,
$\mathrm{H}^{i}(\widehat{{S}^{tor}_{K,\Sigma}}^{x},\mathcal{V}_{\kappa,K,\Sigma}(-nD))$
is a direct factor of
$\mathrm{H}^{i}(\widehat{{S}^{tor}_{K,\Sigma}}^{x},\mathcal{V}_{\kappa,K,\Sigma}(-sD^{\prime}))$.
Taking $C$ large enough, this last group vanishes for $i>0$.

We now prove the claim about the existence of $\psi$. Our proof follows
[AIP15], p. 679. We will construct everything on
$\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$. For any integer
$\ell$, the multiplication by $\ell$-map on the torus $\mathbf{E}_{K}(\Phi)$
induces a finite morphism
$\psi_{\ell}:\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))\rightarrow\mathbf{S}_{K_{\Phi}}(Q_{\Phi},D_{\Phi},\Sigma(\Phi))$
which is $\Delta_{K}(\Phi)$ equivariant and for which
$\psi_{\ell}^{\star}\overline{g}^{\star}\mathcal{V}_{\kappa,K}=\overline{g}^{\star}\mathcal{V}_{\kappa,K}$.
The morphisms $\psi_{\ell}$ induces a finite morphism
$\widehat{{S}^{tor}_{K,\Sigma}}^{x}\rightarrow\widehat{{S}^{tor}_{K,\Sigma}}^{x}$.
We have $D^{\prime}_{0}=\sum_{\sigma\in\Sigma(\Phi)(1)}a_{\rho}D_{\rho}$ where
$\Sigma(\Phi)(1)$ is the set of one dimensional faces in $\Sigma(\Phi)$ and
$D_{0}=\sum_{\sigma\in\Sigma(\Phi)(1)}D_{\rho}$. Since $D^{\prime}_{0}$ is
$\Delta_{K}(\Phi)$-equivariant and $\Sigma(\Phi)(1)/\Delta_{K}(\Phi)$ is
finite, we deduce that for any $s\geq 1$, there exists $\ell$ such that
$0<sa_{\rho}\leq\ell$. It follows that the round-down of the
$\mathbb{Q}$-divisor $-\ell^{-1}D^{\prime}_{0}$ is $-D_{0}$. We deduce from
[CLS11], lem. 9.3.4 that
$\mathscr{O}(-D_{0})\hookrightarrow(\psi_{\ell})_{\star}\mathscr{O}(-sD^{\prime}_{0})$
is a split injection. Pulling back this morphism by $\psi_{n}$, we deduce that
$\mathscr{O}(-nD_{0})\hookrightarrow(\psi_{\ell})_{\star}\mathscr{O}(-snD^{\prime}_{0})$
is a split injection. ∎

We let
$\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}}:S_{K,\Sigma}^{tor}\rightarrow
S_{K^{\prime},\Sigma^{\prime}}^{tor}$ be the map associated to a change of
level (for $K\subseteq K^{\prime}$) and cone decomposition.

######  Theorem 4.1.8.

We have
$\mathrm{R}^{i}(\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}})_{\star}\mathcal{V}_{\kappa,K,\Sigma}=0$
and
$\mathrm{R}^{i}(\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}})_{\star}\mathcal{V}_{\kappa,K,\Sigma}(-D_{K,\Sigma})=0$
for all $i>0$. Moreover if $K\subseteq K^{\prime}$ is normal then we have

$((\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}})_{\star}\mathcal{V}_{\kappa,K,\Sigma})^{K^{\prime}/K}=\mathcal{V}_{\kappa,K^{\prime},\Sigma^{\prime}}$

and

$((\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}})_{\star}\mathcal{V}_{\kappa,K,\Sigma}(-D_{K,\Sigma}))^{K^{\prime}/K}=\mathcal{V}_{\kappa,K^{\prime},\Sigma^{\prime}}(-D_{K^{\prime},\Sigma^{\prime}}).$

###### Proof.

This is easily extracted from [Har90b], section 2 (look in particular at
proposition 2.4 and proposition 2.6 and their proofs). In the PEL case, this
is explicitly [Lan17], prop. 7.5. ∎

In particular from the case that $K=K^{\prime}$ we deduce that the
cohomologies
$\mathrm{R}\Gamma(S_{K,\Sigma}^{tor},\mathcal{V}_{\kappa,K,\Sigma})$ and
$\mathrm{R}\Gamma(S_{K,\Sigma}^{tor},\mathcal{V}_{\kappa,K,\Sigma}(-D_{K,\Sigma}))$
do not depend on $\Sigma$.

We also recall that the maps $\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}}$ have
fundamental classes in the sense of [FP19] section 2.3. Namely we have maps
$\mathscr{O}_{S_{K,\Sigma}^{tor}}\to\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}}^{!}\mathscr{O}_{S_{K^{\prime},\Sigma^{\prime}}^{tor}}$
and
$\mathscr{O}_{S_{K,\Sigma}^{tor}}(-D_{K,\Sigma})\to\pi_{K,K^{\prime},\Sigma,\Sigma}^{!}\mathscr{O}_{S_{K^{\prime},\Sigma^{\prime}}^{tor}}(-D_{K^{\prime},\Sigma^{\prime}})$
or equivalently, by adjunction, trace maps
$\mathrm{R}(\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}})_{\star}\mathscr{O}_{S_{K,\Sigma}^{tor}}=(\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}})_{\star}\mathscr{O}_{S_{K,\Sigma}^{tor}}\to\mathscr{O}_{S_{K^{\prime},\Sigma^{\prime}}^{tor}}$
and
$\mathrm{R}(\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}})_{\star}(\mathscr{O}_{S_{K,\Sigma}^{tor}}(-D_{K,\Sigma}))=(\pi_{K,K^{\prime},\Sigma,\Sigma^{\prime}})_{\star}(\mathscr{O}_{S_{K,\Sigma}^{tor}}(-D_{K,\Sigma}))\to\mathscr{O}_{S_{K^{\prime},\Sigma^{\prime}}^{tor}}(-D_{K^{\prime},\Sigma^{\prime}})$,
both of which extend the trace map for the finite étale morphism
$\pi_{K,K^{\prime}}:S_{K}\to S_{K^{\prime}}$.

###  4.2. Action of the Hecke algebra

####  4.2.1. Action of Hecke correspondences on cohomology

Let $K_{1},K_{2}\subseteq G(\mathbb{A}_{f})$ be open compact subgroups, and
let $g\in G(\mathbb{A}_{f})$. To this data we associate a Hecke correspondence

where $p_{1}$ is the forgetful map corresponding to the inclusion $K_{1}\cap
gK_{2}g^{-1}\subseteq K_{1}$, and $p_{2}$ is the composition of the action map
$[g]:S_{K_{1}\cap gK_{2}g^{-1}}\to S_{g^{-1}K_{1}g\cap K_{2}}$ and the
forgetful map corresponding to the inclusion $g^{-1}K_{1}g\cap K_{2}\subseteq
K_{2}$.

For any weight $\kappa\in X^{\star}(T^{c})^{M_{\mu},+}$ we have a
cohomological correspondence
$(p_{1})_{\star}p_{2}^{\star}\mathcal{V}_{\kappa,K_{2}}\to\mathcal{V}_{\kappa,K_{1}}$.
This is obtained by combining the trace map
$\mathrm{tr}_{p_{1}}:(p_{1})_{\star}\mathcal{O}_{S_{K_{1}\cap
gK_{2}g^{-1}}}\to\mathcal{O}_{S_{K_{1}}}$ for the finite étale map $p_{1}$
with the isomorphism $p_{2}^{\star}\mathcal{V}_{\kappa,K_{2}}\simeq
p_{1}^{\star}\mathcal{V}_{\kappa,K_{1}}$, which is itself composed of the
isomorphism
$p_{1}^{\star}\mathcal{V}_{\kappa,K_{1}}\simeq\mathcal{V}_{\kappa,K_{1}\cap
gK_{2}g^{-1}}$ and the similar isomorphism for the forgetful map
$S_{g^{-1}K_{1}g\cap K_{2}}\to S_{K_{2}}$, as well as the action map
$[g]^{\star}\mathcal{V}_{\kappa,g^{-1}K_{1}g\cap
K_{2}}\simeq\mathcal{V}_{\kappa,K_{1}\cap gK_{2}g^{-1}}$.

One readily verifies that the cohomological correspondence $((S_{K_{1}\cap
gK_{2}g^{-1}},p_{1},p_{2}),(p_{1})_{\star}p_{2}^{\star}\mathcal{V}_{\kappa,K_{2}}\to\mathcal{V}_{\kappa,K_{1}})$
only depends on the double coset $K_{1}gK_{2}$, up to canonical isomorphism.

Now for suitable choices of cone decomposition we have a diagram

and we claim that our cohomological correspondence on the interior extends to
a cohomological correspondence
$\mathrm{R}(p_{1})_{\star}p_{2}^{\star}\mathcal{V}_{\kappa,K_{2},\Sigma^{\prime}}=(p_{1})_{\star}p_{2}^{\star}\mathcal{V}_{\kappa,K_{2},\Sigma^{\prime}}\to\mathcal{V}_{\kappa,K_{1},\Sigma^{\prime\prime}}$
as well as a cuspidal version
$\mathrm{R}(p_{1})_{\star}p_{2}^{\star}\mathcal{V}_{\kappa,K_{2},\Sigma^{\prime}}(-D_{K_{2},\Sigma^{\prime}})=(p_{1})_{\star}p_{2}^{\star}\mathcal{V}_{\kappa,K_{2},\Sigma^{\prime}}(-D_{K_{2},\Sigma^{\prime}})\to\mathcal{V}_{\kappa,K_{1},\Sigma^{\prime\prime}}(-D_{K_{1},\Sigma})$.
We have already discussed the extensions of the trace map in the previous
section. As for the action map, it also induces an isomorphism of canonical
extensions $p_{2}^{\star}\mathcal{V}_{\kappa,K_{2},\Sigma}\to
p_{1}^{\star}\mathcal{V}_{\kappa,K_{1},\Sigma}$ as well as a morphism of
subsheaves
$p_{2}^{\star}(\mathcal{V}_{\kappa,K_{2},\Sigma}(-D_{K_{2},\Sigma^{\prime}}))\to(p_{1}^{\star}\mathcal{V}_{\kappa,K_{1},\Sigma})(-D_{K_{1}\cap
gK_{2}g^{-1},\Sigma^{\prime\prime}})$.

Finally these cohomological correspondences induces maps on cohomology in the
usual way, namely we denote by $[K_{1}gK_{2}]$ the composition

$\mathrm{R}\Gamma(S_{K_{2},\Sigma^{\prime}}^{tor},\mathcal{V}_{\kappa,K_{2},\Sigma^{\prime}})\to\mathrm{R}\Gamma(S_{K_{1}\cap
gK_{2}g^{-1},\Sigma^{\prime\prime}}^{tor},p_{2}^{*}\mathcal{V}_{\kappa,K_{2},\Sigma^{\prime}})=$
$\mathrm{R}\Gamma(S_{K_{1},\Sigma}^{tor},\mathrm{R}(p_{1})_{*}p_{2}^{*}\mathcal{V}_{\kappa,K_{2},\Sigma^{\prime}})\to\Gamma(S_{K_{1},\Sigma}^{tor},\mathcal{V}_{\kappa,K_{1},\Sigma})$

where the first map is $p_{2}^{*}$ and the last map uses the cohomological
correspondence. We have a similar definition for cuspidal cohomology.

####  4.2.2. Composition of Hecke correspondences

Let us briefly explain the point of this section. We have defined an action of
individual Hecke operators on coherent cohomology, and we would like to show
that this actually gives an action of the Hecke algebra. The standard proof
(see [Har90b, Prop. 2.6]) is to pass to a limit over all $K$ to obtain a
representation of $G(\mathbb{A}_{f})$, and then use the purely group theoretic
fact that Hecke algebras act on its invariants. However in this paper we will
also study coherent cohomology with support conditions which are not
$G(\mathbb{Q}_{p})$ invariant, and so we no longer expect an action of the
full Hecke algebra, but only of certain Hecke operators which preserve the
support conditions in a suitable sense. We would still like to know that these
Hecke operators compose according to relations in the abstract Hecke algebra.

For this reason we develop a different approach to composition of Hecke
operators, by directly studying the geometric composition of Hecke
correspondences. This material is presumably well known but we lack a
reference (see [FC90, VII §3] for a closely related discussion.)

We recall the formalism of double coset multiplication. Fix a Haar measure on
$G(\mathbb{A}_{f})$ and let $C_{c}^{\infty}(G(\mathbb{A}_{f}),\mathbb{R})$ be
the Hecke algebra with its convolution product. For $K_{1},K_{2}\subset
G(\mathbb{A}_{f})$ we consider the free group $\mathbb{Z}[K_{1}\backslash
G(\mathbb{A}_{f})/K_{2}]$ and we write the basis elements as $[K_{1}gK_{2}]$.
We have an embedding $i_{K_{1},K_{2}}:\mathbb{Z}[K_{1}\backslash
G(\mathbb{A}_{f})/K_{2}]\to C_{c}^{\infty}(G(\mathbb{A}_{f}),\mathbb{R})$
which sends $[K_{1}gK_{2}]$ to
$\frac{1}{\sqrt{\mathrm{vol}(K_{1})\mathrm{vol}(K_{2})}}\mathbf{1}_{K_{1}gK_{2}}$,
where $\mathbf{1}_{K_{1}gK_{2}}$ denotes the characteristic function of the
double coset $K_{1}gK_{2}$.

For $K_{1},K_{2},K_{3}\subset G(\mathbb{A}_{f})$ open compact, there is a
product map

$\mathbb{Z}[K_{1}\backslash
G(\mathbb{A}_{f})/K_{2}]\times\mathbb{Z}[K_{1}\backslash
G(\mathbb{A}_{f})/K_{2}]\to\mathbb{Z}[K_{1}\backslash
G(\mathbb{A}_{f})/K_{3}]$

which can be defined by

$i_{K_{1},K_{3}}([K_{1}gK_{2}][K_{2}hK_{3}])=i_{K_{1},K_{2}}([K_{1}gK_{2}])\star
i_{K_{2},K_{3}}([K_{2}hK_{3}])$

To see that the right hand side is in the image of $i_{K_{1},K_{3}}$ note that

$(\mathbf{1}_{K_{1}gK_{2}}\star\mathbf{1}_{K_{2}hK_{3}})(x)=\mathrm{vol}(K_{1}gK_{2}\cap
xK_{3}h^{-1}K_{2})$

is an integer multiple of $\mathrm{vol}(K_{2})$. We also note that this
definition is independent of the choice of Haar measure.

It follows from the definition that double coset multiplication is associative
and satisfies

$([K_{1}gK_{2}][K_{2}hK_{3}])^{t}=[K_{2}hK_{3}]^{t}[K_{1}gK_{2}]^{t}$

where the transpose map $(-)^{t}:\mathbb{Z}[K_{1}\backslash
G(\mathbb{A}_{f})/K_{2}]\to\mathbb{Z}[K_{2}\backslash
G(\mathbb{A}_{f})/K_{1}]$ is defined by
$[K_{1}gK_{2}]^{t}=[K_{2}g^{-1}K_{1}]$. This corresponds to transposes of
Hecke correspondences.

We now give a formula for double coset multiplication which is closely related
to geometric composition of Hecke correspondences.

######  Proposition 4.2.3.

Let $k_{1},\ldots,k_{n}\in K_{2}$ be a set of representatives for the double
cosets $(K_{2}\cap g^{-1}K_{1}g)\backslash K_{2}/(K_{2}\cap hK_{3}h^{-1})$.
Then we have

$[K_{1}gK_{2}][K_{2}hK_{3}]=\sum_{i=1}^{n}c(k_{i})[K_{1}gk_{i}hK_{3}]$

where $c(k_{i})=[g^{-1}K_{1}g\cap(k_{i}h)K_{3}(k_{i}h)^{-1}:g^{-1}K_{1}g\cap
K_{2}\cap(k_{i}h)K_{3}(k_{i}h)^{-1}]$

###### Proof.

We first note that for $x,y\in G(\mathbb{A}_{f})$ we have
$\mathbf{1}_{K_{1}x}\star\mathbf{1}_{yK_{3}}=\mathrm{vol}(x^{-1}K_{1}x\cap
yK_{3}y^{-1})\mathbf{1}_{K_{1}xyK_{3}}$. Decomposing $K_{1}gK_{2}$ into right
$K_{1}$ cosets and $K_{2}hK_{3}$ into left $K_{3}$ cosets we have

$\mathbf{1}_{K_{1}gK_{2}}\star\mathbf{1}_{K_{2}hK_{3}}=\sum_{\begin{subarray}{c}x\in
K_{1}\backslash K_{1}gK_{2}\\\ y\in
K_{2}hK_{3}/K_{3}\end{subarray}}\mathrm{vol}(x^{-1}K_{1}x\cap
yK_{3}y^{-1})\mathbf{1}_{K_{1}xyK_{3}}.$

Now note that the terms of this sum are constant in $K_{2}$ orbits, for the
action $k\cdot(K_{1}x,yK_{3})=(K_{1}xk^{-1},kyK_{3})$. We have a bijection

$(K_{2}\cap g^{-1}K_{1}g)\backslash K_{2}/(K_{2}\cap hK_{3}h^{-1})\to
K_{2}\backslash((K_{1}\backslash K_{1}gK_{2})\times(K_{2}hK_{3}/K_{3}))$

sending the double coset of $k$ to the orbit of $(K_{1}g,khK_{3})$. Moreover
the stabilizer of this orbit is $K_{2}\cap
g^{-1}K_{1}g\cap(kh)K_{3}(kh)^{-1}$. It follows that

$\mathbf{1}_{K_{1}gK_{2}}\star\mathbf{1}_{K_{2}hK_{3}}=\sum_{i=1}^{n}\frac{\mathrm{vol}(K_{2})\mathrm{vol}(g^{-1}K_{1}g\cap(k_{i}h)K_{3}(k_{i}h)^{-1})}{\text{vol}(K_{2}\cap
g^{-1}K_{1}g\cap(k_{i}h)K_{3}(k_{i}h)^{-1})}\mathbf{1}_{K_{1}gk_{i}hK_{3}}.$

which translated back into double coset multiplication gives the proposition.
∎

Suppose we have $K_{1},K_{2},K_{3}\subset G(\mathbb{A}_{f})$ open compact
subgroups and $g,h\in G(\mathbb{A}_{f})$. We have a diagram of correspondences

where the middle diamond is cartesian. We denote $s_{1}=p_{1}r_{1}$,
$s_{2}=q_{2}r_{2}$.

It is not true in general that the correspondence $(C,s_{1},s_{2})$ is
isomorphic to a disjoint union of Hecke correspondences. However this is
almost the case in a way we now explain.

We first construct another correspondence
$(C^{\prime},s_{1}^{\prime},s_{2}^{\prime})$ between $S_{K_{1}}$ and
$S_{K_{3}}$. Let $k_{1},\ldots,k_{n}$ be as in proposition 4.2.3. Now let
$C^{\prime}=\coprod_{i}S_{K_{1}\cap
gK_{2}g^{-1}\cap(gk_{i}h)K_{3}(gk_{i}h)^{-1}}$, let
$s_{1}^{\prime}:C^{\prime}\to S_{K_{1}}$ be the forgetful map on each
component, and let $s_{2}^{\prime}:C^{\prime}\to S_{K_{3}}$ be the action map
$[gk_{i}h]$ followed by the forgetful map. Then we note that while the
components of $C^{\prime}$ are not necessarily Hecke correspondences
themselves, we nonetheless have commutative diagrams

where the vertical arrow and rightward arrows are forgetful maps and the
leftward arrows are the action maps for $[gk_{i}h]$ followed by forgetful
maps. In particular the bottom correspondence is exactly the Hecke
correspondence corresponding to the double coset $K_{1}gk_{i}hK_{3}$.

######  Proposition 4.2.4.

There is an isomorphism of correspondences
$(C,s_{1},s_{2})\simeq(C^{\prime},s_{1}^{\prime},s_{2}^{\prime})$.

We will use the following group theoretic lemma.

######  Lemma 4.2.5.

Let $K$ be a group and $H_{1},H_{2}\subseteq K$ subgroups. Let $X$ be a right
$K$-torsor. Then there is a bijection

$\displaystyle\coprod_{H_{1}gH_{2}\in H_{1}\backslash
K/H_{2}}X/gH_{1}g^{-1}\cap H_{2}$ $\displaystyle\to$ $\displaystyle
X/H_{1}\times X/H_{2}$ $\displaystyle x(gH_{1}g^{-1}\cap H_{2})$
$\displaystyle\mapsto$ $\displaystyle(xgH_{1},xH_{2})$

###### Proof of proposition 4.2.4.

We recall that a point of the Shimura variety $S_{K}$ over a connected locally
Noetherian base is some data which is independent of $K$ (an isogeny class of
abelian varieties with Hodge tensors) to which one associates a right
$G(\mathbb{A}_{f})$-torsor, and a $K$ level structure is just a $K$ orbit in
this $G(\mathbb{A}_{f})$-torsor. Moreover for any $K^{\prime}\subset K$ the
forgetful map $S_{K^{\prime}}\to S_{K}$ at the level of points sends the
$K^{\prime}$-orbit to the $K$-orbit generated by it, and the action map
$[g]:S_{K}\to S_{g^{-1}Kg}$ is given by right multiplication by $g$.

With this observation the proof of the proposition proceeds formally. We can
consider a slightly expanded diagram.

Here all three quadrilaterals are Cartesian. We can compute $C^{\prime\prime}$
using lemma 4.2.5. Indeed giving a point of $C^{\prime\prime}$ lying over a
point of $S_{K_{2}}$ is the same as giving a pair of an $H_{1}=K_{2}\cap
hK_{3}h^{-1}$ and $H_{2}=g^{-1}K_{1}g\cap K_{2}$ orbit inside a
$K=K_{2}$-torsor $X$. We deduce from lemma 4.2.5 an isomorphism

$C^{\prime\prime}\simeq\coprod_{i=1}^{n}S_{g^{-1}K_{1}g\cap
K_{2}\cap(k_{i}h)K_{3}(k_{i}h)^{-1}}$

where on each factor, the map to $S_{g^{-1}K_{1}g\cap K_{2}}$ is the forgetful
map, while the map to $S_{K_{2}\cap hK_{3}h^{-1}}$ is the action map $[k_{i}$]
followed by the forgetful map. Then the isomorphism $C\simeq C^{\prime}$ is
obtained as the composition

$C\overset{[g]}{\to}C^{\prime\prime}\to\coprod_{i=1}^{n}S_{g^{-1}K_{1}g\cap
K_{2}\cap(k_{i}h)K_{3}(k_{i}h)^{-1}}\overset{[g]^{-1}}{\to}C^{\prime}.$

Moreover from the descriptions of the maps out of $C^{\prime\prime}$ above one
immediately deduces the compatibility between $s_{1},s_{1}^{\prime}$ and
$s_{2},s_{2}^{\prime}$. ∎

The following proposition is readily deduced from [Har90b, Prop. 2.6], but we
give an alternative proof which will also work for cohomology with support.

######  Proposition 4.2.6.

We have $[K_{1}gK_{2}]\circ[K_{2}hK_{3}]=[K_{1}gK_{2}][K_{2}hK_{3}]$ as maps

$\mathrm{R}\Gamma(S_{K_{3},\Sigma^{\prime\prime}}^{tor},\mathcal{V}_{\kappa,K_{3},\Sigma^{\prime\prime}})\to\mathrm{R}\Gamma(S_{K_{1},\Sigma}^{tor},\mathcal{V}_{\kappa,K_{1},\Sigma}).$

###### Proof.

We can choose toroidal compactifications of all the Shimura varieties
occurring in this section: $S_{K_{i}}$, $i=1,2,3$, $S_{K_{1}\cap
gK_{2}g^{-1}}$,$S_{K_{2}\cap hK_{3}h^{-1}}$,
$S_{K_{1}\cap(gk_{i}h)K_{3}(gk_{i}h)^{-1}}$, $S_{K_{1}\cap
gK_{2}g^{-1}\cap(gk_{i}h)K_{3}(gk_{i}h)^{-1}}$ so that all the diagrams
appearing above extend to the compactifications (we will not be particularly
consistent or careful with our labelling of the various cone decompositions in
this argument, and they will be denoted by $\Sigma_{\ast}$ where $\ast$ is an
index). In particular we obtain a compactification $C^{tor}$ of the
correspondence $C$ which is isomorphic to a disjoint union of toroidal
compactifications of Shimura varieties. Moreover using this description, we
can construct cohomological correspondences over $C^{tor}$ as in section 4.2.1
using the action maps for $gk_{i}h$. We first claim that this cohomological
correspondence acts on cohomology by the linear combination of Hecke operators
on the right hand side of the formula in proposition 4.2.3.

Consider a commutative diagram

We claim that the following diagram commutes (where the vertical map is
induced by the adjunction $\mathrm{Id}\Rightarrow c_{\star}c^{\star}$, the
other maps are given by the cohomological correspondence, and $c(k_{i})$ is
the generic degree of the generically finite flat map $c$):

These are maps of locally free sheaves by theorem 4.1.8, and the commutativity
of the diagram follows from its commutativity away from the boundary which is
clear.

Hence to prove the proposition, we need to see that the action of the
cohomological correspondence $C^{tor}$ is equal to
$[K_{1}gK_{2}]\circ[K_{2}hK_{3}]$.

Consider the diamond from the proof of proposition 4.2.4

which is cartesian away from the boundary. We would like to know that

commutes, and similarly for cuspidal cohomology. Note that here what we write
as $(p^{\prime})^{\star}$ is really the composition

$\mathrm{R}\Gamma(S_{K_{2}\cap
hK_{3}h^{-1},\Sigma_{a}}^{tor},\mathcal{V}_{\kappa,K_{2}\cap
hK_{3}h^{-1},\Sigma_{a}})\to\mathrm{R}\Gamma({C^{\prime\prime}}^{tor},(p^{\prime})^{\star}\mathcal{V}_{\kappa,K_{2}\cap
hK_{3}h^{-1},\Sigma_{a}})\to$
$\mathrm{R}\Gamma({C^{\prime\prime}}^{tor},(q^{\prime})^{\star}\mathcal{V}_{\kappa,g^{-1}K_{1}g\cap
K_{2},\Sigma_{b}})$

where the first map is literally $(p^{\prime})^{\star}$ and the second map is
described on each component of ${C^{\prime\prime}}^{tor}$ as the action map
$[k_{i}]$ (see the description of $p^{\prime}$,$q^{\prime}$ given in the proof
of Proposition 4.2.4.)

We claim that the following diagram commutes:

This is a diagram of locally free sheaves by theorem 4.1.8, and therefore the
commutativity can be checked away from the boundary, where this is clear. ∎

######  Remark 4.2.7.

We remark that for $K^{\prime}\subseteq K\subset G(\mathbb{A}_{f})$ open
compact subgroups, the Hecke operator $[K^{\prime}1K]$ and $[K1K^{\prime}]$
are the pullback and trace for the forgetful map
$S_{K^{\prime},\Sigma^{\prime}}\to S_{K,\Sigma}$. Moreover if $g\in
G(\mathbb{A}_{f})$ then $[Kg(g^{-1}Kg)]$ is simply the action map
$[g]^{\star}$. Finally for $K_{1},K_{2}\subset G(\mathbb{A}_{f})$ open compact
subgroups and $g\in G(\mathbb{A}_{f})$, we have a factorization

$[K_{1}gK_{2}]=[K_{1}1(K_{1}\cap gK_{2}g^{-1})][(K_{1}\cap
gK_{2}g^{-1})g(g^{-1}K_{1}g\cap K_{2})][(g^{-1}K_{1}g\cap K_{2})1K_{2}]$

of $[K_{1}gK_{2}]$ into a pullback map, an action map, and a trace, which is
essentially the definition of $[K_{1}gK_{2}]$.

####  4.2.8. Serre duality

The dualizing sheaf of $S_{K,\Sigma}$ is
$\mathcal{V}_{-2\rho_{nc},K,\Sigma}(-D_{K,\Sigma})$ where $\rho_{nc}$ is half
the sum of the non-compact positive roots. Indeed, recall our choice of non-
compact positive roots being in $\mathfrak{g}/\mathfrak{p}_{\mu}^{std}$ and
that $\mathfrak{g}/\mathfrak{p}_{\mu}^{std}$ is the tangent space at the
identity of $FL_{G,\mu}^{std}$ and then apply [Har90b], proposition 2.2.6.

The Serre dual of the automorphic sheaf $\mathcal{V}_{\kappa,K,\Sigma}$ is
therefore $\mathcal{V}_{-2\rho_{nc}-w_{0,M}\kappa,K,\Sigma}(-D_{K,\Sigma})$
where $w_{0,M}$ is the longest element of the Weyl group of $M_{\mu}$. Serre
duality is:

######  Proposition 4.2.9.

There is a Serre duality isomorphism

$D_{F}(\mathrm{R}\Gamma(S_{K,\Sigma}^{tor},\mathcal{V}_{\kappa,K,\Sigma}))[-d]\simeq\mathrm{R}\Gamma(S_{K,\Sigma}^{tor},\mathcal{V}_{-2\rho_{nc}-w_{0,M_{\mu}}\kappa,K,\Sigma}(-D_{K,\Sigma}))$

where $D_{F}(-)=\mathrm{RHom}_{F}(-,F)$ is the dualizing functor for
$F$-vector spaces and $d$ is the dimension of $S_{K}$. Moreover this
isomorphism, is compatible with the Hecke action in the sense the action of
$[KgK]$ on the left matches the action of $[KgK]^{t}=[Kg^{-1}K]$ on the right.

###### Proof.

For the existence of the duality pairing we refer to [Har66]. We simply prove
the formula for the adjoint. We denote by

$D_{K^{\prime}}(-)=\underline{\mathrm{RHom}}(-,\mathcal{V}_{-2\rho_{nc},K^{\prime},\Sigma^{\prime\prime\prime}}(-D_{K^{\prime},\Sigma^{\prime\prime\prime}}))$

the dualizing functor on $S_{K^{\prime},\Sigma^{\prime\prime\prime}}^{tor}$
for any compact open $K^{\prime}$ (and cone decomposition
$\Sigma^{\prime\prime\prime}$). Let $f=[KgK]$ be a characteristic function to
which we associate a Hecke correspondence
$S^{tor}_{K,\Sigma}\stackrel{{\scriptstyle p_{1}}}{{\leftarrow}}S^{tor}_{K\cap
gKg^{-1},\Sigma^{\prime\prime}}\stackrel{{\scriptstyle
p_{2}}}{{\rightarrow}}S^{tor}_{K,\Sigma^{\prime}}$. The action of $f$ on the
cohomology arises from a cohomological correspondence (see section 4.2.1):

$f:p_{2}^{\star}\mathcal{V}_{\kappa,K,\Sigma}\stackrel{{\scriptstyle
f_{\kappa}}}{{\rightarrow}}p_{1}^{\star}\mathcal{V}_{\kappa,K,\Sigma}\stackrel{{\scriptstyle
Id\otimes\mathrm{tr}_{p_{1}}}}{{\rightarrow}}p_{1}^{!}\mathcal{V}_{\kappa,K,\Sigma}.$

We find (since duality switches ⋆ and !) that

$D_{K\cap
gKg^{-1}}(f):p_{1}^{\star}D_{K}(\mathcal{V}_{\kappa,K,\Sigma})\stackrel{{\scriptstyle
Id\otimes\mathrm{tr}_{p_{1}}}}{{\rightarrow}}p_{1}^{!}D_{K}(\mathcal{V}_{\kappa,K,\Sigma})\rightarrow
p_{2}^{!}D_{K}(\mathcal{V}_{\kappa,K,\Sigma}).$

Remark that

$\displaystyle p_{1}^{!}D_{K}(\mathcal{V}_{\kappa,K,\Sigma})$ $\displaystyle=$
$\displaystyle p_{1}^{\star}\mathcal{V}_{\kappa,K,\Sigma}^{\vee}\otimes
p_{1}^{!}\mathcal{V}_{-2\rho_{nc},K,\Sigma}(-D_{K,\Sigma})$ $\displaystyle
p_{2}^{!}D_{K}(\mathcal{V}_{\kappa,K,\Sigma^{\prime}})$ $\displaystyle=$
$\displaystyle
p_{2}^{\star}\mathcal{V}_{\kappa,K,\Sigma^{\prime}}^{\vee}\otimes
p_{2}^{!}\mathcal{V}_{-2\rho_{nc},K,\Sigma}(-D_{K,\Sigma^{\prime}})$

We have a canonical isomorphism

$\mathrm{Id}:p_{1}^{!}\mathcal{V}_{-2\rho_{nc},\Sigma}(-D_{K,\Sigma})=p_{2}^{!}\mathcal{V}_{-2\rho_{nc},K,\Sigma^{\prime}}(-D_{K,\Sigma^{\prime}}),$

as both sheaves identify with the canonical sheaf of $S^{tor}_{K\cap
gKg^{-1},\Sigma^{\prime\prime}}$. The map
$p_{1}^{!}D_{K}(\mathcal{V}_{\kappa,K,\Sigma})\rightarrow
p_{2}^{!}D_{K}(\mathcal{V}_{\kappa,K,\Sigma})$ therefore writes
$f_{\kappa}^{\vee}\otimes\mathrm{Id}$. Let $f^{t}=[Kg^{-1}K]$. We observe that
$f_{\kappa}^{\vee}=(f^{t})_{-w_{0,M}\kappa}$ by definition (this identity
boils down to $A^{t}=((A^{-1})^{-1})^{t}$ for a matrix in $\mathrm{GL}_{n}$).
We now claim that we have a commutative diagram:

This implies that $D_{K\cap gKg^{-1}}(f)=f^{t}$. The commutativity of the
diagram now boils down to the commutativity of:

It is sufficient to prove the commutativity outside of the boundary. For any
level $K$, the identification of the canonical sheaf of $S_{K}$ with
$\mathcal{V}_{-2\rho_{nc},K}$ is functorial in the tower of Shimura varieties.
Therefore, for the action map $[g]:S_{K\cap gKg^{-1}}\rightarrow S_{K\cap
g^{-1}Kg}$, we find that the map $[g]^{\star}\mathcal{V}_{-2\rho_{nc},K\cap
g^{-}1Kg}\rightarrow\mathcal{V}_{-2\rho_{nc},K\cap gKg^{-1}}$ is the canonical
isomorphism between canonical sheaves. We finally deduce that the map
$(f^{t})_{-2\rho_{nc}}:p_{1}^{\star}\mathcal{V}_{-2\rho_{nc},K}\rightarrow
p_{2}^{\star}\mathcal{V}_{-2\rho_{nc},K}$ decomposes as:

$p_{1}^{\star}\mathcal{V}_{-2\rho_{nc},K}\rightarrow\mathcal{V}_{-2\rho_{nc},K\cap
gKg^{-1}}\rightarrow g^{\star}\mathcal{V}_{-2\rho_{nc},K\cap
g^{-1}Kg}\rightarrow p_{2}^{\star}\mathcal{V}_{-2\rho_{nc},K}$

where the first map is induced by $\mathrm{tr}_{p_{1}}$, the second map is the
canonical isomorphism, and the last map is induced by
$g^{\star}(\mathrm{tr}_{p^{\prime}_{2}})^{-1}$ where $p^{\prime}_{2}:S_{K\cap
g^{-1}Kg}\rightarrow S_{K}$. This is telling us that the diagram commutes. ∎

######  Remark 4.2.10.

One proves more generally that the adjoint of an Hecke operator
$[K_{1}gK_{2}]$ for two (not necessarily equal) compact open subgroup $K_{1}$
and $K_{2}$ is $[K_{1}gK_{2}]^{t}=[K_{2}g^{-1}K_{1}]$. Details are left to the
reader.

####  4.2.11. The finite slope part of classical cohomology

We now assume that $G_{\mathbb{Q}_{p}}$ is quasi-split. We assume that
$K=K^{p}\times K_{p}$ where $K_{p}=K_{p,m,b}$ for $m\geq b\geq 0$, $m>0$ is
one of the subgroups with an Iwahori decomposition introduced in section
3.5.1. We recall that for a choice of $+$ or $-$ we have commutative sub-
algebras $\mathcal{H}_{p,m,b}^{\pm}$ of $\mathbb{Z}[K_{p,m,b}\backslash
G(\mathbb{Q}_{p})/K_{p,m,b}]$. The subalgebra $\mathcal{H}_{p,m,b}^{\pm}$ is
generated by the double cosets $[K_{p,m,b}tK_{p,m,b}]$ with $t\in T^{\pm}$. We
have isomorphisms $\mathcal{H}_{p,m,b}^{\pm}=\mathbb{Z}[T^{\pm}/T_{b}]$. We
also have the ideals $\mathcal{H}_{p,m,b}^{\pm\pm}$ generated by the double
cosets $[K_{p,m,b}tK_{p,m,b}]$ with $t\in T^{\pm\pm}$. The anti-involution of
$\mathbb{Z}[K_{p,m,b}\backslash G(\mathbb{Q}_{p})/K_{p,m,b}]$ defined by
inversion exchanges $\mathcal{H}_{p,m,b}^{+}$ with $\mathcal{H}_{p,m,b}^{-}$
and $\mathcal{H}_{p,m,b}^{++}$ with $\mathcal{H}_{p,m,b}^{--}$.

We let

$\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})^{\pm,fs}=$
$\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})\otimes^{L}_{\mathbb{Q}[T^{\pm}/T_{b}]}\mathbb{Q}[T(\mathbb{Q}_{p})/T_{b}]$

be the finite slope direct factor of
$\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})$
for the operators in $[K_{p,m,b}tK_{p,m,b}]$ for $t\in T^{\pm}$. We have a
similar definition for cuspidal cohomology. We note that the monoids $T^{\pm}$
have the property that for any $t\in T^{\pm\pm}$ and $s\in T^{\pm}$ there is
an $s^{\prime}\in T^{\pm}$ and an $n>0$ such that $ss^{\prime}=t^{n}$. It
follows that the finite slope part can also be described as the finite slope
part fo the single operator $[K_{p,m,b}tK_{p,m,b}]$ for any $t\in T^{\pm\pm}$.

We note that the Serre duality pairing of Proposition 4.2.9 restricts to a
duality

$D_{F}(\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})^{\pm,fs})[-d]\simeq\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{-2\rho_{nc}-w_{0,M}\kappa,K^{p}K_{p,m,b},\Sigma}(-D)^{\mp,fs})$

on finite slope parts.

######  Remark 4.2.12.

Assume that the group $G_{\mathbb{Q}_{p}}$ is unramified. Then, as explained
in remark 3.5.4 we can arrange so that the compact $K_{p,1,0}$ is an Iwahori,
and the compact $K_{p,1,1}$ is a pro-$p$ Iwahori. Let $K_{p}=K_{p,1,1}$ or
$K_{p,1,0}$. All the Hecke operators $[K_{p}tK_{p}]$ for $t\in T^{+}$ or $t\in
T^{-}$ are already invertible in $\mathbb{Q}[K_{p}\backslash
G(\mathbb{Q}_{p})/K_{p}]$ (see [Vig05, Cor. 1]) and so

$\mathrm{R}\Gamma(S_{K^{p}K_{p},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p},\Sigma})^{\pm,fs}=\mathrm{R}\Gamma(S_{K^{p}K_{p},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p},\Sigma})$

and similarly for cuspidal cohomology.

Next we recall how the finite slope part behaves under certain changes of
level. We first recall some classical relations in these Hecke algebras.

######  Lemma 4.2.13.

Let $K_{1},K_{2},K_{3}\subseteq G(\mathbb{Q}_{p})$ be open compact subgroups
with Iwahori decompositions $K_{i}=K_{i}^{-}K_{i}^{0}K_{i}^{+}$. Let
$t_{1},t_{2}\in T(\mathbb{Q}_{p})$. Suppose that $t_{1}^{-1}K_{1}^{-}t_{1}\cap
t_{2}K_{3}^{-}t_{2}^{-1}\subseteq K_{2}^{-}\subseteq
t_{1}^{-1}K_{1}^{-}t_{1}$, $t_{1}^{-1}K_{1}^{+}t_{1}\cap
t_{2}K_{3}^{+}t_{2}^{-1}K_{2}^{+}\subseteq t_{2}K_{3}^{+}t_{3}$ and
$K_{1}^{0}\cap K_{3}^{0}\subseteq K_{2}^{0}\subseteq K_{1}^{0}K_{3}^{0}$. Then
$[K_{1}t_{1}K_{2}][K_{2}t_{2}K_{3}]=[K_{1}t_{1}t_{2}K_{3}]$

###### Proof.

This is an immediate consequence of proposition 4.2.3 upon noting that our
hypotheses imply $K_{2}=(K_{2}\cap t_{1}^{-1}K_{1}t_{1})(K_{2}\cap
t_{2}K_{3}t_{2}^{-1})$ and $t_{1}^{-1}K_{1}t_{1}\cap
t_{2}K_{3}t_{2}^{-1}\subseteq K_{2}$. ∎

######  Lemma 4.2.14.

Let $m^{\prime}\geq b^{\prime}\geq 0$ and $m\geq b\geq 0$ satisfy
$m^{\prime}\geq m>0$ and $b^{\prime}\geq b$.

  1. (1)

For all $t\in T^{+}$ we have

$[K_{p,m^{\prime},b^{\prime}}tK_{p,m^{\prime},b^{\prime}}][K_{p,m^{\prime},b^{\prime}}1K_{p,m,b}]=[K_{p,m^{\prime},b^{\prime}}1K_{p,m,b}][K_{p,m,b}tK_{p,m,b}].$

  2. (2)

For all $t\in T^{-}$ we have

$[K_{p,m,b}tK_{p,m,b}][K_{p,m,b}1K_{p,m^{\prime},b^{\prime}}]=[K_{p,m,b}1K_{p,m^{\prime},b^{\prime}}][K_{p,m^{\prime},b^{\prime}}tK_{p,m^{\prime},b^{\prime}}].$

  3. (3)

For all $t\in T^{++}$ with $\min(t)\geq 1$ (see 3.4) we have factorizations:

$[K_{p,m,b}tK_{p,m,b}]=[K_{p,m,b}tK_{p,m+1,b}][K_{p,m+1,b}1K_{p,m,b}]$

and

$[K_{p,m+1,b}tK_{p,m+1,b}]=[K_{p,m+1,b}1K_{p,m,b}][K_{p,m,b}tK_{p,m+1,b}].$

  4. (4)

For all $t\in T^{--}$ with $\min(t)\geq 1$ we have factorizations:

$[K_{p,m,b}tK_{p,m,b}]=[K_{p,m,b}1K_{p,m+1,b}][K_{p,m+1,b}tK_{p,m,b}]$

and

$[K_{p,m+1,b}tK_{p,m+1,b}]=[K_{p,m+1,b}tK_{p,m,b}][K_{p,m,b}1K_{p,m+1,b}].$

###### Proof.

We note that the second and fourth points are just the transposes of the first
and third. The first and third points are immediate consequences of lemma
4.2.13. ∎

Here are the consequences on cohomology:

######  Corollary 4.2.15.

Let $m^{\prime}\geq b^{\prime}\geq 0$ and $m\geq b\geq 0$ satisfy
$m^{\prime}\geq m>0$ and $b^{\prime}\geq b$.

  1. (1)

For all $t\in T^{+}$, the following diagram commutes:

  2. (2)

For all $m^{\prime}\geq m$ and $t\in T^{-}$, the following diagram commutes:

  3. (3)

For all $m$ and $t\in T^{++}$ with $\min(t)\geq 1$, there is a factorization:

  4. (4)

For all $m$ and $t\in T^{--}$ with $\min(t)\geq 1$, there is a factorization:

  5. (5)

We have the same results for cuspidal cohomology.

We deduce the following classical corollary

######  Corollary 4.2.16.

  1. (1)

For all $m^{\prime}\geq m\geq b$ with $m>0$, the pullback map

$\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})^{+,fs}\rightarrow\mathrm{R}\Gamma(S_{K^{p}K_{p,m^{\prime},b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m^{\prime},b},\Sigma})^{+,fs}$

and the trace map

$\mathrm{R}\Gamma(S_{K^{p}K_{p,m^{\prime},b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m^{\prime},b},\Sigma})^{-,fs}\rightarrow\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})^{-,fs}$

are quasi-isomorphisms, compatible with the action of
$\mathbb{Q}[T(\mathbb{Q}_{p})/T_{b}]$, and the same statements are true for
cuspidal cohomology. Moreover these isomorphisms are compatible with Serre
duality.

  2. (2)

For all $m\geq b^{\prime}\geq b$ with $m>0$, the pullback map

$\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})^{+,fs}\rightarrow(\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b^{\prime}},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b^{\prime}},\Sigma})^{+,fs})^{T_{b}/T_{b^{\prime}}}$

and the trace map

$(\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b^{\prime}},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b^{\prime}},\Sigma})^{-,fs})^{T_{b}/T_{b^{\prime}}}\rightarrow\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})^{-,fs}$

are quasi-isomorphisms, compatible with the action of
$\mathbb{Q}[T(\mathbb{Q}_{p})/T_{b}]$, and the same statements are true for
cuspidal cohomology. Moreover these isomorphisms are compatible with Serre
duality.

###### Proof.

Immediate from corollary 4.2.15 and theorem 4.1.8. ∎

Now let $\chi:T(\mathbb{Z}_{p})\to\overline{F}^{\times}$ be a finite order
character. For all $m\geq b\geq\mathrm{cond}(\chi)$ with $m>0$, the spaces
$\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})^{+,fs}[\chi]$
are canonically isomorphic. We denote this space by
$\mathrm{R}\Gamma(K^{p},\kappa,\chi)^{+,fs}$. We define in the same way
$\mathrm{R}\Gamma(K^{p},\kappa,\chi)^{-,fs}$, and the cuspidal versions
$\mathrm{R}\Gamma(K^{p},\kappa,\chi,cusp)^{+,fs}$ and
$\mathrm{R}\Gamma(K^{p},\kappa,\chi,cusp)^{-,fs}$

These are the classical finite slope cohomologies, for the tame level $K^{p}$,
$M_{\mu}$-dominant algebraic weight $\kappa$, and nebentypus $\chi$. They
satisfy a Serre duality:

$D_{F}(\mathrm{R}\Gamma(K^{p},\kappa,\chi)^{\pm,fs})[-d]\simeq\mathrm{R}\Gamma(K^{p},-2\rho_{nc}-w_{0,M_{\mu}}\kappa,\chi^{-1},cusp)^{\mp,fs}.$

######  Remark 4.2.17.

We explain some constraints on the possible nebentypus $\chi$ for which we
have non trivial cohomology in the case that the subgroup $Z_{s}(G)$ of the
center $Z(G)$ of $G$ is non-trivial (a case that may only occur for non Hodge
type Shimura datum). For simplicity we put $Z=Z(G)$ and $Z_{s}=Z_{s}(G)$.
Recall that $S_{K}(\mathbb{C})=G(\mathbb{Q})\backslash X\times
G(\mathbb{A}_{f})/K$. Passing to the limit over $K$, we find that
$S(\mathbb{C})=\lim_{K}S_{K}(\mathbb{C})=G(\mathbb{Q})\backslash X\times
G(\mathbb{A}_{f})/\overline{Z(\mathbb{Q})}$ where $\overline{Z(\mathbb{Q})}$
is the closure of $Z(\mathbb{Q})$ in $G(\mathbb{A}_{f})$ (see [Del79], sect.
2.19). Let $K^{\prime}\subseteq K$ be two compact open subgroups, with
$K^{\prime}$ normal in $K$. The map $S_{K^{\prime}}(\mathbb{C})\rightarrow
S_{K}(\mathbb{C})$ is a covering with Galois group
$K/(K^{\prime}.\overline{Z(\mathbb{Q})}\cap K)$. If $K$ is neat,
$\overline{Z(\mathbb{Q})}\cap K$ is a subgroup of
$\overline{Z_{s}(\mathbb{Q})}$. In the case that $Z_{s}=\\{1\\}$, we deduce
that $S_{K^{\prime}}(\mathbb{C})\rightarrow S_{K}(\mathbb{C})$ is a Galois
covering with group $K/K^{\prime}$. In general, it follows that the action of
$T(\mathbb{Z}_{p})$ on
$\mathrm{R}\Gamma(S_{K^{p}K_{p,m,b},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p,m,b},\Sigma})^{+,fs}$
factors through an action of $T(\mathbb{Z}_{p})/Z^{\prime}(K^{p})$, where
$Z^{\prime}(K^{p})=\mathrm{Im}((\overline{Z_{s}(\mathbb{Q})}\cap
K^{p}K_{p,m,0})\rightarrow T(\mathbb{Z}_{p}))$. We deduce that when $Z_{s}\neq
1$, if $\chi$ does not factor through $T(\mathbb{Z}_{p})/Z^{\prime}(K^{p})$,
then $\mathrm{R}\Gamma(K^{p},\kappa,\chi)^{\pm,fs}=0$ for all $\kappa$. To
avoid this situation, we will often impose that $\chi$ factors through a
character of $T^{c}(\mathbb{Z}_{p})$.

###  4.3. Jacquet Modules

In this section we translate the finite slope condition into more
representation theoretic terms. We keep assuming that $G_{\mathbb{Q}_{p}}$ is
quasi-split with Borel $B$. We let $U$ be the unipotent radical of $B$. Let
$\pi$ be a smooth admissible representation of $G(\mathbb{Q}_{p})$ with
coefficient in a field of characteristic $0$. We let $\pi(U)\subseteq\pi$ be
the submodule generated by the elements $n\cdot v-v$ for $n\in
U(\mathbb{Q}_{p})$ and $v\in\pi$. We let $\pi_{U}=\pi/\pi(U)$ be the Jacquet
module of $\pi$ (with respect to $U$). This is a smooth admissible
representation of $T(\mathbb{Q}_{p})$ by [Cas], thm. 3.3.1. Moreover, the
functor $\pi\mapsto\pi_{U}$ is an exact functor by [Cas], prop. 3.3.2. We can
define similarly the Jacquet module $\pi_{\overline{U}}$ with respect to
$\overline{U}$. Note that conjugation by the longest element $w_{0}$ of the
Weyl group realizes an isomorphism from $\pi_{U}$ to $\pi_{\overline{U}}$. Let
$\psi:T(\mathbb{Q}_{p})\rightarrow\mathbb{C}^{\times}$ be a continuous
character. We let
$\iota_{B}^{G}(\psi)=\\{f:G(\mathbb{Q}_{p})\rightarrow\mathbb{C},~{}\textrm{smooth},~{}f(bg)=\psi(b)f(g)\\}$,
equipped with the left action induced by right translation of
$G(\mathbb{Q}_{p})$ on itself. We define similarly
$\iota_{\overline{B}}^{G}(\psi)$.

The adjunction formula of [Cas], thm. 3.2.4 states that

$\mathrm{Hom}_{G(\mathbb{Q}_{p})}(\pi,\iota_{B}^{G}(\psi))=\mathrm{Hom}_{T(\mathbb{Q}_{p})}(\pi_{U},\psi)$

and

$\mathrm{Hom}_{G(\mathbb{Q}_{p})}(\pi,\iota_{\overline{B}}^{G}(\psi))=\mathrm{Hom}_{T(\mathbb{Q}_{p})}(\pi_{\overline{U}},\psi).$

Let $K=K_{p,m,b}$. The algebra
$\mathcal{H}_{p,m,b}^{\pm}=\mathbb{Z}[T^{\pm}/T_{b}]$ (by lemma 3.5.8) acts on
$\pi^{K_{p,m,b}}$, and we can define
$\pi^{K_{p,m,b},\pm,fs}\subseteq\pi^{K_{p,m,b}}$ as the sub-vector space where
the operators $[K_{p,m,b}tK_{p,m,b}]$ for $t\in T^{\pm}$ act invertibly.

######  Proposition 4.3.1.

The natural map $\pi^{K_{p,m,b},+,fs}\rightarrow\pi_{U}^{T_{b}}$ is an
isomorphism which is $T^{+}/T_{b}$ equivariant, and the natural map
$\pi^{K_{p,m,b},-,fs}\rightarrow\pi_{\overline{U}}^{T_{b}}$ is an isomorphism
which is $T^{-}/T_{b}$ equivariant.

###### Proof.

See [Cas], Lemma 4.1.1 and proposition 4.1.4. ∎

######  Proposition 4.3.2.

For a smooth irreducible representation $\pi$ of $G(\mathbb{Q}_{p})$, the
following properties are equivalent:

  1. (1)

There exists $m\geq b\geq 0$ such that $\pi^{K_{p,m,b},+,fs}\neq 0$,

  2. (2)

There exists $m\geq b\geq 0$ such that $\pi^{K_{p,m,b},-,fs}\neq 0$,

  3. (3)

There exists a character $\psi$ of $T(\mathbb{Q}_{p})$ such that
$\pi\hookrightarrow\iota_{B}^{G}\psi$,

  4. (4)

There exists a character $\psi^{\prime}$ of $T(\mathbb{Q}_{p})$ such that
$\pi\hookrightarrow\iota_{\overline{B}}^{G}\psi^{\prime}$.

###### Proof.

The points $(1)$ and $(2)$ are equivalent because the non-vanishing of
$\pi^{K_{p,m,b},\pm,fs}$ is equivalent to the non-vanishing of
$\pi_{U}^{T_{b}}$ and $\pi_{\overline{U}}^{T_{b}}$ respectively by proposition
4.3.1. But conjugation by $w_{0}$ realizes an isomorphism between these
spaces. Similarly $(3)$ is equivalent to $(4)$. If we assume $(3)$, the
adjunction formula shows that $\pi_{U}\neq 0$, hence there exists $b$ such
that $\pi_{U}^{T_{b}}\neq 0$ and $\pi^{K_{p,m,b},+,fs}\neq 0$ for any $m\geq
b$ by proposition 4.3.1. Conversely, if $\pi^{K_{p,m,b},+,fs}\neq 0$, then
$\pi_{U}^{T_{b}}\neq 0$ and by adjunction, there is a non zero map:
$\pi\rightarrow\iota_{B}^{G}\psi$ for a character $\psi$. Since $\pi$ is
irreducible, this map is injective. ∎

If one of the equivalent properties of the proposition is satisfied, we say
that an irreducible smooth representation $\pi$ is a _finite slope_
representation.

Let $\pi$ be an admissible representation of $G(\mathbb{Q}_{p})$. By
adjunction, we have a morphism $\pi\rightarrow\iota_{B}^{G}\pi_{U}$ and we let
$\pi^{fs}$ be the image of this morphism. We call $\pi^{fs}$ the finite slope
part of $\pi$.

######  Proposition 4.3.3.

The following properties are satisfied:

  1. (1)

The $G(\mathbb{Q}_{p})$-representations $\pi^{fs}$ is a direct summand of
$\pi$.

  2. (2)

Any irreducible factor of $\pi^{fs}$ is a finite slope representation.

  3. (3)

Any irreducible factor of $\pi$ which is a finite slope representation lies in
$\pi^{fs}$.

  4. (4)

$\pi^{fs}$ is the sub-representation of $\pi$ generated by the
$\pi^{K_{p,m,b},\pm,fs}$ for all $m\geq b\geq 0$.

###### Proof.

By the Bernstein decomposition of the category of smooth representations
([Ren10], VI.7.2) we find that $\pi=\pi^{\prime}\oplus\pi^{\prime\prime}$
where $\pi^{\prime}$ satisfies the properties $(1)$, $(2)$ and $(3)$ of the
representation $\pi^{fs}$ of the proposition. We have that
$\pi^{\prime\prime}_{U}=0$ because the Jacquet functor is exact and
$\pi^{\prime\prime}$ has no finite slope irreducible sub-quotient. We deduce
that $\pi_{U}=\pi^{\prime}_{U}$. Moreover the morphism
$\pi\rightarrow\iota_{B}^{G}\pi_{U}$ factors into
$\pi^{\prime}\rightarrow\iota_{B}^{G}\pi_{U}$ and it follows again from the
exactness of the Jacquet functor that the map
$\pi^{\prime}\rightarrow\iota_{B}^{G}\pi_{U}$ is injective. Therefore
$\pi^{\prime}=\pi^{fs}$. Let $\pi^{\prime\prime\prime}$ be the sub-
representation of $\pi$ generated by $\pi^{K_{p,m,b},\pm,fs}$ for all $m\geq
b\geq 0$. We see that $\pi^{\prime\prime\prime}\subseteq\pi^{fs}$. But it
follows from proposition 4.3.1 that
$\pi^{\prime\prime\prime}_{U}=\pi^{fs}_{U}$. Therefore
$\pi^{fs}/\pi^{\prime\prime\prime}$ has trivial Jacquet module, hence contains
no finite slope sub-quotient and has to be trivial. ∎

Let us denote by

$\mathrm{H}^{i}(K^{p},\kappa)=\mathrm{colim}_{K_{p}}\mathrm{H}^{i}(S_{K^{p}K_{p},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p},\Sigma}),$
$\mathrm{H}^{i}(K^{p},\kappa,cusp)=\mathrm{colim}_{K_{p}}\mathrm{H}^{i}(S_{K^{p}K_{p},\Sigma}^{tor},\mathcal{V}_{\kappa,K^{p}K_{p},\Sigma}(-D_{K^{p}K_{p},\Sigma}))$

and

$\overline{\mathrm{H}}^{i}(K^{p},\kappa)=\mathrm{Im}(\mathrm{H}^{i}(K^{p},\kappa,cusp)\rightarrow\mathrm{H}^{i}(K^{p},\kappa)).$

These are smooth admissible $G(\mathbb{Q}_{p})$-representations. We can
consider their finite slope parts $\mathrm{H}^{i}(K^{p},\kappa)^{fs}$,
$\mathrm{H}^{i}(K^{p},\kappa,cusp)^{fs}$ and
$\overline{\mathrm{H}}^{i}(K^{p},\kappa)^{fs}$. These are direct summands of
$\mathrm{H}^{i}(K^{p},\kappa)$, $\mathrm{H}^{i}(K^{p},\kappa,cusp)$ and
$\overline{\mathrm{H}}^{i}(K^{p},\kappa)$ respectively, and are generated
respectively as $G(\mathbb{Q}_{p})$-representations by the vector spaces
$\mathrm{H}^{i}(K^{p},\kappa,\chi)^{\pm,fs}$,
$\mathrm{H}^{i}(K^{p},\kappa,\chi,cusp)^{\pm,fs}$ and
$\overline{\mathrm{H}}^{i}(K^{p},\kappa,\chi)^{\pm,fs}=\mathrm{Im}(\mathrm{H}^{i}(K^{p},\kappa,\chi,cusp)^{\pm,fs}\rightarrow\mathrm{H}^{i}(K^{p},\kappa,\chi)^{\pm,fs})$
for all characters $\chi:T(\mathbb{Z}_{p})\rightarrow\overline{F}^{\times}$.

###  4.4. The Hodge-Tate period morphism

In this section we recall a number of results concerning the Hodge-Tate period
morphism and infinite level Shimura varieties. We now assume (unless
explicitly mentioned) that $F$ is a finite extension of $\mathbb{Q}_{p}$ such
that we have an embedding $E\hookrightarrow F$ and such that $G$ splits over
$F$. In this paper, the rationality questions with respect to $E$ are not very
important. We will frequently allow ourselves to enlarge $F$ if necessary. Let
$\mathcal{S}^{an}_{K}=(S_{K}\times\mathrm{Spec}~{}F)^{an}$,
$\mathcal{S}^{\star}_{K}=(S^{\star}_{K}\times\mathrm{Spec}~{}F)^{{an}}$,
$\mathcal{S}^{tor}_{K,\Sigma}=(S^{tor}_{K,\Sigma}\times\mathrm{Spec}~{}F)^{{an}}$,
$\mathcal{FL}_{G,\mu}=(FL_{G,\mu}\times\mathrm{Spec}~{}F)^{{an}}$ (see section
3.3 for the meaning of the superscript $an$). The first of these spaces is not
quasi-compact if the Shimura variety is not proper, the other three spaces are
quasi-compact. We will also consider the groups
$\mathcal{G}^{an}=(G\times\mathrm{Spec}~{}\mathbb{Q}_{p})^{an}$,
$\mathcal{P}_{\mu}^{an}=(P_{\mu}\times\mathrm{Spec}~{}F)^{an}$,
$\mathcal{M}_{\mu}^{an}=(M_{\mu}\times\mathrm{Spec}~{}F)^{an}$.

####  4.4.1. Inverse limit of adic spaces

We start by a definition following [SW13], sect. 2.4. Our definition is
slightly more restrictive but fits in our setting.

######  Definition 4.4.2.

Let $\\{\mathcal{X}_{i}\\}_{i\in I}$ be a cofiltered inverse system of locally
of finite type adic spaces over $\mathrm{Spa}(F,\mathcal{O}_{F})$, with finite
transition maps. Let $\mathcal{X}$ be a perfectoid space with compatible maps
$\mathcal{X}\rightarrow\mathcal{X}_{i}$.

We say that $\mathcal{X}\sim\lim_{i\in I}\mathcal{X}_{i}$ if:

  1. (1)

The maps $\mathcal{X}\rightarrow\mathcal{X}_{i}$ induces an homeomorphism of
topological spaces $|\mathcal{X}|=\lim_{i}|\mathcal{X}_{i}|$.

  2. (2)

There is a covering of $\mathcal{X}$ by open affinoids
$U=\mathrm{Spa}(A,A^{+})$ such that $U$ is the preimage of an affinoid
$U_{i}=\mathrm{Spa}(A_{i},A_{i}^{+})\subseteq\mathcal{X}_{i}$ for a cofinal
subset of $I$ and the map $\mathrm{colim}_{i}A_{i}\rightarrow A$ has dense
image.

If $\mathcal{X}\sim\lim_{i\in I}\mathcal{X}_{i}$, then the diamond
$\lim_{i}\mathcal{X}_{i}^{\Diamond}$ is representable by the perfectoid space
$\mathcal{X}$ by [SW13], prop. 2.4.5. In particular, $\mathcal{X}$ is unique
up to a unique isomorphism.

In the notation of point $(2)$, we see that $A^{0}$ is a ring of definition of
$A$ (because $A$ is uniform) and $A^{0}$ is the completion of
$\mathrm{colim}_{i}A^{0}_{i}$ with respect to the $p$-adic topology.

######  Definition 4.4.3.

Let $\\{\mathcal{X}_{i}\\}_{i\in I}$ be a cofiltered inverse system of locally
of finite type adic spaces over $\mathrm{Spa}(F,\mathcal{O}_{F})$ with finite
transition maps. Let $\mathcal{X}$ be a perfectoid space and assume that
$\mathcal{X}\sim\lim_{i}\mathcal{X}_{i}$. We say that an open affinoid subset
$U\hookrightarrow\mathcal{X}$ is good if it satisfies the second property of
definition 4.4.2. We say that an open affinoid
$U_{i}\hookrightarrow\mathcal{X}_{i}$ is pregood if the open subset
$\mathcal{X}\times_{\mathcal{X}_{i}}\mathcal{U}_{i}$ of $\mathcal{X}$ is good.

######  Remark 4.4.4.

We remark that a rational subset of a good open affinoid is also good by
[Sch13b], proposition 2.22.

######  Remark 4.4.5.

It is conjectured in [Sch13b], conjecture 2.24 and proposition 2.26 that any
open affinoid in $\mathcal{X}_{i}$ is pregood.

We end this paragraph with two useful lemmas.

######  Lemma 4.4.6.

Let $\\{\mathcal{X}_{i}\\}_{i\in I}$ be a cofiltered inverse system of locally
of finite type adic spaces over $\mathrm{Spa}(F,\mathcal{O}_{F})$ with finite
transition maps. For each $i$ let $\Pi_{i}$ be the set of connected components
of $\mathcal{X}_{i}$ which we assume to be finite. Let $\Pi=\lim_{i}\Pi_{i}$.
For any $e\in\Pi$ we get a cofiltered inverse system
$\\{\mathcal{X}_{i,e}\\}$. If there is a perfectoid space $\mathcal{X}$ such
that $\mathcal{X}\sim\lim_{i}\mathcal{X}_{i}$ then for all $e\in\Pi$, there is
a perfectoid space $\mathcal{X}_{e}\sim\lim_{i}\mathcal{X}_{i,e}$.

###### Proof.

We reduce to the affine case. Let
$\mathcal{X}_{i}=\mathrm{Spa}(A_{i},A_{i}^{+})$. For each $e\in\Pi_{i}$, we
have $\mathcal{X}_{i,e}=\mathrm{Spa}(A_{i,e},A_{i,e}^{+})$ and
$A_{i}=\prod_{e\in\Pi_{i}}A_{i,e}$. We may assume that all rings $A_{i}$ are
reduced (by taking the reduction). In particular $A_{i}^{0}$ is open and
bounded. This does not affect the $\sim$-limit because perfectoid spaces are
reduced. We may also assume that all maps $A_{i}\rightarrow A_{j}$ are