††thanks: The author was supported by the Swiss National Science Foundation
grant #P2ELP2-191672.

# A tale of parahoric–Hecke algebras, Bernstein and Satake homomorphisms.

Reda Boumasmoud<EMAIL_ADDRESS>&<EMAIL_ADDRESS>Institut de Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG)
Sorbonne Université and Université de Paris, CNRS, F-75006 Paris, France.

###### Abstract.

Let $\mathbf{G}$ be a connected reductive group over a non-archimedean local
field $F$. Let $K_{\mathcal{F}}$ be the parahoric subgroup attached to a facet
$\mathcal{F}$ in the Bruhat–Tits building of $\mathbf{G}$. The ultimate goal
of the present paper is to describe the center of the parahoric–Hecke algebra
$\mathcal{H}(\mathbf{G}(F)\sslash{K_{\mathcal{F}}},\mathds{Z}[q^{-1}])$ with
level $K_{\mathcal{F}}$ and prove the compatibility of generalized (twisted)
Bernstein and Satake homomorphisms.

###### 2020 Mathematics Subject Classification:

11E95, 20E42, 20G25 and 20C08 (primary).

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
      1. 1.1.1 The Iwahori–Matsumoto presentation
      2. 1.1.2 Freeness of the universal unramified principal series module
      3. 1.1.3 Bernstein presentation of the Iwahori–Hecke algebra
      4. 1.1.4 Twisted Satake isomorphism with $\mathds{Z}$-coefficients
      5. 1.1.5 Centers of parahoric–Hecke algebras
      6. 1.1.6 Compatiblity of twisted Satake and Bernstein morphisms for parahoric levels
    2. 1.2 Acknowledgements
  2. 2 A review of Bruhat–Tits theory
    1. 2.1 Notations
    2. 2.2 The standard apartment
    3. 2.3 A discrete valuation of the generating root datum
    4. 2.4 Affine roots
    5. 2.5 Affine Weyl groups
    6. 2.6 The reduced Bruhat–Tits building
    7. 2.7 The extended Bruhat–Tits building
      1. 2.7.1 The extended apartment
      2. 2.7.2 The extended building
    8. 2.8 Parahoric subgroups
      1. 2.8.1 Further notations
      2. 2.8.2 The Kottwitz homomorphism
      3. 2.8.3 Passage to completion and descent
      4. 2.8.4 $F$-Parahoric subgroups
      5. 2.8.5 The scheme $\mathcal{Z}$
      6. 2.8.6 Decompositions
      7. 2.8.7 The subgroup generated by all parahoric subgroups
    9. 2.9 A double Tits system
    10. 2.10 The subgroup $\mathbf{G}(F)_{1}$
    11. 2.11 More on $\mathbf{G}(F)^{1}$-fixators
  3. 3 Preliminary results around (Iwahori)–Hecke algebras
    1. 3.1 $\flat$ is everywhere !
    2. 3.2 Dominance in $\Lambda_{M}^{\flat}$
    3. 3.3 Iwahori–Weyl group
    4. 3.4 Bruhat and Iwasawa decompositions
    5. 3.5 Iwahori decompositions
    6. 3.6 Conjugations
    7. 3.7 Double cosets
    8. 3.8 Volumes and $q$-powers
    9. 3.9 Double cosets decomposition
    10. 3.10 The modular function
    11. 3.11 Relative Hecke algebras
    12. 3.12 ”Averaging” maps are homomorphisms
    13. 3.13 Iwahori–Matsumoto presentation and other properties
    14. 3.14 Root data
    15. 3.15 Properties of dominance $\Lambda_{M}^{\flat}$
    16. 3.16 The Weyl dot-action
  4. 4 Twisted integral Bernstein morphism
    1. 4.1 Freeness of the $\mathcal{R}^{\flat}$-module $\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$
    2. 4.2 Properties of the $\mathcal{H}_{I^{\flat}}(\mathds{Z})$-module $\mathcal{M}_{I^{\flat}}(\mathds{Z})$
    3. 4.3 Decomposition of $\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$ and the twisted Bernstein map
    4. 4.4 Further properties of $\dot{\Theta}^{\flat}$-elements
    5. 4.5 $\dot{W}$-invariant elements are central
  5. 5 Twisted integral Satake morphisms
    1. 5.1 Twisted Satake homomorphism: first basic properties
      1. 5.1.1 Algebraic argument
      2. 5.1.2 Analytical argument
    2. 5.2 Twisted Satake isomorphism
      1. 5.2.1 Algebraic argument
      2. 5.2.2 Combinatorial argument
    3. 5.3 Positivity properties of the Satake isomorphism
  6. 6 A compressible algebra and centers of parahoric-Hecke algebras
    1. 6.1 Normalized Intertwiners
    2. 6.2 A skew group ring
    3. 6.3 An Azumaya algebra
    4. 6.4 The center of parahoric–Hecke algebras
    5. 6.5 Compatibility of the twisted Satake and twisted Bernstein isomorphisms
  7. .6 $G$-Galois extensions of commutative rings
  8. .7 Properties
  9. .8 An obvious corollary
  10. .9 Compressibility

##  1\. Introduction

Congruences between modular forms, automorphic forms or automorphic
representations have received a significant attention in the past few decades
for several reasons. Their study has helped investigating numerous questions
that have been wandering in number theory’s paysage, e.g. first and foremost,
Andrew Wiles’ proof of Fermat’s last theorem.

An omnipresent theme occurring in this landscape are Hecke algebras. These are
as fundamental as Galois theory is. Actually, in more than one respect, the
two go hand in hand. These special algebras are concerned with the following
kind of situation. We are given a reductive group $\mathbf{G}$ defined over a
$p$-adic field $F$ and a smooth111Every vector $v\in V$ has an open stabilizer
in $\mathbf{G}(F)$. complex representation of $\mathbf{G}(F)$, say $(V,r)$. If
$C\subset\mathbf{G}(F)$ is an open compact subgroup, then we introduce the
Hecke algebra $\mathcal{H}(\mathbf{G}(F)\sslash C,\mathds{C})$, which is the
convolution of double cosets in $C\backslash\mathbf{G}(F)/C$. Similar to how
the group algebra of a finite group allows us to translate representation
theory questions into module-theoretic ones, $\mathcal{H}(\mathbf{G}(F)\sslash
C,\mathds{C})$ provides a mechanism that allows us to dissect the
representations $(V,r)$. In more concrete terms, we have a bijection between
the set of isomorphism classes of irreducible smooth complex
$\mathbf{G}(F)$-representations $V$ such that $V^{C}\neq 0$ and the set of
isomorphism classes of simple smooth $\mathcal{H}(\mathbf{G}(F)\sslash
C,\mathds{C})$-modules.

The purpose of this process is to obtain algebraic or number theoretical
information which may yield new insight about $\mathbf{G}(F)$ or about its
representations theory. Therefore, it is decisive to further deepen our
understanding of Hecke algebras and classify their categories of modules.

An important step toward this goal is offered by Satake’s transform, which
yields an explicit description of $\mathcal{H}(\mathbf{G}(F)\sslash
C,\mathds{C})$ for two crucial cases: maximal open compact subgroups by Satake
[Sat63] and special maximal parahoric subgroups by Haines–Rostami [HR10].
Besides, Hecke algebras for smaller levels also hold a special place and have
intriguing applications, for instance when $C=K_{\mathcal{F}}$ is a parahoric
subgroup, Pappas and Zhu proved in [PZ13] a conjecture of Kottwitz which
asserts that the semi-simple trace of Frobenius on the nearby cycles of a
Shimura variety gives a function which is central in
$\mathcal{H}(\mathbf{G}(F)\sslash K_{\mathcal{F}},\mathds{C})$. Thanks to the
characterization of the minuscule central Bernstein basis functions by Haines
[Hai01], one can actually compute these traces. However, there remains the
task of comprehending the center of these parahoric Hecke algebras. Such an
understanding, is provided by the framework of Bernstein presentations for
Iwahori–Hecke algebra when $C=I$ is an Iwahori subgroup, which is due to
[Lus89] for the extended affine case and more recently to Rostami [Ros15] and
Vigneras [Vig16] for the general reductive case.

This paper attempts to extend and generalize to general reductive $p$-adic
groups various results known for Hecke algebras for split, unramified or semi-
simple and simply connected $p$-adic groups. In this direction, we are
particularly interested in the study of these Hecke algebras when their ring
of coefficients is as small as possible; $\mathds{Z}$ or $\mathds{Z}[q^{-1}]$.
Our methods are of geometric flavor and involve extensively Bruhat–Tits
theory. They provide a uniform treatment for Hecke algebras with parahoric
levels and their $\flat$-avatars (§3.1). This allows us to take full advantage
of Lusztig’s work [Lus89] and extend his results to the general reductive
case. Following the ideas of Bernstein [BD84], we focus our attention on the
universal unramified principal series module, which leads us naturally to the
Bernstein elements $\dot{\Theta}_{m}$ and accordingly the Bernstein
presentation of the Iwahori–Hecke algebras. The payoff for this, we obtain an
explicit description of the center of general parahoric Hecke algebras with
coefficients in $\mathds{Z}[q^{-1}]$, and also integral twisted Satake
isomorphism thus generalizing [HR10] and [Sat63, §9] together with
compatibilities between integral twisted Satake isomorphism and twisted
Bernstein isomorphism. Along the way, we also prove various auxiliary results
that may be useful in other contexts. Furthermore, we give as much as we can
original proofs for all the previously known results.

A hidden motivation for this paper, is to prepare the ground field for
introducing and studying the ring of $\mathbb{U}$-operators, which is the
subject of the sequel part II of this paper [Bou21e]. This is the first step
of an ongoing project exploring arithmetic application of these operators in
Euler systems theory [Bou21a, Bou21b] and Eichler–Shimura congruence relations
for general Shimura varieties [Bou21d, Bou21c].

###  1.1. Main results

Let $\mathbf{G}$ be a reductive group defined over a non-archimedean local
field $F$. Let $\mathbf{S}$ be maximal $F$-split subtorus of $\mathbf{G}$ and
$\mathcal{A}$ its corresponding apartment in the reduced Bruhat–Tits building
of $\mathbf{G}$. Set $\mathbf{M}$ and $\mathbf{N}$ for the centralizer and the
normalizer of $\mathbf{S}$ in $\mathbf{G}$, respectively. We have a
homomorphism of groups
$\nu_{N}\colon\mathbf{N}(F)\to\textit{\emph{\text{Aff}}}(\mathcal{A})$. Let
$\mathfrak{a}\subset\mathcal{A}$ be an alcove, and
$a_{\circ}\in\overline{\mathfrak{a}}$ a special point. Set $I\subset K$ for
the Iwahori subgroup and the special maximal parahoric subgroup corresponding
to ${\mathfrak{a}}$ and $a_{\circ}$, respectively.

Let $M_{1}=\ker\kappa_{M}$ and $G_{1}=\ker\kappa_{G}$ be the kernels of the
Kottwitz map for $\mathbf{M}$ and $\mathbf{G}$ (see §2.8.2). Fix any open
compact subgroup $M_{1}\subset M^{\flat}\subset M$. By [Lan96, Proposition
1.2], $M^{\flat}$ is contained in the maximal open compact subgroup $M^{1}$.

For any $X\subset G$, set $X^{\flat}=M^{\flat}X$. Whenever we use this
notation, the structure of the object we are interested in is not altered,
e.g. $K^{\flat}$ or $G_{1}^{\flat}$ are still a groups.

For any facet $\mathcal{F}\subset\mathcal{A}$, set $K_{\mathcal{F}}$ for the
parahoric subgroup attached to it, $N_{1,\mathcal{F}}:=\mathbf{N}(F)\cap
K_{\mathcal{F}}$ and
$W_{\mathcal{F}}^{\flat}=N_{1,\mathcal{F}}^{\flat}/M^{\flat}$. Because
$a_{\circ}$ is special, $W_{a_{\circ}}^{\flat}\simeq
W:=\mathbf{M}(F)/\mathbf{N}(F)$.

Set $\Lambda_{M}^{\flat}:=\mathbf{M}(F)/M^{\flat}$,
$\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}:=(\mathbf{M}(F)\cap
G_{1})^{\flat}/M^{\flat}$,
$\mathcal{R}^{\flat}:=\mathds{Z}[\Lambda_{M}^{\flat}]$ and
$\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}:=\mathds{Z}[\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}]$.

We write $\dot{W}$, for the Weyl group to specify that it acts on
$\mathcal{R}^{\flat}$ by the dot-action, this is the standard action twisted
by a factor coming form the modulus function (see Definition 3.1).

If $A$ is a commutative ring and $H\subset\mathbf{G}(F)$ an open compact
subgroup, define $\mathcal{H}_{H}(A)$ to be the $A$-algebra of functions
$f\colon\mathbf{G}(F)\to A$, that are $H$-invariant on the right and the left.

####  1.1.1. The Iwahori–Matsumoto presentation

We start with a well known result, at least for Iwahori–Hecke attached to
extended affine Weyl groups. See for example [Gar97, §6.3]. For the general
case, it was showed in [Vig16, Theorem 2.1] or [Ros15, Proposition 4.1.1] with
$\mathds{C}$-coefficients. We will reprove it here using a different argument.

######  Theorem 3.1 (The Iwahori–Matsumoto Presentation).

The Iwahori–Hecke ring $\mathcal{H}_{I^{\flat}}(\mathds{Z})$ is the free
$\mathds{Z}$-module with basis $\\{i_{w}={\bf
1}_{I^{\flat}wI^{\flat}},{w\in\widetilde{W}^{\flat}}:=N/M^{\flat}\\}$, endowed
with the unique ring structure satisfying

  * •

The braid relations:
$i_{w}^{\flat}*_{I^{\flat}}i_{w^{\prime}}^{\flat}=i_{ww^{\prime}}^{\flat}\text{
if }w,w^{\prime}\in\widetilde{W}^{\flat}\text{ such that
}\ell(w)+\ell(w^{\prime})=\ell(ww^{\prime})$.

  * •

The quadratic relations:
$(i_{s}^{\flat})^{2}=q_{s}i_{1}^{\flat}+(q_{s}-1)i_{s}^{\flat}\text{ if
}s\in\mathcal{S}^{\flat}$, where $q_{s}:=[IsI:I]$ and
$\mathcal{S}^{\flat}\subset\widetilde{W}^{\flat}$ corresponds to the set of
orthogonal reflections with respect to the walls of the fixed alcove.

####  1.1.2. Freeness of the universal unramified principal series module

Let $\mathcal{F}\subset\mathcal{A}$ be any fixed facet. For any commutative
ring $A$, define the universal unramified principal series right
$(\mathcal{R}^{\flat},\mathcal{H}_{K_{\mathcal{F}}^{\flat}}(A))$-bi-module
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(A)=\mathcal{C}_{c}(M^{\flat}U^{+}\backslash
G/K_{\mathcal{F}},A)$, this is the set of $A$-valued functions supported on
finitely many double cosets.

######  Proposition 4.4.

The $\mathcal{R}^{\flat}$-module
$\mathcal{M}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z})$ is free of rank
$r_{\mathcal{F}}:=|W/\jmath_{W}(W_{\mathcal{F}}^{\flat})|$, with basis
$\\{v_{w,\mathcal{F}}^{\flat}:={\bf 1}_{U^{+}wK_{\mathcal{F}}^{\flat}},\,w\in
D_{\mathcal{F}}^{\flat}\\}$ where $D_{\mathcal{F}}^{\flat}\subset
W_{a_{\circ}}$ is any fixed set of representatives for
$W/\jmath_{W}(W_{\mathcal{F}}^{\flat})$ and $\jmath_{W}\colon N\to W=N/M$ is
the quotient map. In particular, $\\{v_{w,\mathfrak{a}}^{\flat},\,w\in
W_{a_{\circ}}^{\flat}\\}$ is a canonical basis for the
$\mathcal{R}^{\flat}$-module $\mathcal{M}_{I^{\flat}}(\mathds{Z})$.

####  1.1.3. Bernstein presentation of the Iwahori–Hecke algebra

Following the ideas of Bernstein [BD84], as exposed by the approach of [HKP10]
(which treats the split case), we use the universal unramified principal
series module $\mathcal{M}_{I^{\flat}}(\mathds{Z})$ to study the Iwahori–Hecke
algebra. A first result into this direction is:

######  Theorem 4.3.

The following homomorphism of right
$\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}])$-modules is an isomorphism

$\displaystyle\hbar_{I^{\flat}}\colon\mathcal{H}_{I^{\flat}}({\mathds{Z}[q^{-1}]})\longrightarrow\mathcal{M}_{I^{\flat}}({\mathds{Z}[q^{-1}]}),\quad
h\longmapsto v_{1,\mathfrak{a}}^{\flat}*_{I^{\flat}}h.$

In §4.3, we construct an embedding
$\dot{\Theta}_{Bern}^{\flat}\colon\mathcal{R}^{\flat}\hookrightarrow\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}]),m\mapsto\dot{\Theta}_{m}^{\flat}$.
This provides an alternative description for the Iwahori–Hecke algebra:

######  Corollary 4.6.

The algebra $\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$ is a free left
(and right)
$\mathcal{R}^{\flat}\otimes_{\mathds{Z}}{\mathds{Z}[q^{-1}]}$-module, with
canonical basis $\\{i_{w}^{\flat}\colon w\in W_{a_{\circ}}^{\flat}\\}$. The
sets $\\{\dot{\Theta}_{m}^{\flat}*_{{I^{\flat}}}i_{w}^{\flat}\colon
m\in\Lambda_{M}^{\flat},w\in W_{a_{\circ}}^{\flat}\\}$ and
$\\{i_{w}^{\flat}*_{{I^{\flat}}}\dot{\Theta}_{m}^{\flat}\colon
m\in\Lambda_{M}^{\flat},w\in W_{a_{\circ}}^{\flat}\\}$ are both
${\mathds{Z}[q^{-1}]}$-basis for
$\mathcal{H}_{{I^{\flat}}}({\mathds{Z}[q^{-1}]})$.

Set
$\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}(\mathds{Z}):=\mathcal{H}(G_{1}^{\flat}\sslash
I^{\flat},\mathds{Z})$ and let $\widetilde{\Upomega}^{\flat}$ denote the
subgroup of stabilizers of $\mathfrak{a}$ in $\widetilde{W}^{\flat}$. The
following diagram is commutative (see Proposition 3.4)

${\mathcal{H}_{I}(\mathds{Z})}$${\mathds{Z}[\widetilde{\Upomega}]\otimes^{\prime}\mathcal{H}_{I}^{\textit{\emph{\text{aff}}}}(\mathds{Z})}$${\mathcal{H}_{I^{\flat}}(\mathds{Z})}$${\mathds{Z}[\widetilde{\Upomega}^{\flat}]\otimes^{\prime}\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}(\mathds{Z})}$$\scriptstyle{\simeq}$$\scriptstyle{\iota^{\flat}}$$\scriptstyle{\square^{\flat}\otimes\iota^{\flat}}$$\scriptstyle{\simeq}$

where, $\otimes^{\prime}$ indicates that the tensor modules is endowed with a
”untwisted” multiplication: $(\sigma\otimes i_{w})(\sigma^{\prime}\otimes
i_{w^{\prime}})=\sigma\sigma^{\prime}\otimes
i_{\sigma^{\prime-1}w\sigma^{\prime}}*_{I^{\flat}}i_{w^{\prime}}$ for any
$\sigma,\sigma^{\prime}\in\widetilde{\Upomega}^{\flat}$ and any
$w,w^{\prime}\in\widetilde{W}^{\flat}$. The restriction of the surjective
$\iota^{\flat}$ to $\mathcal{H}_{I}^{\textit{\emph{\text{aff}}}}(\mathds{Z})$
is injective (Remark 3.4) and yields an isomorphism of algebras
$\mathcal{H}_{I}^{\textit{\emph{\text{aff}}}}(\mathds{Z})\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}\mathcal{H}_{I^{\flat}}^{\textit{\emph{\text{aff}}}}(\mathds{Z})$.

This fact, allows us to extend various results due to Lusztig in [Lus89] for
the affine extended case ($M^{\flat}=M^{1}$) to the general reductive case:

######  Proposition 4.3.

For any $s\in\mathcal{T}_{a_{\circ}}^{\flat}=\mathcal{S}^{\flat}\cap
W_{a_{\circ}}^{\flat}$ and any $m\in\Lambda_{M}^{\flat}$, we have

$\dot{\Theta}_{m}^{\flat}*_{I^{\flat}}(i_{s}^{\flat}+1)-(i_{s}^{\flat}+1)*_{I^{\flat}}\dot{\Theta}_{\dot{s}(m)}^{\flat}=(\dot{\Theta}_{m}^{\flat}-\dot{\Theta}_{\dot{s}(m)}^{\flat})\mathcal{G}(\alpha)\in\mathcal{R}^{\flat}$

where,
$\mathcal{G}(\alpha)\in\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}$ is
the element defined in [Lus89, §3.8 & §3.13] for the algebra
$\mathcal{H}(\mathcal{D}_{\textit{\emph{\text{aff}}}},{W}_{\textit{\emph{\text{aff}}}},v=q,L)$
(see 3.14).

This result was also obtained by Rostami [Ros15, Proposition 5.4.2] for the
untwisted $\Theta$-elements, using a slightly different argument.

####  1.1.4. Twisted Satake isomorphism with $\mathds{Z}$-coefficients

Motivated by arithmetic problems for Shimura varieties, Haines and Rostami
established in [HR10],
$\mathcal{S}_{M}^{G}\colon\mathcal{H}_{K}(\mathds{C})\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}\mathcal{R}^{W}\otimes_{\mathds{Z}}\mathds{C}$
for the untwisted Satake morphism. In §5, considering the twisted Satake
isomorphism, allows us to extend their result to the Hecke algebra with
integral coefficients $\mathcal{H}_{K}(\mathds{Z})$.

The twisted Satake transform
$\dot{\mathcal{S}}_{M}^{G^{\flat}}\colon\mathcal{H}_{K^{\flat}}\to\mathcal{R}^{\flat}$
is characterized by
$v_{1,a_{\circ}}^{\flat}*_{K^{\flat}}h=\dot{\mathcal{S}}_{M}^{G^{\flat}}(h)\cdot
v_{1,a_{\circ}}^{\flat},\text{ for all
}{h\in\mathcal{H}_{K^{\flat}}(\mathds{Z})}$.

######  Theorem 5.1.

The twisted Satake homomorphism
$\dot{\mathcal{S}}_{M}^{G^{\flat}}\colon\mathcal{H}_{K^{\flat}}(\mathds{Z})\to{\mathcal{R}^{\flat}}^{\dot{W}}$
is a canonical isomorphism of ${\mathds{Z}}$-algebras.

We propose two proofs for this result; the first is ”algebraic” while the
second is of ”combinatorial” taste. Upon completion of the first version of
this paper, the author learned that Henniart and Vignéras already proved
(differently) this in [HV15].

We also show the natural generalization of Rapoport’s positivity result for
unramified groups [Rap00, Theorem 1.1], see also Haines’ [Hai00]. The key tool
is due to C. Schwer [Sch06, Corollary 4.8], in which he gives the desired
positivity statement for the extended affine case.

######  Theorem 5.1 (Positivity of the twisted Satake isomorphism).

For any $x\in\Lambda_{M}^{-}$, write
$\dot{\mathcal{S}}_{M}^{G}(h_{x})=\sum_{m\in\Lambda_{M}^{-}}s_{x,m}r_{m}$. The
following statements are equivalent

  1. (1)

$s_{x,m}>0$,

  2. (2)

$U^{+}m\cap KxK\neq\emptyset$,

  3. (3)

$U^{+}m\cap Iwxw^{\prime}I\neq\emptyset$, for some $w,w^{\prime}\in
W_{a_{\circ}}$,

  4. (4)

${x}\succsim{m}$, i.e. $xm^{-1}\in\mathbf{M}(F)\cap G_{1}$ and
$\nu_{N}(x{m}^{-1})=\sum n_{\alpha}\alpha^{\vee},n_{\alpha}\in\mathds{N}$, the
sum being over the simple relative coroots.

####  1.1.5. Centers of parahoric–Hecke algebras

Using normalized intertwiners in §6.1, we prove the following result, which is
the key ingredient to describe centers of parahoric–Hecke algebras.

######  Theorem 6.1& 6.1& 6.2.

Let $\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}$ be the fraction field
of $\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}$ and
$\mathcal{L}^{\flat}:=\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{R}^{\flat}$.

  1. (1)

We have an isomorphism of $\mathcal{L}^{\flat}$-algebras:
$\mathcal{L}^{\flat}*\dot{W}\to{\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}\otimes_{{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}^{\dot{W}}}\mathcal{H}_{{I}^{\flat}}({\mathds{Z}[q^{-1}]})$,
where $\mathcal{L}^{\flat}*\dot{W}$ is the skew group ring of $\dot{W}$ over
$\mathcal{L}^{\flat}$ (See Definition .1)

  2. (2)

The ring extension $\mathcal{L}^{\flat}/{\mathcal{L}^{\flat}}^{\dot{W}}$ is
${\dot{W}}$-Galois.

  3. (3)

The $\mathcal{L}^{\dot{W}}$-algebra
$A=\mathcal{L}_{\textit{\emph{\text{aff}}}}^{\flat}\otimes_{\mathcal{R}_{\textit{\emph{\text{aff}}}}^{\flat}}\mathcal{H}_{{I}^{\flat}}({\mathds{Z}[q^{-1}]})$
is compressible, that is $Z(eAe)=eZ(A)=e\mathcal{L}^{\dot{W}}$ for any
idempotent $e$ of $A$.

With this result in hand, we generalize Haines’s [Hai09b, Theorem 3.1.1] which
treats the case of unramified $\mathbf{G}$:

######  Theorem 6.1.

For any facet $\mathcal{F}\subset\overline{\mathfrak{a}}$, we have

$Z\left(\mathcal{H}_{K_{\mathcal{F}}^{\flat}}({\mathds{Z}[q^{-1}]})\right)=\dot{\Theta}_{\text{Bern}}^{\flat}({\mathcal{R}^{\flat}}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}])*_{I^{\flat}}{\bf
1}_{K_{\mathcal{F}}^{\flat}}.$

In particular, the set $\\{r_{m}^{\flat}*_{I^{\flat}}{\bf
1}_{K_{\mathcal{F}}^{\flat}}:m\in\Lambda_{M}^{-,\flat}\\}$ forms a basis for
the $\mathds{Z}[q^{-1}]$-module
$Z\left(\mathcal{H}_{K_{\mathcal{F}}^{\flat}}({\mathds{Z}[q^{-1}]})\right)$.

I would like to point out that once the first version of this article was
completed, the author learned of Haines’s paper [Hai14]. Within the Appendix
of loc. cit., some general facts about the center of parahoric Hecke algebras
(i.e. $M^{\flat}=M_{1}$) with $\mathds{C}$-coefficients, compatibility with
constant terms and change of parahorics are treated based on the theory of
Bushnell–Kutzko types. In loc. cit., Haines focuses on Iwahori levels
($\mathcal{F}=\mathfrak{a}$) and affirms the passage to general parahoric
levels should rely on other properties of intertwiners for principal series
representations as in [HKP10]. However, the only written reference for these
properties that the author is aware of treat only the split case [HKP10] or
the unramified case [Hai07]. The more elementary approach of the present paper
treats all parahoric groups and their $\flat$-avatars directly and works with
integral coefficients.

####  1.1.6. Compatiblity of twisted Satake and Bernstein morphisms for
parahoric levels

For any facet $\mathcal{F}\subset\overline{\mathfrak{a}}$, define the
generalized twisted Satake transform
$\dot{\mathcal{S}}_{\mathcal{F}}^{\flat}\colon
Z(\mathcal{H}_{K_{\mathcal{F}}^{\flat}}({\mathds{Z}[q^{-1}]}))\to{\mathcal{R}^{\flat}}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}]$
to be the morphism of $\mathds{Z}[q^{-1}]$-algebras that is characterized by
$\dot{\mathcal{S}}_{\mathcal{F}}^{\flat}(h)*_{I^{\flat}}{\bf
1}_{K_{\mathcal{F}}^{\flat}}=h$. When $\mathcal{F}=\\{a_{\circ}\\}$, then
$\dot{\mathcal{S}}_{\mathcal{F}}^{\flat}=\dot{\mathcal{S}}_{M}^{G^{\flat}}$.

######  Theorem 6.1.

Let
$\mathcal{F}\subset\overline{\mathcal{F}}^{\prime}\subset\overline{\mathfrak{a}}$,
be two facets, i.e. $K_{\mathcal{F}}\supset K_{\mathcal{F}^{\prime}}\supset
I$. The following diagram of $\mathds{Z}[q^{-1}]$-algebras is commutative:

${Z(\mathcal{H}_{K_{\mathcal{F}}^{\flat}}(\mathds{Z}[q^{-1}]))}$${{\mathcal{R}^{\flat}}^{\dot{W}}\otimes_{\mathds{Z}}\mathds{Z}[q^{-1}]}$${Z(\mathcal{H}_{K_{\mathcal{F}^{\prime}}^{\flat}}(\mathds{Z}[q^{-1}]))}$${Z(\mathcal{H}_{I^{\flat}}(\mathds{Z}[q^{-1}]))}$$\scriptstyle{\dot{\mathcal{S}}_{\mathcal{F}}^{\flat}}$$\scriptstyle{\simeq}$$\scriptstyle{\dot{\Theta}_{Bern}^{\flat}}$$\scriptstyle{\simeq}$$\scriptstyle{-*_{K_{\mathcal{F}^{\prime}}^{\flat}}{\bf
1}_{K_{\mathcal{F}}}^{\flat}}$$\scriptstyle{\simeq}$$\scriptstyle{-*_{I^{\flat}}{\bf
1}_{K_{\mathcal{F}^{\prime}}^{\flat}}}$$\scriptstyle{\simeq}$

This commutative diagram is also compatible with change of $\flat$-avatars
$M^{\flat}\subset{M^{{\flat}^{\prime}}}$ (See remark 5.6). To our knowledge
the first proof of the compatibility between the Satake and Bernstein
isomorphism, i.e. commutativity of the diagram above is due to J.-F. Dat
[Dat99, §3] for split reductive groups and for a fixed Iwahori level, i.e.
$\mathcal{F}=\mathcal{F}^{\prime}=\\{a_{\circ}\\}$.

###  1.2. Acknowledgements

Some parts of this article originated (partially) from my doctoral thesis,
directed by Dimitar Jetchev, to whom I am very grateful. I am thankful to
Christophe Cornut for his support and meticulous reading, Anne-Marie Aubert
and Marie-France Vigneras for many helpful comments and suggestions and Thomas
Haines, for constructive advice and pointing me to his paper [Hai14].

##  2\. A review of Bruhat–Tits theory

Bruhat and Tits made in their seminal work [BT72, BT84] a profound exploration
of reductive groups over local fields by constructing for them a combinatorial
avatar; for a reductive group $\mathbf{G}$ over a local field $F$, they
associate to $\mathbf{G}^{ad}(F)$ an affine building
$\mathcal{B}(\mathbf{G},F)_{red}$ and to $\mathbf{G}(F)$ a building
$\mathcal{B}(\mathbf{G},F)_{ext}$. The first one is called the reduced
Bruhat–Tits building, the latter one is called the extended Bruhat–Tits
building. Here is a gentle mise-en-bouche: the building
$\mathcal{B}(\mathbf{G},F)_{red}$ is a complete metric space, that has a
structure of a poly-simplicial complex. This building is obtained by ”gluing”
a family of distinguished subsets, called apartments, these are affine spaces
for some real affine space. This inner poly-simplicial structure comes with an
isometric action of the group $\mathbf{G}(F)$.

The purpose of this section is to extract a reasonably brief exposition of
Bruhat–Tits theory and expose the properties that will be needed in the proofs
of our results. Moreover, most of the notations introduced for this goal will
be used later.

###  2.1. Notations

  * •

$F$ a finite extension of $\mathds{Q}_{p}$ for some prime $p$,
$\mathcal{O}_{F}$ its ring of integers, $\varpi$ a fixed uniformizer in
$\mathcal{O}_{F}$, $\kappa(F)$ its residue field and $q$ its cardinality,

  * •

$\omega\colon F^{\times}\to\mathds{Z}$ the normalized discrete valuation, i.e.
$\omega(\varpi)=1$ and $|\cdot|_{F}=q^{-\omega(\cdot)}$ for the corresponding
normalized absolute value,

  * •

$F^{sep}$ a fixed separable closure,

  * •

For any connected reductive $F$-group $\mathbf{H}$, there exists a
homomorphism of groups
$\nu_{H}\colon\mathbf{H}(F)\to\text{Hom}_{\mathds{Z}}(X^{*}(\mathbf{H})_{F},\mathds{Z})$
characterized by

$\langle\nu_{H}(h),\chi\rangle=\nu_{H}(h)(\chi)=-\omega(\chi(h)),\text{ for
all }h\in\mathbf{H}(F)\text{ and }\chi\in X^{*}(\mathbf{H})_{F}.$

  * •

Fix $\mathbf{G}$ a connected reductive group over $F$, we use the convention
that all reductive groups are connected,

  * •

Let $\mathbf{Z}_{c}$ denote the maximal central F-torus of $\mathbf{G}$ and
$\mathbf{Z}_{c,sp}$ its maximal $F$-split $F$-subtorus,

  * •

Let $\mathbf{S}$ be a maximal $F$-split subtorus of $\mathbf{G}$,

  * •

For any algebraic subgroup $\mathbf{H}\subset\mathbf{G}$, we will denote by
$N_{\mathbf{G}}(\mathbf{H})$ and $Z_{\mathbf{G}}(\mathbf{H})$ the normalizer
and the centralizer of $\mathbf{H}$ in $\mathbf{G}$ respectively,

  * •

The root system of $\mathbf{G}$ with respect to $\mathbf{S}$ will be denoted
$\Phi:=\Phi(\mathbf{G},\mathbf{S})$,
$\Phi_{red}=\\{\alpha\in\Phi\colon\frac{1}{2}\alpha\notin\Phi\\}$, $\Phi^{+}$
a system of positive roots in $\Phi$, $\Delta$ the associated base of simple
roots, $\Delta^{\vee}$ its dual and the Weyl group
$W=W(\mathbf{G},\mathbf{S})(F)=N_{\mathbf{G}}(\mathbf{S})(F)/Z_{\mathbf{G}}(\mathbf{S})(F)$,

  * •

Let $\Psi$ be a closed222A subset $\Psi\subset\Phi_{+}\cap\Phi_{red}$ such
that for any $\alpha,\beta\in\Psi$ one has
$[\alpha,\beta]:=\\{n\alpha+m\beta\colon\text{ for all
}n,m\in\mathds{Z}_{>0}\\}\cap\Phi\subset\Psi$. subset of
$\Phi_{red}^{+}:=\Phi^{+}\cap\Phi_{red}$, e.g. $\Psi=\\{\alpha,2\alpha\\}$ if
$2\alpha\in\Phi$. There exists a unique closed, connected, unipotent
$F$-subgroup $\mathbf{U}_{\Psi}\subset\mathbf{G}$, that is normalized by
$Z_{\mathbf{G}}(\mathbf{S})$ and such that the product morphism

$\prod_{\alpha\in\Psi\cap\Phi_{red}}\mathbf{U}_{\alpha}\to\mathbf{U}_{\Psi},$
($\ast$)

is an isomorphism of $F$-varieties, here the product is taken in any order for
$\Psi$

  * •

Any subset $J\subset\Delta$ is a base for a root system
$\Phi_{J}:=\mathds{Z}J\cap\Phi$. The subgroup $\langle
Z_{\mathbf{G}}(\mathbf{S}),\mathbf{U}_{\Phi_{J}\cup\Phi^{+}}\rangle$ of
$\mathbf{G}$ is a standard parabolic subgroup, we denote it by
$\mathbf{P}_{J}$. We have
$R_{u}(\mathbf{P}_{J})=\mathbf{U}_{\Phi^{+}\setminus\Phi_{J}^{+}}$, where
$\Phi_{J}^{\pm}:=\mathds{Z}J\cap\Phi^{\pm}$.

  * •

The Levi factor $\mathbf{M}_{J}$ of $\mathbf{P}_{J}$ is the subgroup generated
by $\langle Z_{\mathbf{G}}(\mathbf{S}),\mathbf{U}_{\Phi_{J}}\rangle$.
Actually, this Levi subgroup is the centralizer of the split torus
$\mathbf{S}_{J}:=(\cap_{\alpha\in\Phi_{J}}\ker\alpha)^{\circ}$. When
$J=\emptyset$ we get a minimal parabolic subgroup $\mathbf{P}_{\emptyset}$
with Levi factor $Z_{\mathbf{G}}(\mathbf{S})$. If $J=\Delta$, then
$\mathbf{P}_{\Delta}=\mathbf{M}_{\Delta}=\mathbf{G}$ and $\mathbf{S}_{\Delta}$
is the unique maximal split torus of the center $Z_{\mathbf{G}}$. We refer the
reader to [Bor91, §21] for more details.

  * •

Set $\mathbf{M}:=\mathbf{M}_{\emptyset}=Z_{\mathbf{G}}(\mathbf{S})$ for the
centralizer (a minimal Levi $F$-subgroup of $\mathbf{G}$),
$\mathbf{N}:=N_{\mathbf{G}}(\mathbf{S})$ for the normalizer and
$\mathbf{B}:=\mathbf{P}_{\emptyset}$ for the minimal parabolic with Levi
factor $\mathbf{M}$ and unipotent radical
$\mathbf{U}^{+}=\mathbf{U}_{\Phi^{+}}$, we have
$\mathbf{B}=\mathbf{U}_{\Phi^{+}}\rtimes\mathbf{M}$.

  * •

We will sometimes use the notation $\square\otimes k=\square_{k}$ for a field
$k$.

###  2.2. The standard apartment

We have a commutative diagram

${X_{*}(\mathbf{M})_{F}}$${\times}$${X^{*}(\mathbf{M})_{F}}$${\mathds{Z}}$${X_{*}(\mathbf{S})}$${\times}$${X^{*}(\mathbf{S})}$${\mathds{Z}}$

where only the bottom line is a perfect pairing. In the following lemma we
will show that, the injective finite-cokernel group homomorphism
$X^{*}(\mathbf{M})_{F}\hookrightarrow X^{*}(\mathbf{S})$ induces a unique
homomorphism $\mathbf{M}(F)\to
X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q}$ that extend
$\nu_{M}\colon\mathbf{M}(F)\to\text{Hom}_{\mathds{Z}}(X^{*}(\mathbf{M})_{F},\mathds{Z})$.

######  Lemma 2.1.

There exists a unique homomorphism $\nu_{M}\colon\mathbf{M}(F)\to
X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q}$ such that

$\langle\nu_{M}(z),\chi|_{\mathbf{S}}\rangle=-\omega(\chi(z)),\text{ for all
}z\in\mathbf{M}(F)\text{ and }\chi\in X^{*}(\mathbf{M})_{F}(\hookrightarrow
X^{*}(\mathbf{S})).$

###### Proof.

Because the restriction map $X^{*}(\mathbf{M})_{F}\to X^{*}(\mathbf{S})$ is
injective and has an image of finite index, i.e. for any $\chi\in
X^{*}(\mathbf{S})$, the character
$[X^{*}(\mathbf{S}):X^{*}(\mathbf{M})_{F}]\chi\in X^{*}(\mathbf{S})$ extends
to a unique $\widetilde{\chi}\in X^{*}(\mathbf{M})_{F}$. Consider the
following map

$M(F)\to\text{Hom}(X^{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q},\mathds{Q}),\quad
z\mapsto\left(\chi\otimes
r\mapsto-r\frac{1}{[X^{*}(\mathbf{S}):X^{*}(\mathbf{M})_{F}]}\omega(\widetilde{\chi}(z))\right).$

This is clearly a homomorphism of groups. Moreover,
$X^{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q}$ and
$X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q}$ remain a dual pair of
$\mathds{Q}$-modules for the canonical pairing
$\langle\,,\rangle_{\mathds{Q}}$ that extend the pairing
$\langle\,,\rangle\colon X_{*}(\mathbf{S})\times
X^{*}(\mathbf{S})\to\mathds{Z}$. Thus, any homomorphism in
$\text{Hom}(X^{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q},\mathds{Q})$ is of
the form $\chi\otimes r\mapsto\langle\chi,\lambda\rangle rr^{\prime}$ for some
$\lambda\otimes r^{\prime}\in
X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q}$. This defines a unique
homomorphism of groups $\nu_{M}\colon\mathbf{M}(F)\to
X^{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q}$ verifying

$\langle\nu_{M}(z),\chi\otimes
1\rangle_{\mathds{Q}}=-\frac{1}{[X^{*}(\mathbf{S}):X^{*}(\mathbf{M})_{F}]}\omega(\widetilde{\chi}(z))\quad\text{for
all }z\in\mathbf{M}(F)\text{ and }\chi\in X^{*}(\mathbf{S}),$

or equivalently (since $X^{*}(\mathbf{M})_{F}$ generates
$X^{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q}$)

$\displaystyle\langle\nu_{M}(z),\chi_{|_{\mathbf{S}}}\otimes
1\rangle_{\mathds{Q}}$ $\displaystyle=-\omega({\chi}(z))\quad\text{for all
}z\in\mathbf{M}(F)\text{ and }\chi\in X^{*}(\mathbf{M})_{F}.\qed$

By [Lan96, Proposition 1.2], the subgroup $\mathbf{M}(F)^{1}:=\ker\nu_{M}$ is
the maximal compact open subgroup of $\mathbf{M}(F)$. We then have a short
exact sequence of groups

$0\to\mathbf{M}(F)/\mathbf{M}(F)^{1}\to\mathbf{N}(F)/\mathbf{M}(F)^{1}\to
W=\mathbf{N}(F)/\mathbf{M}(F)\to 0,$

in which $\mathbf{M}(F)/\mathbf{M}(F)^{1}$ is a free abelian group containing
$X_{*}(\mathbf{S})$ and having same rank [Lan96, Lemma 1.3].

Consider the $\mathds{R}$-vector space
$V=X_{*}(\mathbf{S}/\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{R}$ and
identify its dual space $V^{*}$ with
$X^{*}(\mathbf{S}/\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{R}$ using the
canonical extended pairing $\langle\,,\rangle_{\mathds{R}}$. We denote by

$\nu\colon\mathbf{M}(F)\to V,$

the composition of the map $\nu_{M}\colon\mathbf{M}(F)\rightarrow
X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}$ with the natural projection
$X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}\twoheadrightarrow V$.

The action of $W$ on $\mathbf{S}$ by conjugation induces a faithful linear
action on $X_{*}(\mathbf{S})$, which induces a canonical homomorphism of
groups

$j\colon W\to GL(X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}),$

and accordingly a homomorphism $W\to GL(V)$ since $\mathbf{N}(F)$ acts
trivially on $\mathbf{Z}_{c,sp}$.

######  Proposition 2.2.

There is a canonical affine space $\mathds{A}_{red}(\mathbf{G},\mathbf{S})$
under $V$ (unique up to a unique isomorphism of affine spaces) together with a
group homomorphism
$\nu_{N}\colon\mathbf{N}(F)\to\textit{\emph{\text{Aff}}}(\mathds{A}_{red}(\mathbf{G},\mathbf{S}))$
extending $\nu$. Accordingly, we have a commutative diagram

${0}$${\mathbf{M}(F)/\mathbf{M}(F)^{1}}$${\mathbf{N}(F)/\mathbf{M}(F)^{1}}$${W}$${0}$${0}$${V}$${\textit{\emph{\text{Aff}}}(\mathds{A}_{red}(\mathbf{G},\mathbf{S}))}$${GL(V)}$${0}$$\scriptstyle{\nu}$$\scriptstyle{\nu_{N}}$$\scriptstyle{j}$

For all $z\in\mathbf{M}(F)$, $\nu_{N}(z)$ is the translation
$a\mapsto\nu(z)+a$ and for any $n\in\mathbf{N}(F)$ the linear part of
$\nu_{N}(n)$ is equal to $j(\jmath_{W}(n))$, where $\jmath_{W}(n)$ denotes the
image of $n$ in $W$.

###### Proof.

See the proof of [Lan96, Proposition 1.8]. ∎

######  Remark 2.3.

(i) The kernel $\ker\nu_{N}$ contains $\mathbf{M}(F)^{1}$ and the center of
$\mathbf{G}(F)$. (ii) The reductivity of $\mathbf{G}$ forces the action of the
Weyl group $W$ on $V_{0}=X_{*}(Z_{\mathbf{G}})\otimes_{\mathds{Z}}\mathds{R}$
to be trivial, which implies the injectivity of the morphism $W\hookrightarrow
GL(V)$ and accordingly $\ker\nu_{N}=\ker\nu$. (iii) All possible extensions of
$\nu$ are of the form

$\nu_{N,v}(n)\colon a\mapsto-v+\nu_{N}(n)(v+a),\,\forall
a\in\mathds{A}_{red}(\mathbf{G},\mathbf{S}),$

for some fixed $v\in V$.

We are now ready to define the central combinatorial object of this section

######  Definition 2.4.

The affine space $\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ together with the
group homomorphism
$\nu_{N}\colon\mathbf{N}(F)\to\textit{\emph{\text{Aff}}}(\mathds{A}_{red}(\mathbf{G},\mathbf{S}))$
is called the standard apartment of $\mathbf{G}$ with respect to $\mathbf{S}$.

###  2.3. A discrete valuation of the generating root datum

Define for each $\alpha\in\Phi$ the set $M_{\alpha}$ of elements
$n\in\mathbf{N}(F)$ whose image in the Weyl group $\jmath_{W}(n)\in W$ is the
reflection $s_{\alpha}\colon V\to V,v\mapsto v-\langle
v,\alpha^{\vee}\rangle\alpha$, where (by abuse of notation) $\alpha^{\vee}$
denotes the image of the co-root in $V$.

A good part of the theory developed by Borel and Tits in [BT65] (see [BT72,
§6.1]) may be condensed in the following fundamental theorem:

######  Theorem 2.1.

The datum
$\left(\mathbf{M}(F),\left(\mathbf{U}_{\alpha}(F),M_{\alpha}\right)_{\alpha\in\Phi}\right)$
is a generating root datum of type $\Phi$ in $\mathbf{G}(F)$.

The nomenclature ”generating root datum” means that the above datum verifies
the list of axioms [BT72, §6.1.1].

######  Corollary 2.2.

For any $\alpha\in\Phi$ and any $u\in\mathbf{U}_{\alpha}(F)\setminus\\{1\\}$
there exists a unique pair

$(u^{\prime},u^{\prime\prime})\in{\mathbf{U}_{-\alpha}(F)\setminus\\{1\\}\times\mathbf{U}_{-\alpha}(F)\setminus\\{1\\}}$

such that $m_{\alpha}(u):=u^{\prime}uu^{\prime\prime}\in\mathbf{N}(F)$ and
verifies
$m_{\alpha}(u)\mathbf{U}_{\alpha}(F)m_{\alpha}(u)^{-1}=\mathbf{U}_{-\alpha}(F)$.

In particular,
$\mathbf{U}_{-\alpha}(F)u\mathbf{U}_{-\alpha}(F)\cap\mathbf{N}(F)=\\{m_{\alpha}(u)\\}$.
Moreover, $m_{\alpha}(u)$ normalizes $\mathbf{S}$ and its image in $W$ acts on
$V$ by the reflexion $s_{\alpha}$. Accordingly, the set $M_{\alpha}$ defined
above Theorem 2.1 is precisely the right coset $m_{\alpha}(u)\mathbf{M}(F)$.

###### Proof.

See [BT72, §6.1.2]. ∎

Fix any point (origin) $a_{\circ}$ in
$\mathds{A}_{red}(\mathbf{G},\mathbf{S})$. For all $\alpha\in\Phi$ and all
$u\in\mathbf{U}_{\alpha}(F)\setminus\\{1\\}$ we have

$\nu_{N}(m_{\alpha}(u))(a)=s_{\alpha}(a-a_{\circ})+\nu_{N}(m_{\alpha}(u))(a_{\circ})=-\alpha(a-a_{\circ})\alpha^{\vee}+a-a_{\circ}+\nu_{N}(m_{\alpha}(u))(a_{\circ})$

The set of fixed points of $\nu_{N}(m_{\alpha}(u))$ is an affine hyperplane in
$\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ with direction
$\ker(s_{\alpha}-\text{Id}_{V})$. Let $b_{a_{\circ},\alpha,u}$ be any fixed
element of this hyperplane, one has

$\nu_{N}(m_{\alpha}(u))(a_{\circ})=\alpha(b_{a_{\circ},\alpha,u}-a_{\circ})\alpha^{\vee}+a_{\circ}.$

Therefore,

$\nu_{N}(m_{\alpha}(u))(a)=s_{\alpha}(a-a_{\circ})+\alpha(b_{a_{\circ},\alpha,u}-a_{\circ})\alpha^{\vee}+a_{\circ}.$

Now, by setting
$\varphi_{\alpha}^{a_{\circ}}(u):=-\alpha(b_{a_{\circ},\alpha,u}-a_{\circ})\in\mathds{R}$
one can rewrite $\nu_{N}(m_{\alpha}(u))$ as follows:

$\nu_{N}(m_{\alpha}(u))(a)=-\left(\alpha(a-a_{\circ})+\varphi_{\alpha}^{a_{\circ}}(u)\right)\alpha^{\vee}+a.$

Accordingly, the element $\nu_{N}(m_{\alpha}(u))$ acts as a reflection with
respect to the hyperplane
$\\{a\in\mathds{A}_{red}(\mathbf{G},\mathbf{S})\colon\alpha(a-a_{\circ})=-\varphi_{\alpha}^{a_{\circ}}(u)\\}$
(See [BT72, Remark (b) 6.2.12]).

One of the fundamental results obtained by Bruhat and Tits is

######  Theorem 2.3.

The family
$\varphi^{a_{\circ}}=(\varphi_{\alpha}^{a_{\circ}}\colon\mathbf{U}_{\alpha}(F)\to\mathds{R}\cup\\{\infty\\})_{\alpha\in\Phi}$
is a discrete valuation of the generating root datum
$\left(\mathbf{M}(F),\left(\mathbf{U}_{\alpha}(F),M_{\alpha}\right)_{\alpha\in\Phi}\right)$,
meaning that $\varphi^{a_{\circ}}$ verifies the properties listed in [BT72,
§6.2.1 & §6.2.21].

###### Proof.

For a proof we refer the reader to [BT84, §5.1.20 & §5.1.23] and [BT72,
Remarque 6.2.12 (b)].∎

Consequently, for all $\alpha\in\Phi$ and $r\in\mathds{R}\cup\\{\infty\\}$,
the set
$U_{\alpha+r}^{a_{\circ}}:=(\varphi_{\alpha}^{a_{\circ}})^{-1}([r,\infty])$ is
a compact open subgroup of $\mathbf{U}_{\alpha}(F)$ and
$U_{\alpha+\infty}^{a_{\circ}}=\\{1\\}$. Moreover, if $2\alpha\in\Phi$ then
$\varphi_{2\alpha}^{a_{\circ}}$ is the restriction of
$2\varphi_{\alpha}^{a_{\circ}}$ to $U_{2\alpha}$.

For any $v\in V$, the map $a_{\circ}\mapsto v+a_{\circ}$ yields the map

$\varphi^{a_{\circ}}\mapsto
v+\varphi^{a_{\circ}}:=\varphi^{v+a_{\circ}}=\left(\varphi^{v+a_{\circ}}_{\alpha}\colon\mathbf{U}_{\alpha}(F)\to\mathds{R}\cup\\{\infty\\},u\mapsto\varphi^{a_{\circ}}(u)+\alpha(v)\right)_{\alpha\in\Phi}.$

Two discrete valuations in
$\mathds{A}_{\varphi^{a_{\circ}}}:=\\{v+\varphi^{a_{\circ}}\colon v\in V\\}$
are called equipollent. As noted by [BT72, 6.2.6], the action of $V$ on
$\mathds{A}_{\varphi^{a_{\circ}}}$ described above endow this latter with a
structure of an affine space under $V$.

######  Remark 2.4.

Under the isomorphism of affine spaces
$\mathds{A}_{\varphi^{a_{\circ}}}\simeq\mathds{A}_{red}(\mathbf{G},\mathbf{S})$,
the action of $N$ on $\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ described in
Proposition 2.2 corresponds to the following action of $N$ on
$\mathds{A}_{\varphi^{a_{\circ}}}$:

$N\times\mathds{A}_{\varphi^{a_{\circ}}}\to\mathds{A}_{\varphi^{a_{\circ}}},\quad[n,v+\varphi^{a_{\circ}}]\mapsto\jmath_{W}(n)(v)+\varphi^{\nu_{N}(n)(a_{\circ})}.$

For any $\alpha\in\Phi$, the valuation $\varphi^{a_{\circ}}$ is discrete;
$\Gamma_{\alpha}(a_{\circ}):=\varphi_{\alpha}^{a_{\circ}}(\mathbf{U}_{\alpha}(F)\setminus\\{1\\})$
is a discrete subset of $\mathds{R}$ [BT72, §6.2.21] and
$\Gamma_{\alpha}(a_{\circ})=\Gamma_{-\alpha}(a_{\circ})$. Set

$\Gamma_{\alpha}^{\prime}(a_{\circ}):=\\{\varphi_{\alpha}^{a_{\circ}}(u)\in\Gamma_{\alpha}(a_{\circ})\colon\varphi_{\alpha}^{a_{\circ}}(u)=\sup\varphi_{\alpha}^{a_{\circ}}(u\mathbf{U}_{2\alpha}(F))\\}$

(If $2\alpha\not\in\Phi$ we put $U_{2\alpha}=\\{1\\}$, i.e.
$\Gamma_{\alpha}^{\prime}(a_{\circ})=\Gamma_{\alpha}(a_{\circ})$). We have
$\Gamma_{\alpha}(a_{\circ})=\Gamma_{\alpha}^{\prime}(a_{\circ})\cup\frac{1}{2}\Gamma_{2\alpha}(a_{\circ})$
[BT72, §6.2.2].

######  Remark 2.5.

The set $\Gamma_{\alpha}^{\prime}(a_{\circ})$ can be empty, in our discrete
case, this happens if and only if $U_{2\alpha}$ is dense in $U_{\alpha}$
[BT72, §6.2.2].

######  Definition 2.6.

We say that a discrete valuation
$\varphi^{a}\in\mathds{A}_{\varphi^{a_{\circ}}}$ (for some
$a\in\mathds{A}_{red}(\mathbf{G},\mathbf{S})$) is special if, for any
$\alpha\in\Phi_{red}$, one has $0\in\Gamma_{\alpha}(a)$.

######  Remark 2.7.

By definition $\varphi^{a}=(\varphi_{\alpha}^{a}\colon
u\mapsto-\alpha(b_{a,\alpha,u}-a))$, we see then that $\varphi^{a}$ is special
if and only if for any $\alpha\in\Phi_{red}$, there is a
$u_{\alpha}\in\mathbf{U}(F)$ such that $a=b_{a,\alpha,u_{\alpha}}$, this is
equivalent to saying that $a$ is fixed by $m_{\alpha}(u_{\alpha})$ for all
$\alpha\in\Phi_{red}$, in which case we will say that $a$ is a special point
of $\mathds{A}_{red}(\mathbf{G},\mathbf{S})$.

######  Lemma 2.8.

There exists a special discrete valuation in
$\mathds{A}_{\varphi^{a_{\circ}}}$.

###### Proof.

This is [BT72, Corollaire 6.2.15].∎

By the previous lemma we may and will assume from now on that $a_{\circ}$ is a
special point. From now on we will omit indicating $a_{\circ}$ in the notation
of
$\varphi^{a_{\circ}},\Gamma_{\alpha}(a_{\circ}),\Gamma_{\alpha}^{\prime}(a_{\circ}),\varphi_{\alpha}^{a_{\circ}}$
and $U_{\alpha+r}^{a_{\circ}}$ for $\alpha\in\Phi,r\in\mathds{R}$, i.e. we
will write $\varphi,\varphi_{\alpha}$ and $U_{\alpha+r}$ …

######  Proposition 2.9.

For any $\alpha\in\Phi$, there exists a positive integer $n_{\alpha}$ such
that $\Gamma_{\alpha}=n_{\alpha}^{-1}\mathds{Z}$, which satisfies the
following properties: $n_{\alpha}=n_{-\alpha}$, $n_{w(\alpha)}=n_{\alpha}$ for
any $w\in W=W(\Phi)$ and $n_{\alpha}$ is divisible by $n_{2\alpha}$ if
$\alpha,2\alpha\in\Phi$.

###### Proof.

This follows from [SS97, Lemma I.2.10] and [BT72, 6.2.23]. ∎

###  2.4. Affine roots

In the notation $U_{\alpha+r}$ for any $\alpha\in\Phi$ and any
$r\in\Gamma_{\alpha}$, $\alpha+r$ should be interpreted as the closed half-
space of $\mathds{A}_{red}(\mathbf{G},\mathbf{S})$, on which the affine
mapping $a\mapsto\alpha(a-a_{\circ})+r$ is positive.

######  Definition 2.1.

Let $\Phi_{\textit{\emph{\text{aff}}}}$ denote the set of all affine roots
$\left\\{\alpha+r\colon\alpha\in\Phi,r\in\Gamma_{\alpha}^{\prime}\right\\}$
[BT72, 6.2.6].

For use later, we describe in the following lemma the action of
$\mathbf{N}(F)$ on root groups $U_{\alpha+r}$ for affine roots.

######  Lemma 2.2.

Let $n\in\mathbf{N}(F)$, with image $w$ in $W$. We have for all
$\alpha\in\Phi$ and all $r\in\Gamma_{\alpha}^{\prime}$:

$nU_{\alpha+r}n^{-1}=U_{\beta},$

where, $\beta=w(\alpha)+r-w(\alpha)(\nu_{N}(n)(a_{\circ})-a_{\circ})$. In
particular, $mU_{\alpha+r}m^{-1}=U_{\alpha+r-\langle\nu(m),\alpha\rangle}$ for
$m\in\mathbf{M}(F)$.

###### Proof.

Let $\alpha+r\in\Phi_{\textit{\emph{\text{aff}}}}$ and $n\in N$ of image $w$
in $W$. As suggested by the proof of [BT72, 6.2.10. (iii)]:

$\displaystyle nU_{\alpha+r}n^{-1}$
$\displaystyle=(\varphi_{\alpha})^{-1}([r,\infty])$
$\displaystyle=n\\{u\in\mathbf{U}_{\alpha}(F)\colon\varphi_{\alpha}(u)\geq
r\\}n^{-1}$
$\displaystyle=\\{u\in\mathbf{U}_{w(\alpha)}(F)\colon\varphi_{\alpha}(n^{-1}un)\geq
r\\}$
$\displaystyle=\\{u\in\mathbf{U}_{w(\alpha)}(F)\colon(\nu_{N}(\varphi))_{w(\alpha)}(u)\geq
r\\}$
$\displaystyle{=}\\{u\in\mathbf{U}_{w(\alpha)}(F)\colon(w(0_{V})+\varphi^{\nu_{N}(a_{\circ})})_{w(\alpha)}(u)\geq
r\\}$
$\displaystyle=\\{u\in\mathbf{U}_{w(\alpha)}(F)\colon(\varphi^{\nu_{N}(a_{\circ})})_{w(\alpha)}(u)\geq
r\\}$
$\displaystyle{=}\\{u\in\mathbf{U}_{w(\alpha)}(F)\colon(\nu_{N}(n)(a_{\circ})-a_{\circ}+\varphi)_{w(\alpha)}(u)\geq
r\\}$
$\displaystyle=\\{u\in\mathbf{U}_{w(\alpha)}(F)\colon\varphi_{w(\alpha)}(u)+w(\alpha)(\nu_{N}(n)(a_{\circ})-a_{\circ})\geq
r\\}$
$\displaystyle=U_{\beta=w(\alpha)+r-w(\alpha)(\nu_{N}(n)(a_{\circ})-a_{\circ})}\qed$

For each affine root $\alpha+r\in\Phi_{\textit{\emph{\text{aff}}}}$ consider
the affine hyperplane
$H_{\alpha+r}:=\big{\\{}a\in\mathds{A}_{red}(\mathbf{G},\mathbf{S})\colon\alpha(a-a_{\circ})=-r\\}\big{\\}}$.
For any $u_{r}\in\varphi_{\alpha}^{-1}(r)$, one has
$H_{\alpha+r}=\\{a\in\mathds{A}_{red}(\mathbf{G},\mathbf{S})\colon
m_{\alpha}(u_{r})(a)=a\\}$. If $\alpha,2\alpha\in\Phi$, then for any
$r\in\Gamma_{2\alpha}$, we have $H_{2\alpha+r}=H_{\alpha+r/2}$. These
hyperplanes define a poly-simplicial structure on the standard apartment in
the following way:

######  Definition 2.3.

Define the equivalence relation on $\mathds{A}_{red}(\mathbf{G},\mathbf{S})$
by $a\sim b$ if for any affine root $\beta=\alpha+r$ the sign of
$\alpha(a-a_{\circ})+r$ and $\alpha(b-a_{\circ})+r$ is the same or are both
equal to zero; 333Equivalently, $a\sim b$ if for any affine hyperplane $H$,
either $a,b\in H$ or they are in the same connected component of
$\mathds{A}_{red}(\mathbf{G},\mathbf{S})\setminus H$. the equivalence classes
are called the facets. A vertex is a point which is a facet, e.g. the point
$a_{\circ}$. A facet with maximal dimension is called an alcove, it is also a
connected component
$\mathds{A}_{red}(\mathbf{G},\mathbf{S})\setminus\bigcup_{\beta\in\Phi_{\textit{\emph{\text{aff}}}}}H_{\beta}$.

From now on, we fix an alcove
$\mathfrak{a}\subset\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ containing in its
closure the special vertex $a_{\circ}$.

######  Definition 2.4.

Let $\alpha\in\Phi$ and $r\in\Gamma_{\alpha}$. We say that $\alpha+r$ is
$\mathfrak{a}$-positive (resp. $\mathfrak{a}$-negative) if
$\alpha(a-a_{\circ})+r>0$ (resp. $<0$) for some $a\in\mathfrak{a}$ (then for
all, since the sign does not depend on the choice of $a\in\mathfrak{a}$). Let
$\Phi_{\textit{\emph{\text{aff}}}}^{+}$ (resp.
$\Phi_{\textit{\emph{\text{aff}}}}^{-}$) denote the set of affine
$\mathfrak{a}$-positive (resp. negative) affine roots.

For any non-empty subset
$\Upomega\subset\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ and $\alpha\in\Phi$,
we define $f_{\Upomega}\colon\Phi\to\mathds{R}\cup\\{\infty\\}$,

$f_{\Upomega}(\alpha):=\inf\\{r\in\Gamma_{\alpha}\colon(\alpha+r)(\Upomega)\subset\mathds{R}^{+}\\}.$

Define also the group $U_{\Upomega}$ to be the subgroup of $\mathbf{G}(F)$
generated by $\cup_{\alpha\in\Phi_{red}}U_{\alpha+f_{\Upomega}(\alpha)}$. Note
that for $a\in\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ and $\alpha\in\Phi$,
the real $f_{\\{a\\}}(\alpha)$ depends only on the facet containing $a$.

######  Proposition 2.5.

For any non-empty subset
$\Upomega\subset\mathds{A}_{red}(\mathbf{G},\mathbf{S})$, the groups defined
above have the following important properties:

  1. (a)

For any $n\in\mathbf{N}(F)$ we have
$nU_{\Upomega}n^{-1}=U_{\nu_{N}(n)\cdot\Upomega}$, so in particular
$N_{\Upomega}:=\\{n\in\mathbf{N}(F)\colon\nu_{N}(n)\cdot x=x\text{ for all
}x\in\Upomega\\}$ normalizes $U_{\Upomega}$.

  2. (b)

For any $\alpha\in\Phi_{red}$ we have $U_{\Upomega}\cap
U_{\alpha}=U_{\alpha+f_{\Upomega}(\alpha)}$.

  3. (c)

The set $P_{\Upomega}:=N_{\Upomega}U_{\Upomega}=U_{\Upomega}N_{\Upomega}$ is a
group. We have $P_{\Upomega}\cap\mathbf{N}(F)=N_{\Upomega}$.

  4. For any decomposition into positive and negative roots $\Phi=\Phi^{+}\sqcup\Phi^{-}$, we have

  5. (d)

The following product map is an

${\prod_{\alpha\in\Phi_{red}^{\pm}}U_{\alpha+f_{\Upomega}(\alpha)}}$${U_{\Upomega}\cap\mathbf{U}(F)^{\pm}=:U_{\Upomega}^{\pm}.}$$\scriptstyle{\simeq}$

homeomorphism whatever ordering of the factors we take.

  6. (e)

$U_{\Upomega}=U_{\Upomega}^{+}U_{\Upomega}^{-}(U_{\Upomega}\cap\mathbf{N}(F))$.

###### Proof.

See [BT72, 6.2.10(iii),6.4.9, 7.1.3& 7.1.8]. We warn the reader that the
groups denoted here $N_{\Upomega}$ and $N_{\Upomega}$ are denoted by
$\widehat{N}_{\Upomega}$ and $\widehat{P}_{\Upomega}$. ∎

From now, we will adopt the following notation: when $\Upomega=\\{x\\}$ we
will write $\square_{x}$ instead of $\square_{\\{x\\}}$ for
$\square\in\\{f,U,N,P\\}$.

######  Example 2.6.

1 - For $\Upomega=\\{a_{\circ}\\}$, we have $f_{\\{a_{\circ}\\}}(\alpha)=0$
for all $\alpha\in\Phi$. Now, let us see the case $\Upomega=\mathfrak{a}$. We
first have

$\mathfrak{a}=\\{a\in\mathds{A}_{red}(\mathbf{G},\mathbf{S})\colon
0<\alpha(a-a_{\circ})<n_{\alpha}^{-1}\text{ for all }\alpha\in\Phi\text{ that
are $\mathfrak{a}$-positive}\\}.$

Therefore, if $\alpha\in\Phi$ is $\mathfrak{a}$-positive, then
$f_{\mathfrak{a}}(\alpha)=0$. If now $\alpha\in\Phi_{red}$ is
$\mathfrak{a}$-negative, then $-\alpha$ is $\mathfrak{a}$-positive, hence for
all $a\in\mathfrak{a}$ we have $0<-\alpha(a-a_{\circ})<n_{-\alpha}^{-1}$. The
real $f_{\mathfrak{a}}(\alpha)\in\Gamma_{\alpha}$ being the smallest element
such that $\alpha(a-a_{\circ})\geq-f_{\mathfrak{a}}(\alpha)$, we see that
$f_{\mathfrak{a}}(\alpha)=n_{-\alpha}^{-1}$. We know that
$\Gamma_{\alpha}=\Gamma_{-\alpha}$, this implies that
$n_{\alpha}^{-1}=n_{-\alpha}^{-1}$. In conclusion:

$f_{\mathfrak{a}}(\alpha)=\begin{cases}0&\text{ if $\alpha\in\Phi$ is
$\mathfrak{a}$-positive},\\\ n_{\alpha}^{-1}&\text{ if $\alpha\in\Phi$ is
$\mathfrak{a}$-negative}\end{cases}$

2 - If $\Upomega_{1}\subset\Upomega_{2}$, then $f_{\Upomega_{1}}(\alpha)\geq
f_{\Upomega_{2}}(\alpha)$, for all $\alpha\in\Phi$. For example, if
$\mathcal{F}$ is facet of $\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ such that
$a_{\circ}\in\overline{\mathcal{F}}\subset\overline{\mathfrak{a}}$, then

$f_{\mathcal{F}}(\alpha)=0\text{ if $\alpha\in\Phi$ is
$\mathfrak{a}$-positive},\text{ and
}f_{\mathcal{F}}(\alpha)\in\\{0,n_{\alpha}^{-1}\\}\text{ if $\alpha\in\Phi$ is
$\mathfrak{a}$-negative}.$

3 - For any non-empty subset
$\Upomega\subset\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ and any
$n\in\widetilde{W}$ whose image in $W$ is $w$, we have by Lemma 2.2

$f_{\nu_{N}(n)\cdot\Upomega}(\alpha)=w(\alpha)(\nu_{N}(n)(a_{\circ})-a_{\circ})+f_{\Upomega}(w^{-1}(\alpha)).$

###  2.5. Affine Weyl groups

For any affine root $\beta=\alpha+r\in\Phi_{\textit{\emph{\text{aff}}}}$, let
$s_{\beta}\in\nu_{N}(\mathbf{N}(F))$ denote the orthogonal reflection with
respect to the hyperplane $H_{\beta}$:

$s_{\beta}\colon
a\mapsto-\left(\alpha(a-a_{\circ})+r\right)\alpha^{\vee}+a,\,\forall
a\in\mathds{A}_{red}(\mathbf{G},\mathbf{S}).$

######  Definition 2.1.

We define the affine Weyl group
$W_{\textit{\emph{\text{aff}}}}\subset\nu_{N}(\mathbf{N}(F))$ to be the group
generated by the reflections $s_{\beta}$ for all
$\beta\in\Phi_{\textit{\emph{\text{aff}}}}$. It is a normal subgroup of
$\nu_{N}(\mathbf{N}(F))$ [BT72, 6.2.11].

By [BT84, §4.2.21 & §5.1.19], we have
$\Phi^{\prime}:=\\{\alpha\in\Phi\colon\Gamma_{\alpha}^{\prime}\neq 0\\}=\Phi$.
The set $\Sigma:=\\{n_{\alpha}\alpha\colon\alpha\in\Phi\\}$ is a root system
and the map

${\Phi_{red,\textit{\emph{\text{aff}}}}:=\cup_{\alpha\in\Phi_{red}}(\alpha+\Gamma_{\alpha}^{\prime})}$${\Sigma_{\textit{\emph{\text{aff}}}}:=\cup_{\alpha\in\Sigma}(\alpha+\mathds{Z});\quad(\alpha+r)}$${n_{\alpha}(\alpha+r),}$$\scriptstyle{\simeq}$

is bijective and respects positivity. The group
$W_{\textit{\emph{\text{aff}}}}=W_{\textit{\emph{\text{aff}}}}(\Sigma)$ is the
affine Weyl group associated with $\Sigma$ and
$\Phi_{red,\textit{\emph{\text{aff}}}}$ is the corresponding affine roots
system [BT72, 6.2.22]. This is also the normal subgroup of
$\nu_{N}(\mathbf{N}(F))$ generated by the set of orthogonal reflections
$s_{\alpha+r}$ for all $\alpha\in\Phi$ and all $r\in\Gamma_{\alpha}$.

Define $\Lambda_{\textit{\emph{\text{aff}}}}$ to be the subgroup of
translations in $W_{\textit{\emph{\text{aff}}}}$, we can then identify
$\Lambda_{\textit{\emph{\text{aff}}}}$ with the coroot lattice
$\mathds{Z}[\Sigma^{\vee}]$ [BT72, Proposition 6.2.20]. The affine Weyl group
$W_{\textit{\emph{\text{aff}}}}$ acts simply transitively on the set of
alcoves in $\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ (VI §2.1 loc. cit.).

Let $a$ be vertex in $\mathds{A}_{red}(\mathbf{G},\mathbf{S})$, we denote by
$\Phi_{\textit{\emph{\text{aff}}}}^{a}$ the set of affine roots that vanish at
$a$, set $W_{\textit{\emph{\text{aff}}}}^{a}=\langle
s_{\beta}\colon\beta\in\Phi_{\textit{\emph{\text{aff}}}}^{a}\rangle\subset
W_{\textit{\emph{\text{aff}}}}$. The vertex $a$ is special in the sense of
Remark 2.7 if and only if the composition of the following maps

${W_{\textit{\emph{\text{aff}}}}^{a}}$${W_{\textit{\emph{\text{aff}}}}}$${\nu_{N}(\mathbf{N}(F))}$${W,}$

is an isomorphism444This is equivalent to
$W_{\textit{\emph{\text{aff}}}}^{a}\simeq W$ or just to
$\\#W_{\textit{\emph{\text{aff}}}}^{a}=\\#W$ since the induced map
$W_{\textit{\emph{\text{aff}}}}^{a}\hookrightarrow
W_{\textit{\emph{\text{aff}}}}\to W$ is always injective.. Special vertices
exists by [BT72, Corollaire 6.2.16]. We have then by [BT72, 1.3 & 6.2.19] a
decomposition for the affine Weyl group

$W_{\textit{\emph{\text{aff}}}}\simeq\Lambda_{\textit{\emph{\text{aff}}}}\rtimes
W_{\textit{\emph{\text{aff}}}}^{a}\simeq\mathds{Z}[\Sigma^{\vee}]\rtimes
W_{\textit{\emph{\text{aff}}}}^{a},$

for any fixed special vertex $a\in\mathds{A}_{red}(\mathbf{G},\mathbf{S})$.

######  Definition 2.2.

A face (resp. facet) of the alcove $\mathfrak{a}$ is a facet contained in a
single affine hyperplane (resp. in $\overline{\mathfrak{a}}$). A wall of the
alcove $\mathfrak{a}$ is a hyperplane containing a face of $\mathfrak{a}$. Set

$S(\mathfrak{a}):=\\{s_{H}\colon\text{$H$ a wall of $\mathfrak{a}$}\\}.$

Define the type of a facet $\mathcal{F}$ of $\mathfrak{a}$ to be the set

$\mathcal{T}_{\mathcal{F}}=\\{s_{H}\colon\text{$H$ a wall of $\mathfrak{a}$,
}\mathcal{F}\subset H\\}\subset S(\mathfrak{a}).$

Let $W_{\textit{\emph{\text{aff}}}}^{\mathcal{F}}$ be the group generated by
$\mathcal{T}_{\mathcal{F}}$. Then
$(W_{\textit{\emph{\text{aff}}}}^{\mathcal{F}},\mathcal{T}_{\mathcal{F}})$ is
a finite Coxeter system.

######  Remark 2.3.

A facet is determined by its type

$\mathcal{F}=\\{a\in\overline{\mathfrak{a}}\colon(a\in
H\Longleftrightarrow\mathcal{F}\subset H)\text{ for any wall $H$ of
$\mathfrak{a}$}.\\}$

Since a facet of $\mathcal{F}$ is the image by an element of
$W_{\textit{\emph{\text{aff}}}}$ of a unique facet of $\mathfrak{a}$ and each
facet of $\mathfrak{a}$ is determined by its type, one can then define the
type for all facets [BT72, 1.3.5]. So alcoves have empty types
($\mathcal{T}_{\mathfrak{a}}=\emptyset$) and special points have full type and
$W_{\textit{\emph{\text{aff}}}}^{a_{\circ}}$ is generated by
$\mathcal{T}_{a_{\circ}}$.

###  2.6. The reduced Bruhat–Tits building

We arrive now at the most important object of this section, that is the
reduced Bruhat–Tits building:

######  Definition 2.1.

Define the set

$\mathcal{B}(\mathbf{G},F)_{red}:=\mathbf{G}(F)\times\mathds{A}_{red}(\mathbf{G},\mathbf{S})/\sim,$

where $\sim$ is the equivalence relation on
$\mathbf{G}(F)\times\mathds{A}_{red}(\mathbf{G},\mathbf{S})$ defined by

$(g,x)\sim(h,y)\,\text{ if }\exists n\in\mathbf{N}(F)\text{ such that
}nx=y\text{ and }g^{-1}hn\in U_{x}.$

The group $\mathbf{G}(F)$ acts on $\mathcal{B}(\mathbf{G},F)_{red}$ on the
left:

$g\cdot[(h,y)]:=[(gh,y)]\quad\text{ for }g\in\mathbf{G}(F)\text{ and
}(h,y)\in\mathbf{G}(F)\times\mathds{A}_{red}(\mathbf{G},\mathbf{S}).$

Moreover, the map
$\mathds{A}_{red}(\mathbf{G},\mathbf{S})\to\mathcal{B}(\mathbf{G},F)_{red}$
given by $x\mapsto[(1,x)]$ is an $\mathbf{N}(F)$-equivariant embedding. We
will denote its image by $\mathcal{A}$. We have
$g\cdot\mathcal{A}=\mathcal{A}$ (resp. $g\cdot x=x,\,\forall x\in\mathcal{A}$)
if and only if $g\in\mathbf{N}(F)$ (resp. $g\in\ker\nu_{N}$) [BT72, 7.4.10].

######  Definition 2.2.

An apartment of the building $\mathcal{B}(\mathbf{G},F)_{red}$ is a subset of
the form $g\mathcal{A}$ for some $g\in\mathbf{G}(F)$. A facet (resp. alcove)
of $\mathcal{B}(\mathbf{G},F)_{red}$ is a subset of the form $g\mathcal{F}$
for some $g\in\mathbf{G}(F)$ and a facet (resp. alcove)
$\mathcal{F}\subset\mathcal{A}$.

Since all maximal $F$-split $F$-tori of $\mathbf{G}$ are
$\mathbf{G}(F)$-conjugate [Bor91, Theorem 20.9], all apartments of
$\mathcal{B}(\mathbf{G},F)_{red}$ are in bijection with maximal split tori of
$\mathbf{G}$.

######  Proposition 2.3.

In this proposition, we will collect some important properties regarding the
building $\mathcal{B}(\mathbf{G},F)_{red}$:

  1. (a)

(Fixators) Let $\Upomega\subset\mathcal{A}$. We have an alternative
characterization for the subgroups $P_{\Upomega}=N_{\Upomega}U_{\Upomega}$
(defined in Proposition 2.5 \- c):

$P_{\Upomega}=\text{Fix}(\mathbf{G}(F),\Upomega):=\\{g\in\mathbf{G}(F)\colon
gx=x\quad\forall x\in\Upomega\\},$

in other words $P_{\Upomega}$ is the subgroup that fixes all points of
$\Upomega$ [BT72, 7.4.4].

  2. (b)

For any $g\in\mathbf{G}(F)$ we can find a $n\in\mathbf{N}(F)$ verifying
$gx=nx$ for all $x$ in the closed subset $\mathcal{A}\cap g^{-1}\mathcal{A}$
[BT72, 7.4.8].

  3. (c)

(Transitivity) Let $\Upomega\subset\mathcal{A}$. The group $U_{\Upomega}$ acts
transitively on the set of all apartments containing $\Upomega$ [BT72, 7.4.9].

  4. (d)

For any two facets in the building, there exists an apartment that contains
both of them [BT72, 7.4.18].

  5. (e)

Let us fix an $W$-invariant euclidean metric $d$ on $\mathcal{A}$. The
previous properties ensure that the distance $d$ extends uniquely to a
$\mathbf{G}(F)$-invariant metric on the set
$(\mathcal{B}(\mathbf{G},F)_{red},d)$.

######  Remark 2.4.

The term ”fixators” refers to pointwise stabilizers, in contrast the term
”stabilizers” will be used for setwise stabilizers.∎

######  Definition 2.5.

The reduced Bruhat–Tits building of $\mathbf{G}(F)$ is the pair
$(\mathcal{B}(\mathbf{G},F)_{red},d)$ together with its isometric
$\mathbf{G}(F)$-action and the poly-simplicial structure defined by its
facets.

###  2.7. The extended Bruhat–Tits building

####  2.7.1. The extended apartment

When the center $Z(\mathbf{G})$ has split rank $>0$, the stabilizer $P_{x}$ of
a point $x\in\mathcal{A}\subset\mathcal{B}(\mathbf{G},F)_{red}$ is no longer a
compact subgroup of $\mathbf{G}(F)$. To remedy this ”issue” we define in this
section a larger building following [BT84, 4.2.16].

We have a decomposition
$X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{Q}=(X_{*}(\mathbf{S}^{der})\otimes_{\mathds{Z}}\mathds{Q})\oplus(X_{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{Q})$,
where $\mathbf{S}^{der}=(\mathbf{G}^{der}\cap\mathbf{S})^{\circ}$. This is the
”dual” statement of the decomposition (2) in Lemma [Con19, Lemma 11.3.3]. This
decomposition allows us to inject $\Phi^{\vee}$ in $V^{\vee}$. So, the ”dual”
statement of (3) in loc. cit. together with [Bor91, Theorem 201.6 & Remark
21.1] imply that
$V=\text{span}_{\mathds{R}}(\Phi^{\vee})=X_{*}(\mathbf{S}^{der})\otimes_{\mathds{Z}}\mathds{R}$.
Accordingly, the apartment $\mathcal{A}\subset\mathcal{B}(\mathbf{G},F)_{red}$
corresponding to $\mathbf{S}$ can be viewd as a torsor for the real vector
space

$X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}\big{/}X_{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{R}.$

Let us construct a homomorphism $\nu_{G}\colon\mathbf{G}(F)\to
X_{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{Q}$: Using the isogeny
$\mathbf{Z}_{c}\times\mathbf{G}^{der}\to\mathbf{G}$, we see that
$X^{*}(\mathbf{G})_{F}=X^{*}(\mathbf{G}/\mathbf{G}^{der})_{F}$. Since the
quotient map $\mathbf{Z}_{c}\to\mathbf{G}/\mathbf{G}^{der}$ is an $F$-isogeny
of $F$-tori, we see that $X^{*}(\mathbf{G}/\mathbf{G}^{der})_{F}$ identifies
with a subgroup of $X^{*}(\mathbf{Z}_{c})_{F}$ of finite index. Therefore, we
have a natural isomorphism of $\mathds{Q}$-vector spaces

$X^{*}(\mathbf{G}/\mathbf{G}^{der})_{F}\otimes_{\mathds{Z}}\mathds{Q}\to
X^{*}(\mathbf{Z}_{c})_{F}\otimes_{\mathds{Z}}\mathds{Q}.$

Identify the $\mathds{Q}$-linear dual
$\text{Hom}_{\mathds{Z}}(X^{*}(\mathbf{Z}_{c,sp}),\mathds{Q})$ of
$X^{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{Q}$ with
$X_{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{Q}$. In conclusion, we
get a natural isomorphism

$X^{*}(\mathbf{G})_{F}\otimes_{\mathds{Z}}\mathds{Q}\simeq
X^{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{Q}.$

Consequently, we obtain a canonical isomorphism

$X_{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{Q}\simeq\text{Hom}_{\mathds{Z}}(X^{*}(\mathbf{G})_{F}\otimes_{\mathds{Z}}\mathds{Q},\mathds{Q}).$

This shows that there exists a unique homomorphism

$\mathbf{G}(F)\to X_{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{Q},$

extending
$\nu_{G}\colon\mathbf{G}(F)\to\text{Hom}_{\mathds{Z}}(X^{*}(\mathbf{G})_{F},\mathds{Z})$
(§2.1); this is the unique homomorphism denoted by abuse of notation $\nu_{G}$
and satisfying555We abuse notation and denote also by $\langle\,,\rangle$ the
pairing $X_{*}(\mathbf{Z}_{c,sp})\times
X^{*}(\mathbf{Z}_{c,sp})\to\mathds{Z}$. (see proof of Lemma 2.1)

$\langle\nu_{G}(g),\chi|_{\mathbf{Z}_{c,sp}}\rangle=-\omega(\chi(g)),\text{
for all }\chi\in X^{*}(\mathbf{G})_{F}\text{ and }g\in\mathbf{G}(F).$

Put $\mathbf{G}(F)^{1}:=\ker\nu_{G}$666We refer the reader to §2.8.2 for more
on the homomorphism $\nu_{G}$. Then $\mathbf{G}(F)/\mathbf{G}(F)^{1}$ is a
finitely generated abelian group and there is an isomorphism

$\mathbf{G}(F)/\mathbf{G}(F)^{1}\otimes_{\mathds{Z}}\mathds{Q}\simeq
X_{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{Q}.$

Set $V_{G}:=X_{*}(\mathbf{Z}_{c,sp})\otimes_{\mathds{Z}}\mathds{R}$ and let
$\mathds{A}_{G}$ be a fixed affine space under $V_{G}$. We have a morphism
$\nu_{G}\colon\mathbf{G}(F)\to\textit{\emph{\text{Aff}}}(\mathds{A}_{G})$
sending any $g$ to the translation $(a\mapsto\nu_{G}(g)+a)$.

We define the extended standard apartment
$\mathds{A}_{ext}(\mathbf{G},\mathbf{S})$ to be the product of
$\mathds{A}_{red}(\mathbf{G},\mathbf{S})\times\mathds{A}_{G}$ together with
the group homomorphism

${\nu_{N,ext}\colon\mathbf{N}(F)}$${\textit{\emph{\text{Aff}}}(\mathds{A}_{ext}(\mathbf{G},\mathbf{S})),}$${n}$${\nu_{N}(n)\oplus\nu_{G}(n).}$

The decomposition $X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}=V\oplus
V_{G}$ shows that the extended standard apartment
$\mathds{A}_{ext}(\mathbf{G},\mathbf{S})$ (as defined above) is actually an
affine space under $X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}$ and the
restriction of $\nu_{N,ext}$ to $\mathbf{M}(F)$ corresponds precisely to the
translation action given by the homomorphism $\nu_{M}$ of Lemma 2.1. In
particular,

$\mathbf{M}(F)^{1}=\ker\nu_{M}=\ker\nu\cap\mathbf{G}(F)^{1}=\ker\nu_{N}\cap\mathbf{G}(F)^{1}=\ker\nu_{N,ext}.$

######  Remark 2.1.

While the reduced system $(\mathds{A}_{red}(\mathbf{G},\mathbf{S}),\nu_{N})$
is canonical, that is, unique up to unique isomorphism (Proposition 2.2), the
extended system $(\mathds{A}_{ext}(\mathbf{G},\mathbf{S}),\nu_{N,ext})$ is
only unique up to isomorphism. We ”canonify” it (following G. Rousseau) by
viewing (which we will adopt from now on) $\mathds{A}_{G}$ as $V_{G}$ with a
marked origin $\\{0_{V_{G}}\\}$.

Since there is a one-to-one correspondence between systems of positive roots
for $\Phi(\mathbf{G},\mathbf{S})$ and vectorial chambers in
$\mathds{A}_{ext}(\mathbf{G},\mathbf{S})$ with apex the fixed special point
$a_{\circ}$, we may and will refine our choice of the alcove $\mathfrak{a}$ to
be the unique alcove with apex $a_{\circ}$ such that

$\\{\alpha\in\Phi\colon\alpha\text{ is $\mathfrak{a}$-positive}\\}=\Phi^{+}.$

####  2.7.2. The extended building

######  Definition 2.2.

The extended building is the following product of a poly-simplicial complex
and a real vector space

$\mathcal{B}(\mathbf{G},F)_{ext}=\mathcal{B}(\mathbf{G},F)_{red}\times V_{G}.$

The group $\mathbf{G}(F)$ acts isometrically on it as follows:

$g\cdot(x,v)=(g\cdot x,v+\nu_{G}(g))\quad\forall
g\in\mathbf{G}(F),x\in\mathcal{B}(\mathbf{G},F)_{red},v\in V_{G}.$

A facet (resp. alcove, Weyl chamber, apartment) of
$\mathcal{B}(\mathbf{G},F)_{ext}$ is defined to be a product of a same object
in $\mathcal{B}(\mathbf{G},F)_{red}$ and $V_{G}$. For instance let us fix the
extended apartment $\mathcal{A}_{ext}=\mathcal{A}\times V_{G}$. We identify
$\mathcal{B}(\mathbf{G},F)_{red}$ with the subset
$\mathcal{B}(\mathbf{G},F)_{red}\times V_{G}$ in
$\mathcal{B}(\mathbf{G},F)_{ext}$. Then, the stabilizer of
$\mathcal{B}(\mathbf{G},F)_{red}$ for the action of $\mathbf{G}(F)$ in the
extended building is exactly $\ker\nu_{G}$. In this building the minimal
dimensional facets are of the form $\mathcal{F}_{x}:=\\{x\\}\times V_{G}$ for
some vertex $x\in\mathcal{B}(\mathbf{G},F)_{red}$. The stabilizer of $(x,0)$
is equal to $P_{x}\cap\ker\nu_{G}$, which is also the fixator of
$\mathcal{F}_{x}$, where $P_{x}$ is the subgroup defined in §2.3.

###  2.8. Parahoric subgroups

####  2.8.1. Further notations

Let $F^{un}$ denote the maximal unramified extension of $F$ contained in
$F^{sep}$ and $L$ the completion of $F^{un}$ with respect to the valuation on
$F^{un}$ which extends the normalized valuation on $F$. The residue field of
$F$ is perfect, thus $F^{un}=F^{\text{sh}}$ is the strict henselization of $F$
in the fixed separable closure $F^{sep}$. Let $L^{sep}$ the separable closure
of $L$ containing $F^{sep}$. The arithmetic Frobenius automorphism
$\sigma\in\text{Gal}(F^{un}/F)$ (which induces $x\mapsto x^{q}$ on the residue
field of $F^{un}$) extends continuously to an automorphism of $L$ over $F$,
also denoted $\sigma$. Write $\text{In}=\text{Gal}(F^{sep}/F^{un})$ for the
inertia subgroup of $\text{Gal}(F^{sep}/F)$. Since
${L}^{sep}=L\otimes_{F^{un}}{F}^{sep}$, one can identify the inertia subgroup
In with $\text{Gal}(L^{sep}/L)$.

####  2.8.2. The Kottwitz homomorphism

Let $\mathbf{H}$ be any connected reductive $F$-group, $\widehat{\mathbf{H}}$
be its connected Langlands dual and $Z(\widehat{\mathbf{H}})$ be the center of
$\widehat{\mathbf{H}}$. Kottwitz defines in [Kot97, §7.7] a functorial
surjective homomorphism

$\kappa_{\mathbf{H}}:\mathbf{H}({L})\rightarrow
X^{*}(Z(\widehat{\mathbf{H}})^{\text{In}})=X^{*}(Z(\widehat{\mathbf{H}}))_{\text{In}},$

where, the subscript In indicates coinvariants of the
$\text{Gal}(F^{sep}/F)$-module $X^{*}(Z(\widehat{\mathbf{H}}))$, this latter
is the Borovoi fundamental algebraic group (usually denoted
$\pi_{1}(\mathbf{H})$) and it is isomorphic to the quotient
$X_{*}(\mathbf{T})$ for a maximal $F$-torus $\mathbf{T}$ by the coroots
lattice. The group $\pi_{1}(\mathbf{H})$ acquires an action of
$\text{Gal}(F^{sep}/F)$ via its representation as
$X_{*}(\mathbf{T})/\sum_{\alpha\in\Phi(\mathbf{H}_{F^{sep}},\mathbf{T})}\mathds{Z}\alpha^{\vee}$.
There is a canonical surjective homomorphism

$q_{\mathbf{H}}\colon
X^{*}(Z(\widehat{\mathbf{H}}))_{\text{In}}\to\text{Hom}_{\mathds{Z}}(X^{*}(\mathbf{H})^{\text{In}},\mathds{Z})$

whose kernel is the torsion subgroup of
$X^{*}(Z(\widehat{\mathbf{H}}))_{\text{In}}$ [Kot97, 7.4.4]777The codomain of
(7.4.4) in loc. cit. is actually
$\text{Hom}_{\mathds{Z}}(X_{*}(Z(\widehat{\mathbf{H}}))^{\text{In}},\mathds{Z}))$,
so one needs to use the isomorphism $X_{*}(Z(\widehat{\mathbf{H}}))\simeq
X^{*}(\mathbf{H})$.. Kottwitz shows in [Kot97, §7.4] that the above two
homomorphisms sit in the following commutative diagram:

${\mathbf{H}(L)}$${X^{*}(Z(\widehat{\mathbf{H}}))_{\text{In}}}$${\text{Hom}_{\mathds{Z}}(X^{*}(\mathbf{H})^{\text{In}},\mathds{Z})}$$\scriptstyle{\kappa_{\mathbf{H}}}$$\scriptstyle{-\nu_{\mathbf{H}}}$$\scriptstyle{q_{\mathbf{H}}}$

where, $\nu_{\mathbf{H}}$ is the natural homomorphism characterized by

$\langle\nu_{\mathbf{H}}(h),\chi\rangle=\nu_{\mathbf{H}}(h)(\chi)=-\omega(\chi(h)),\text{
for all }h\in\mathbf{H}(L)\text{ and }\chi\in X^{*}(\mathbf{H})^{\text{In}}.$

Therefore,
$\mathbf{H}(L)_{1}:=\ker\kappa_{\mathbf{H}}\subset\mathbf{H}(L)^{1}:=\ker\nu_{\mathbf{H}}$
and

$\left(X^{*}(Z(\widehat{\mathbf{H}}))_{\text{In}}\right)_{\text{tors}}=\mathbf{H}(L)^{1}/\mathbf{H}(L)_{1}.$

######  Remark 2.1.

Note here that $\nu_{\mathbf{H}}$ differs in a sign from the map $v_{H}$ in
[Kot97, 7.4.3].

Since our connected reductive group $\mathbf{H}$ is defined over $F$, the
restriction of $\kappa_{\mathbf{H}}$ to $\mathbf{H}(F)$ provides a surjective
homomorphism $\kappa_{H}$ that sits in the following commutative diagram (see
[Kot97, §7.7] for the surjectivity of $\kappa_{H}$, see also [Ros15, §2.7])

${\mathbf{H}(F)}$${(X^{*}(Z(\widehat{\mathbf{H}}))_{\text{In}})^{\langle\sigma\rangle}}$${\text{Hom}_{\mathds{Z}}(X^{*}(\mathbf{H})_{F},\mathds{Z})}$$\scriptstyle{\kappa_{H}}$$\scriptstyle{-\nu_{H}}$$\scriptstyle{q_{H}}$

We denote by $\mathbf{H}(F)_{1}=\mathbf{H}(F)\cap\mathbf{H}(L)_{1}$ and
$\mathbf{H}(F)^{1}$ the kernel of $\kappa_{H}$ and $\nu_{H}$, respectively.
Therefore, by [HV15, §3.2, Lemma]

$\mathbf{H}(F)^{1}=\mathbf{H}(L)^{1}\cap\mathbf{H}(F)=\left\\{h\in\mathbf{H}(F)\colon\kappa_{H}(h)\in\left(X^{*}(Z(\widehat{\mathbf{H}}))_{\text{In}}^{\sigma}\right)_{\text{tors}}\right\\}.$

Set $\Lambda_{H}:=\mathbf{H}(F)/\mathbf{H}(F)_{1}$, thus

$\mathbf{H}(F)^{1}/\mathbf{H}(F)_{1}=(\Lambda_{H})_{\text{tors}}\simeq\left(X^{*}(Z(\widehat{\mathbf{H}}))_{\text{In}}^{\sigma}\right)_{\text{tors}}.$

####  2.8.3. Passage to completion and descent

Consider the extended building $\mathcal{B}(\mathbf{G},F^{un})_{ext}$ of the
group $\mathbf{G}_{F^{un}}=\mathbf{G}\times_{F}F^{un}$. This building is
equipped with an action of $\mathbf{G}(F^{un})\rtimes\langle\sigma\rangle$.
Moreover, there is a natural $\mathbf{G}(F)$-equivariant embedding
$\iota\colon\mathcal{B}(\mathbf{G},F)_{ext}\hookrightarrow\mathcal{B}(\mathbf{G},F^{un})_{ext}$
such that [BT84, 5.1.25]

$\iota(\mathcal{B}(\mathbf{G},F)_{ext})=\mathcal{B}(\mathbf{G},F^{un})_{ext}^{\sigma}.$

In other words, the extended building of $\mathbf{G}$ over $F$ is identified
with the fixed points of $\sigma\in\text{Gal}(F^{un}/F)$ (in particular of
$\sigma$) in the building of $\mathbf{G}$ over $F^{un}$.

Since $F^{un}$ is henselian, Rousseau has shown in [Rou77, proposition 2.3.9]
that $\mathbf{G}$ has the same relative rank over $F^{un}$ and over $L$ and
hence by [Rou77, proposition 2.3.9] the building
$\mathcal{B}(\mathbf{G},F^{un})_{ext}$ identifies canonically with
$\mathcal{B}(\mathbf{G},L)_{ext}$ as $\mathbf{G}(F^{un})$-space. Using this
identification, we see that the extended building of $\mathbf{G}$ over $F$,
can also be identified with the fixed points of $\sigma\in\text{Aut}(L/F)$ in
the building of $\mathbf{G}$ over $L$, i.e.

$\iota(\mathcal{B}(\mathbf{G},F)_{ext})=\mathcal{B}(\mathbf{G},L)_{ext}^{\sigma}.$

This gives a bijection between the set of $\sigma$-stable facets in
$\mathcal{B}(\mathbf{G},L)_{ext}$ and the set of facets in
$\mathcal{B}(\mathbf{G},F)_{ext}$. For more details one can consult [Pra17].

####  2.8.4. $F$-Parahoric subgroups

In [BT84, §4.6.28, §5.2.6, §5.1.29 and §5.1.30] (see also [Yu15, §9.3]),
Bruhat and Tits associated two subgroups to any non-empty $\sigma$-stable set
$\Upomega$ that is contained in an apartment of
$\mathcal{B}(\mathbf{G},F^{un})_{ext}$ and with bounded image in the reduced
building:

  1. (1)

The fixator subgroup

$\text{Fix}(\mathbf{G}(F^{un}),\Upomega):=\\{g\in\mathbf{G}(F^{un})\colon
g\cdot a=a,\forall a\in\Upomega\\}\subset\mathbf{G}(F^{un}).$

They also showed in loc. cit., the existence of a unique affine smooth group
scheme $\mathbb{P}_{\Upomega}$ defined over $\text{Spec}\,{\mathcal{O}_{F}}$
with generic fiber $\mathbf{G}$ and which is uniquely characterized by the
property
$\mathbb{P}_{\Upomega}(\mathcal{O}_{F^{un}})=\text{Fix}(\mathbf{G}(F^{un}),\Upomega)$.

  2. (2)

The ”connected fixator” subgroup

$\text{Fix}(\mathbf{G}(F^{un}),\Upomega)^{\circ}:=\mathbb{P}_{\Upomega}^{\circ}(\mathcal{O}_{F^{un}})\subset\text{Fix}(\mathbf{G}(F^{un}),\Upomega),$

where $\mathbb{P}_{\Upomega}^{\circ}$ is the identity component of
$\mathbb{P}_{\Upomega}$.

A $F$-parahoric subgroup of $\mathbf{G}$ is by definition [BT84, under 5.2.6]
the ”connected fixator” of a facet
$\mathcal{F}\subset\iota(\mathcal{B}(\mathbf{G},F)_{ext})$.

######  Remark 2.2.

We warn the reader that, in [BT84, §4 and §5] notations,
$\mathbb{P}_{\Upomega}$, $\mathbb{P}_{\Upomega}^{\circ}$,
$\text{Fix}(\mathbf{G}(F^{un}),\Upomega)$,
$\text{Fix}(\mathbf{G}(F^{un}),\Upomega)^{\circ}$ and $\mathbb{P}_{\Upomega}$
and $\mathbb{P}_{\Upomega}^{\circ}$ are denoted by
$\widehat{\mathfrak{G}}_{\Upomega}^{\natural}$,
${\mathfrak{G}}_{\Upomega}^{0\natural}$, $\widehat{P}_{\Upomega}^{1}$ and
${P}_{\Upomega}^{0}$, , respectively.

######  Lemma 2.3.

Two facets
$\mathcal{F},\mathcal{F}^{\prime}\subset\mathcal{B}(\mathbf{G},F)_{ext}$ are
equal if and only if

$\mathbb{P}_{\iota(\mathcal{F})}^{\circ}(\mathcal{O}_{F})=\text{Fix}(\mathbf{G}(F^{un}),\iota(\mathcal{F}))^{\circ}\cap\mathbf{G}(F)=\text{Fix}(\mathbf{G}(F^{un}),\iota(\mathcal{F}^{\prime}))^{\circ}\cap\mathbf{G}(F)=\mathbb{P}_{\iota(\mathcal{F}^{\prime})}^{\circ}(\mathcal{O}_{F}).$

###### Proof.

See [BT84, 5.2.8]. ∎

This lemma justifies (by misuse of language) the following definition:

######  Definition 2.4.

For any bounded subset $\Upomega$ contained in an apartment of
$\mathcal{B}(\mathbf{G},F)_{red}$, set

$\widetilde{K}_{\Upomega}=\mathbb{P}_{\iota(\Upomega\times
V_{G})}(\mathcal{O}_{F^{un}})\cap\mathbf{G}(F)\text{ and
}K_{\Upomega}:=\mathbb{P}_{\iota(\Upomega\times
V_{G})}^{\circ}(\mathcal{O}_{F})$

A parahoric subgroup of $\mathbf{G}(F)$ is $K_{\mathcal{F}}$ for some facet
$\mathcal{F}\subset\mathcal{B}(\mathbf{G},F)_{red}$. When the facet
$\mathcal{F}$ is an alcove $\mathfrak{b}$, the parahoric subgroup
$K_{\mathcal{F}}$ is called an Iwahori subgroup. The subgroups
$\widetilde{K}_{\mathfrak{b}}$ and ${K}_{\mathfrak{b}}$ will be denoted by
$\widetilde{I}_{\mathfrak{b}}$ and ${I}_{\mathfrak{b}}$, respectively.

######  Remark 2.5.

The subgroup $\mathbb{P}_{\Upomega\times V_{G}}(\mathcal{O}_{F})$ might be
strictly smaller than $\widetilde{K}_{\Upomega}$, see [Hai09a] for examples.

Haines and Rapoport showed in [HR08] that parahoric subgroups over
$\mathbf{G}(F^{un})$ are actually $\mathbf{G}(F^{un})_{1}$-fixators of facets
in the Bruhat–Tits building over $F^{un}$. Accordingly, as we see in the
following proposition, this result descend also to the base field $F$, i.e.
parahoric subgroups of $\mathbf{G}(F)$ are fixators of facets in the kernel of
$\kappa_{G}$, i.e. for any facet
$\mathcal{F}\subset\mathcal{B}(\mathbf{G},F)_{red}$ we have
$K_{\mathcal{F}}=P_{\mathcal{F}}\cap\mathbf{G}(F)_{1}$.

######  Proposition 2.6.

For any bounded subset $\Upomega$ contained in an apartment of
$\mathcal{B}(\mathbf{G},F)_{red}$ we have

$\widetilde{K}_{\Upomega}=P_{\Upomega}\cap\mathbf{G}(F)^{1}\text{ and
}K_{\Upomega}=\widetilde{K}_{\Upomega}\cap\mathbf{G}(F)_{1}=\mathbb{P}_{\iota(\Upomega\times
V_{G})}(\mathcal{O}_{F})\cap\mathbf{G}(F)_{1}$

###### Proof.

On the one hand,

$\text{Fix}(\mathbf{G}(F^{un}),\iota(\Upomega\times
V_{G}))\cap\mathbf{G}(F)=\text{Fix}(\mathbf{G}(F^{un}),\iota(\Upomega))\cap\mathbf{G}(F^{un})^{1}\cap\mathbf{G}(F).$

On the second hand
$\text{Fix}(\mathbf{G}(F^{un}),\iota(\Upomega))\cap\mathbf{G}(F)=P_{\Upomega}$
(because the map $\iota$ is $\mathbf{G}(F)$-invariant and a Proposition 2.3),
which shows the first equality of the Lemma, since
$\mathbf{G}(F^{un})^{1}\cap\mathbf{G}(F)=\mathbf{G}(F)^{1}$. For the remaining
equalities, thanks to [HR08, Proposition 3, remarks 4 and 11], one has
$\mathbb{P}_{\iota(\Upomega\times
V_{G})}^{\circ}(\mathcal{O}_{F^{un}})=\mathbb{P}_{\iota(\Upomega\times
V_{G})}(\mathcal{O}_{F^{un}})\cap\mathbf{G}(F^{un})_{1}$, and accordingly:

$\displaystyle\mathbb{P}_{\iota(\Upomega\times
V_{G})}^{\circ}(\mathcal{O}_{F})$
$\displaystyle=\mathbb{P}_{\iota(\Upomega\times
V_{G})}^{\circ}(\mathcal{O}_{F^{un}})\cap\mathbf{G}(F)$
$\displaystyle=\mathbb{P}_{\iota(\Upomega\times
V_{G})}(\mathcal{O}_{F^{un}})\cap\mathbf{G}(F^{un})_{1}\cap\mathbf{G}(F)=\mathbb{P}_{\iota(\Upomega\times
V_{G})}(\mathcal{O}_{F})\cap\mathbf{G}(F)_{1}.\qed$

The map that associates to a facet $\mathcal{F}$ its parahoric subgroup
$K_{\mathcal{F}}$ is decreasing, in particular, if $\mathcal{F}$ is any facet
of the alcove $\mathfrak{a}\subset\mathcal{B}(\mathbf{G},F)_{red}$ then we
have $I_{\mathfrak{a}}\subset K_{\mathcal{F}}$. In addition, the action of an
element $g\in\mathbf{G}(F)$ on a facet $\mathcal{F}$ translates for parahorics
to $K_{g\cdot\mathcal{F}}=gK_{\mathcal{F}}g^{-1}$.

####  2.8.5. The scheme $\mathcal{Z}$

By replacing $\mathbf{G}$ by $\mathbf{M}$ in all of the above, we see that
$\mathcal{B}(\mathbf{M},F)_{red}$ is reduced to a single point
$\\{a_{\circ,M}\\}$ [BT84, §5.1.26]. The fixator subgroup
$\text{Fix}(\mathbf{M}(F),a_{\circ,M})$ is equal to the
$\mathcal{O}_{F}$-points of a unique smooth $\mathcal{O}_{F}$-group scheme
$\mathcal{Z}$ which is the schematic closure of $\mathbf{M}$ in
$\mathbb{P}_{\iota(\Upomega\times V_{G})}$ for any bounded subset
$\Upomega\subset\mathbb{A}_{red}(\mathbf{G},\mathbf{S})$ (see [BT84, 5.2.1 &
5.2.4]) and the corresponding parahoric subgroup is

$K_{a_{\circ,M}}=\text{Fix}(\mathbf{M}(F),a_{\circ,M})^{\circ}=\mathcal{Z}^{\circ}(\mathcal{O}_{F}).$

Applying Proposition 2.6 to $\mathbf{M}$ and $\Upomega=\\{a_{\circ,M}\\}$ one
shows that $K_{a_{\circ,M}}=\mathbf{M}(F)_{1}$. By [BT84, §5.2.1], one has
$\mathbf{M}(F)^{1}=\ker\nu\cap\mathbf{G}(F)^{1}=\mathcal{Z}(\mathcal{O}_{F})$
and in particular $\mathbf{M}(F)_{1}=\mathcal{Z}^{\circ}(\mathcal{O}_{F})$ is
of finite index in $\mathbf{M}(F)^{1}$. In summary:

######  Corollary 2.7.

The subgroup $\mathbf{M}(F)_{1}$ is the unique parahoric subgroup of
$\mathbf{M}(F)$ and it is a finite index subgroup of
$\mathbf{M}(F)^{1}=\ker\nu_{M}$ (The maximal open compact subgroup of
$\mathbf{M}(F)$).

######  Lemma 2.8.

Let ${\mathfrak{b}}\subset\mathcal{A}$ be an alcove. The
$\mathbf{N}(F)$-fixator of ${\mathfrak{b}}$ (resp. ${\mathfrak{b}}\times
V_{G}$) is $N_{\mathfrak{b}}=\ker\nu_{N}$ (resp.
$\mathbf{N}(F)\cap\widetilde{I}_{\mathfrak{b}}=\mathbf{M}(F)^{1}$). Therefore,

$P_{\mathfrak{b}}=U_{\mathfrak{b}}^{+}U_{\mathfrak{b}}^{-}\ker\nu\,\text{ and
}\widetilde{I}_{\mathfrak{b}}=U_{\mathfrak{b}}^{+}U_{\mathfrak{b}}^{-}\mathbf{M}(F)^{1}.$

###### Proof.

The first equality is stated at the end of [BT72, §4.1.2]. Here is a geometric
argument: Because $\mathbf{N}(F)$ acts transitively on the set of alcoves of
$\mathcal{A}$, it suffices to prove the Lemma for $\mathfrak{b}=\mathfrak{a}$.
Let $n\in\mathbf{N}(F)\cap P_{\mathfrak{a}}$. By assumption $\nu_{N}(n)(a)=a$
for any $a\in\overline{\mathfrak{a}}$, in particular for all
$a\in\text{vert}(\mathfrak{a})=\\{\text{vertices of $\mathfrak{a}$}\\}$. The
set $\left\\{a-a_{\circ}\colon
a\in\text{vert}(\mathfrak{a})\setminus\\{a_{\circ}\\}\right\\}$ forms a basis
for $V$ and hence $\nu_{N}(n)$ must equal the identity on $V$. Therefore,
$\mathbf{N}(F)\cap P_{\mathfrak{a}}=\mathbf{M}(F)\cap
P_{\mathfrak{a}}=\ker\nu$. By c 2.5, this is precisely the
$\mathbf{N}(F)$-fixator of ${\mathfrak{a}}$. Therefore, the first equality in
the displayed equation in the Lemma follows from e Proposition 2.5 and the
second one by Proposition 2.6. ∎

####  2.8.6. Decompositions

By [BT84, §5.2.4], we have the following decomposition of parahoric subgroups:

######  Proposition 2.9.

For any bounded subset $\Upomega\subset\mathcal{A}$, the subgroup
$K_{\Upomega}\subset\mathbf{G}(F)$ is equal to the subgroup generated by
$\mathbf{M}(F)_{1}=\mathcal{Z}^{\circ}(\mathcal{O}_{F})$ and $U_{\Upomega}$,
more precisely it has the following decomposition

$K_{\Upomega}=U_{\Upomega}\mathbf{M}(F)_{1}=U_{\Upomega}^{+}U_{\Upomega}^{-}U_{\Upomega}^{+}\mathbf{M}(F)_{1}=U_{\Upomega}^{+}U_{\Upomega}^{-}(\mathbf{N}(F)\cap
K_{\Upomega}),$

such that the factors commute in the right factorization and the product maps

${\prod_{\alpha\in\Phi_{red}\cap\Phi^{\pm}}U_{\alpha+f_{{\Upomega}}(\alpha)}}$${U_{\Upomega}^{\pm}=U_{\Upomega}\cap\mathbf{U}(F)^{\pm}=K_{\Upomega}\cap\mathbf{U}(F)^{\pm}.}$$\scriptstyle{\sim}$

is an homeomorphism, whatever ordering of the factors we take.

######  Remark 2.10.

As observed by Haines [Hai09a, §6], the reference [BT84, §5.2.4] contains a
typographical error. All of the ”hats” in the four displayed equations in loc.
cit. should be removed.

######  Lemma 2.11.

Let ${\mathfrak{b}}\subset\mathcal{A}$ be an alcove. We have
$\mathbf{N}(F)\cap
I_{\mathfrak{b}}=\mathbf{M}(F)^{1}\cap\mathbf{G}(F)_{1}=\mathbf{M}(F)_{1}$.

###### Proof.

Clearly, $\mathbf{M}(F)^{1}\subset P_{\mathfrak{b}}$, hence
$\mathbf{M}(F)^{1}\cap\mathbf{G}(F)_{1}\subset\mathbf{N}(F)\cap
I_{\mathfrak{b}}$. The opposite inclusion follows from Lemma 2.8. This shows
the first equality. For the second, let
$m\in\mathbf{M}(F)^{1}\cap\mathbf{G}(F)_{1}=\mathbf{N}(F)\cap
I_{\mathfrak{b}}$. Using the decomposition of Proposition 2.9 for
$I_{\mathfrak{b}}$, write $m$ as $u_{1}^{+}u_{1}^{-}u_{2}^{+}m_{1}\in
U_{\mathfrak{b}}^{+}U_{\mathfrak{b}}^{-}U_{\mathfrak{b}}^{+}\mathbf{M}(F)_{1}$,
then, one finds that
$mm_{1}^{-1}=\text{ad}(mm_{1}^{-1})(u_{2}^{+})u_{1}^{+}u_{1}^{-}\in\mathbf{M}(F)\cap\mathbf{U}(F)\mathbf{U}^{-}(F)=\\{1\\}$,
i.e. $m=m_{1}\in\mathbf{M}(F)_{1}$. This concludes the proof of the lemma. ∎

####  2.8.7. The subgroup generated by all parahoric subgroups

As suggested by the condition [BT72, (A 6) §5.2.1], Bruhat and Tits introduces
in [BT84, §5.2.11] the subgroup generated by all parahoric subgroups of
$\mathbf{G}(F)$ denoted here by $\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}$.
They also prove in loc. cit. that
$\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}=\mathbf{M}(F)_{1}\cdot\varphi(\mathbf{G}_{sc}(F))$,
where $\varphi\colon\mathbf{G}_{sc}\to\mathbf{G},$ be the natural homomorphism
from the simply connected cover of $\mathbf{G}^{der}$. For any subgroup
$H\subset\mathbf{G}(F)$ set
$H^{\textit{\emph{\text{aff}}}}=H\cap\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}$.

###  2.9. A double Tits system

Set $H_{1}=G_{1}\cap H$ for any subgroup $H\subset G$. A good part of the
following two sections relies heavily on the following theorem of Bruhat and
Tits:

######  Theorem 2.1.

The homomorphism $\nu_{N}$ induces an isomorphism of groups
$W_{\textit{\emph{\text{aff}}}}\simeq\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}/\mathbf{M}(F)_{1}$.
If
$\mathcal{S}\subset\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}/\mathbf{M}(F)_{1}$
is the subset corresponding to $\mathcal{S}(\mathfrak{a})$, then the quadruple

$(\mathbf{G}(F)^{\textit{\emph{\text{aff}}}},I_{\mathfrak{a}},\mathbf{N}(F)^{\textit{\emph{\text{aff}}}},\mathcal{S})$

is a double Tits system whose Weyl group is $W_{\textit{\emph{\text{aff}}}}$
and the injection $\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}\to\mathbf{G}(F)$
is $I_{\mathfrak{a}}$-$\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}$-adapted of
connected type.

###### Proof.

This is [BT84, Proposition 5.2.12] together with the fact that
$I_{\mathfrak{a}}\cap\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}=\mathbf{M}(F)_{1}$
proved in Lemma 2.11. ∎

The nomenclature ”double Tits system” [BT72, §1.2.6] means that both

$(\mathbf{G}(F)^{\textit{\emph{\text{aff}}}},I_{\mathfrak{a}},\mathbf{N}(F)^{\textit{\emph{\text{aff}}}})\text{
and
}(\mathbf{G}(F)^{\textit{\emph{\text{aff}}}},\mathbf{B}(F)^{\textit{\emph{\text{aff}}}},\mathbf{N}(F)^{\textit{\emph{\text{aff}}}})$

are Tits systems. The second assertion means that [BT72, §1.2.13]: for any
$g\in\mathbf{G}(F)$, there exists
$h\in\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}$ such that
$hI_{\mathfrak{a}}h^{-1}=gI_{\mathfrak{a}}g^{-1}$ and
$h\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}h^{-1}=g\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}g^{-1}$.
We refer the reader to the section [BT72, §5.1] for more details, especially
Theorem 5.1.3 in loc. cit. for equivalent properties. Being of connected type
(Definition 4.1.3 in loc. cit.) shows that the images of $\nu(\mathbf{N}(F))$
and $\nu(\mathbf{N}(F)^{\textit{\emph{\text{aff}}}})$ in $GL(V)$ are equal,
i.e.
$\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}/\mathbf{M}(F)^{\textit{\emph{\text{aff}}}}\simeq
W=\mathbf{N}(F)/\mathbf{M}(F)$. Thanks to Theorem 2.1 we have:

######  Corollary 2.2.

  1. (1)

If $a\in\mathcal{A}$ is a special point, then by the claim below [BT72,
Definition 4.1.3], we have $\mathbf{N}(F)\cap P_{a}=(\mathbf{N}(F)\cap
K_{a})\cdot\ker\nu=(\mathbf{N}(F)\cap U_{a})\cdot\ker\nu$, thus
$P_{a}=\ker\nu\,\cdot U_{a}=\ker\nu\,\cdot K_{a}$.

  2. (2)

For any subset $\Upomega\subset\mathcal{A}$, set $P_{\Upomega}^{\dagger}$ for
its $\mathbf{G}(F)$-stabilizer, i.e. $\\{g\in\mathbf{G}(F)\colon
g\cdot\Upomega=\Upomega\\}$ and
$N_{\Upomega}^{\dagger}=P_{\Upomega}^{\dagger}\cap\mathbf{N}(F)$. We have
$P_{\Upomega}^{\dagger}=N_{\Upomega}^{\dagger}K_{\Upomega}$.

###### Proof.

See [BT72, §4.1] for these statement. ∎

######  Corollary 2.3.

For any bounded subset $\Upomega\subset\mathcal{A}$, we have

$\widetilde{K}_{\Upomega}\cap\mathbf{M}(F)=\mathbb{P}_{\iota(\Upomega\times
V_{G})}(\mathcal{O}_{F})\cap\mathbf{M}(F)=\mathbf{M}(F)^{1}\text{ and
}K_{\Upomega}\cap\mathbf{M}(F)=\mathbf{M}(F)_{1}.$

###### Proof.

The equality $\widetilde{K}_{\Upomega}\cap\mathbf{M}(F)=\mathbf{M}(F)^{1}$ is
clear by double inclusions, since $\mathbf{M}(F)$ acts by translations, so any
element of $\mathbf{M}(F)$ fixing any point fixes all of $\mathcal{A}_{ext}$,
i.e. is in $\mathbf{M}(F)^{1}$. The last equality
$K_{\Upomega}\cap\mathbf{M}(F)=\mathbf{M}(F)_{1}$ follows readily from the
first one using Lemma 2.11.

The last equality, can also be proved using [HR10, Lemma 4.1.1], which shows
that for any facet $\mathcal{F}$ in $\mathcal{A}_{ext}$ the subgroup
$\mathbf{M}(F)\cap K_{\mathcal{F}}$ is a parahoric subgroup of $\mathbf{M}(F)$
and hence must be $\mathbf{M}(F)_{1}$ by Corollary 2.7. ∎

######  Lemma 2.4.

For any special point $a\in\mathcal{A}$, the canonical
injection${\mathbf{N}(F)\cap K_{a}/\mathbf{M}(F)_{1}}$${W}$is an isomorphism.

###### Proof.

Because $a$ is special, we know by Remark 2.7 that for any
$\alpha\in\Phi_{red}$ there exists $m_{\alpha}\in U_{a}\cap\mathbf{N}(F)$
whose image in $W=\langle s_{\alpha}\colon\alpha\in\Phi_{red}\rangle$ is the
reflection $s_{\alpha}$. ∎

######  Lemma 2.5.

Let ${\mathfrak{b}}\subset\mathcal{A}$ be an alcove. The subgroup
${P}_{\mathfrak{b}}^{{\dagger}}=N_{\mathfrak{b}}^{{\dagger}}\cdot
I_{\mathfrak{b}}$ is the $\mathbf{G}(F)$-normalizer of $P_{\mathfrak{b}}$,
$\widetilde{I}_{\mathfrak{b}}$ and $I_{\mathfrak{b}}$. Moreover,

$N_{\mathfrak{b}}^{{\dagger}}\cap\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}=\mathbf{M}(F)_{1}\text{
and }P_{\mathfrak{b}}^{{\dagger}}\cap\mathbf{M}(F)=\ker\nu.$

###### Proof.

The statement on the normalizer is clear. Since
$P_{\mathfrak{b}}^{{\dagger}}\cap\mathbf{G}(F)^{{\textit{\emph{\text{aff}}}}}$
equals the $\mathbf{G}(F)^{{\textit{\emph{\text{aff}}}}}$-stabilizer of
${\mathfrak{b}}$ which is the
$\mathbf{G}(F)^{{\textit{\emph{\text{aff}}}}}$-normalizer of $I$, but this
latter is its own normalizer (in
$\mathbf{G}(F)^{{\textit{\emph{\text{aff}}}}}$), hence
$P_{\mathfrak{b}}^{{\dagger}}\cap\mathbf{G}(F)^{{\textit{\emph{\text{aff}}}}}=I_{\mathfrak{b}}$.
Intersecting this equality with $\mathbf{N}(F)$ shows the first equality
thanks to Lemma 2.11.

Finally, an $m\in{P}_{\mathfrak{b}}^{{\dagger}}\cap\mathbf{M}(F)$ acts by
translation, and we have $\cup_{k\in\mathds{Z}_{\geq
0}}{\mathfrak{b}}+k\nu(m)\subset{\mathfrak{b}}$. But since ${\mathfrak{b}}$ is
bounded, this inclusion can only happen if $\nu(m)=0$, i.e.
${P}_{\mathfrak{b}}^{{\dagger}}\cap\mathbf{M}(F)=\ker\nu$. We can argue also
using $\mathbf{M}(F)=\ker\nu\,\mathbf{M}(F)^{\textit{\emph{\text{aff}}}}$ (by
Corollary 2.6), so
${P}_{\mathfrak{b}}^{{\dagger}}\cap\mathbf{M}(F)=\ker\nu\,({P}_{\mathfrak{b}}^{{\dagger}}\cap\mathbf{M}(F)^{\textit{\emph{\text{aff}}}})$
which equals $\ker\nu$ by the first part of this Lemma. ∎

######  Corollary 2.6.

We have
$\mathbf{M}(F)^{\textit{\emph{\text{aff}}}}/\mathbf{M}(F)_{1}\simeq\Lambda_{\textit{\emph{\text{aff}}}}$
and

$\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}/\mathbf{M}(F)_{1}\simeq
W_{\textit{\emph{\text{aff}}}}\simeq(\mathbf{M}(F)^{\textit{\emph{\text{aff}}}}/\mathbf{M}(F)_{1})\rtimes
W.$

###### Proof.

Given the isomorphism
$\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}/\mathbf{M}(F)_{1}\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}W_{\textit{\emph{\text{aff}}}}$
(Theorem 2.1), $\mathbf{M}(F)^{\textit{\emph{\text{aff}}}}$ is precisely the
subgroup of elements in $\mathbf{N}(F)$ that act on $\mathcal{A}$ by
translation, hence the map $\nu$ induces an isomorphism

$\mathbf{M}(F)^{\textit{\emph{\text{aff}}}}/\mathbf{M}(F)_{1}\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}\Lambda_{\textit{\emph{\text{aff}}}}.$

Accordingly, the isomorphism
$W_{\textit{\emph{\text{aff}}}}\simeq\Lambda_{\textit{\emph{\text{aff}}}}\rtimes
W_{\textit{\emph{\text{aff}}}}^{a}$ for any special point $a\in\mathcal{A}$
(§2.5) corresponds to the decomposition

$\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}/\mathbf{M}(F)_{1}=(\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}\cap\mathbf{M}(F))/\mathbf{M}(F)_{1}\rtimes(\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}\cap
K_{a})/\mathbf{M}(F)_{1}.$

This concludes the proof thanks to Lemma 2.4. ∎

######  Corollary 2.7.

Let ${\mathfrak{b}}\subset\mathcal{A}$ be an alcove. We have,
$\mathbf{G}(F)=N_{\mathfrak{b}}^{{\dagger}}\cdot\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}$
and
$\mathbf{N}(F)=N_{\mathfrak{b}}^{{\dagger}}\cdot\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}$
and
$\mathbf{G}(F)/\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}\simeq\mathbf{N}(F)/\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}\simeq
N_{\mathfrak{b}}^{{\dagger}}/\mathbf{M}(F)_{1}$.

###### Proof.

A immediate consequence of Theorem 2.1 is
$\mathbf{G}(F)\cdot{\mathfrak{b}}=\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}\cdot{\mathfrak{b}}$
and so
$\mathbf{G}(F)=P_{\mathfrak{b}}^{\dagger}\cdot\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}$,
which proves the first equality given that
$P_{\mathfrak{b}}^{\dagger}=N_{\mathfrak{b}}^{\dagger}\cdot I_{\mathfrak{b}}$.
Intersecting this with $\mathbf{N}(F)$ yields the second equality of the
corollary.

This shows that the natural maps
$N_{\mathfrak{b}}^{\dagger}\to\mathbf{G}(F)/\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}$
and
$N_{\mathfrak{b}}^{\dagger}\to\mathbf{N}(F)/\mathbf{N}(F)^{\textit{\emph{\text{aff}}}}$
are both surjective and have the same kernel $\mathbf{M}(F)_{1}$ by Lemma 2.5.
∎

###  2.10. The subgroup $\mathbf{G}(F)_{1}$

Proposition 2.6 shows that
$\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}\subset\mathbf{G}(F)_{1}$ and as we
will see in the following Lemma, this is actually an equality:

######  Lemma 2.1.

We have,
$\mathbf{G}(F)^{1}=(N_{\mathfrak{a}}^{\dagger}\cap\mathbf{G}(F)^{1})\cdot\varphi(\mathbf{G}_{sc}(F))\text{
and
}\mathbf{G}(F)_{1}=\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}=\mathbf{M}(F)_{1}\cdot\varphi(\mathbf{G}_{sc}(F))$.

###### Proof.

The equality $\mathbf{G}(F)_{1}=\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}$
was already proved in [Ric16, Lemma 1.3]. However, we present here another
argument: Intersecting
$\mathbf{G}(F)=N_{\mathfrak{a}}^{\dagger}\cdot\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}$
of Corollary 2.7 with the kernel of $\nu_{G}$ and $\kappa_{G}$ yields

$\mathbf{G}(F)_{1}=(N_{\mathfrak{a}}^{\dagger}\cap\mathbf{G}(F)_{1})\cdot\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}\text{
and
}\mathbf{G}(F)^{1}=(N_{\mathfrak{a}}^{\dagger}\cap\mathbf{G}(F)^{1})\cdot\mathbf{G}(F)^{\textit{\emph{\text{aff}}}}.$

By Lemma 2.5 for $\mathbf{G}_{F^{un}}$ (or by [HR08, Lemma 17]), we have

$P_{\iota(\mathfrak{a})}^{{\dagger}}\cap\mathbf{G}(F^{un})^{\textit{\emph{\text{aff}}}}=P_{\iota(\mathfrak{a})}^{{\dagger}}\cap\mathbf{G}(F^{un})_{1}=\mathbb{P}_{\iota(\mathfrak{a}\times
V_{G})}^{\circ}(\mathcal{O}_{F^{un}}),$

where $P_{\iota(\mathfrak{a})}^{\dagger}$ is the
$\mathbf{G}(F^{un})$-stabilizer of ${\iota(\mathfrak{a})}$. Now since clearly
$P_{\mathfrak{a}}^{\dagger}\subset
P_{\iota(\mathfrak{a})}^{{\dagger}}\cap\mathbf{G}(F)$, we have

$I_{\mathfrak{a}}\subset
P_{\mathfrak{a}}^{\dagger}\cap\mathbf{G}(F)_{1}=P_{\mathfrak{a}}^{\dagger}\cap\mathbf{G}(F^{un})_{1}\cap\mathbf{G}(F)\subset\mathbb{P}_{\iota(\mathfrak{a}\times
V_{G})}^{\circ}(\mathcal{O}_{F^{un}})\cap\mathbf{G}(F)=I_{\mathfrak{a}}.$

In conclusion, this shows
$I_{\mathfrak{a}}=P_{\mathfrak{a}}^{\dagger}\cap\mathbf{G}(F)_{1}$ and so
$\mathbf{M}(F)_{1}=N_{\mathfrak{a}}^{\dagger}\cap\mathbf{G}(F)_{1}$ by Lemma
2.11. Finally, recall that the last one was already seen in §2.8.7, which
concludes the proof of the lemma. ∎

###  2.11. More on $\mathbf{G}(F)^{1}$-fixators

######  Lemma 2.1.

Let $\mathcal{F}\subset\mathcal{A}$ be a facet containing in its closure a
special point $a$, i.e. ${K}_{\mathcal{F}}\subset K_{a}$. We have

$\widetilde{K}_{\mathcal{F}}=\mathbf{M}(F)^{1}K_{\mathcal{F}}\text{ and
}\widetilde{K}_{\mathcal{F}}/{K}_{\mathcal{F}}\simeq\mathbf{M}(F)^{1}/\mathbf{M}(F)_{1}.$

###### Proof.

This Lemma generalizes [Ros15, Lemma 4.2.1] and [HR10, Proposition 11.1.4].
Here we give a different argument: By Corollary 2.2, one has
$P_{a}=K_{a}\cdot\ker\nu$, hence

$\displaystyle\widetilde{K}_{\mathcal{F}}$
$\displaystyle=((K_{a}\cdot\,\ker\nu)\cap\mathbf{G}(F)^{1})\cap
P_{\mathcal{F}}$ $\displaystyle(\text{Prop. \ref{parahr}})$
$\displaystyle=(K_{a}\cdot(\ker\nu\cap\mathbf{G}(F)^{1}))\cap P_{\mathcal{F}}$
$\displaystyle(K_{a}\subset\mathbf{G}(F)^{1})$
$\displaystyle=(K_{a}\cdot\mathbf{M}(F)^{1})\cap P_{\mathcal{F}}$
$\displaystyle=(K_{a}\cap P_{\mathcal{F}})\cdot\mathbf{M}(F)^{1}$
$\displaystyle(\mathbf{M}(F)^{1}\subset P_{\mathcal{F}})$
$\displaystyle=K_{\mathcal{F}}\cdot\mathbf{M}(F)^{1}.$

Consequently, Lemma 2.3 yields
$\widetilde{K}_{\Upomega}/{K}_{\Upomega}\simeq\mathbf{M}(F)^{1}/\mathbf{M}(F)_{1}$
. ∎

######  Remark 2.2.

By Lemma 2.11 we see that $M^{1}\hookrightarrow G^{1}$ induces an inclusion
$(\Lambda_{M})_{\text{tor}}\hookrightarrow(\Lambda_{G})_{\text{tor}}$
compatible with $\kappa_{M}$ and $\kappa_{G}$. We see then:

${(\Lambda_{G})_{\text{tor}}=0}$
${G^{1}=G_{1}}$${(\Lambda_{M})_{\text{tor}}=0}$${M^{1}=M_{1}}$${\widetilde{K}_{\mathcal{F}}=K_{\mathcal{F}}.}$

where $\mathcal{F}\subset\mathcal{B}(\mathbf{G},F)_{red}$ is any facet
containing a special point in its closure. The group $\Lambda_{M}$ (which
plays the role of the cocharacters lattice in the split case §4.) has no
torsion:

  * •

If $\mathbf{M}$ is a torus that splits over an unramified extension of $F$,
i.e. $\mathbf{G}$ is unramified [Bor79, 9.5].

  * •

If $\mathbf{G}$ is semisimple and simply connected then by [BT84, §4.6.32 and
§5.2.9] one has $\widetilde{K}_{\mathcal{F}}=K_{\mathcal{F}}$,

  * •

See [HR10, Corollary 11.1.5] for other situations for which $\Lambda_{M}$ is
torsion free.

##  3\. Preliminary results around (Iwahori)–Hecke algebras

### Notations

In addition to the notation fixed/adopted previously we need to introduce the
following, that we hope will lighten a bit the exposition: For any algebraic
$F$-groups $\mathbf{H}$ (bold style), we denote its group of $F$-points by the
ordinary capital letter $H=\mathbf{H}(F)$. For any subgroup $X\subset G$
different from $M$ set $X_{1}:=X\cap G_{1}$. Recall that Lemma 2.1 says that
$G_{1}=G^{\textit{\emph{\text{aff}}}}$, so for any such a $X$, we have
$X_{1}=X^{\textit{\emph{\text{aff}}}}$. For $M$, one might have
$M_{1}=\ker\kappa_{M}\neq M\cap G_{1}$, for this reason, we keep the notation
$M^{\textit{\emph{\text{aff}}}}$ for $M\cap G_{1}$.

Set $K:=K_{a_{\circ}}$ for the special maximal parahoric subgroup attached to
the special vertex $a_{\circ}\in\mathcal{A}$ and $I:=I_{\mathfrak{a}}$ be the
Iwahori subgroup corresponding to the fixed alcove
${\mathfrak{a}}\subset\mathcal{A}$.

Recall that by Lemma 2.4, the canonical injection $N\cap K/M\cap K\to W$ is an
isomorphism. Therefore, we may and will assume from now on, that any
representative in $N$ of a class in $W=N/M$ lies in $K$.

###  3.1. $\flat$ is everywhere !

Let $M_{1}\subset M^{\flat}\subset M$ be any open compact subgroup, by
maximality of $M^{1}$ [Lan96, Proposition 1.2] we must have $M^{\flat}\subset
M^{1}$.

######  Lemma 3.1.

The group $N$ acts trivially on $\ker\nu/M_{1}$. In particular, $M^{\flat}$ is
normal in $N$.

###### Proof.

Let $n\in N$ and $m\in\ker\nu$. On the one hand, we clearly have
$m^{-1}nmn^{-1}\in\ker\nu_{N}=\ker\nu$. On the other hand, the commutator
element $m^{-1}nmn^{-1}\in G_{1}$. Therefore, Lemma 2.5 implies
$m^{-1}nmn^{-1}\in\ker\nu\cap G_{1}=M_{1}$ and accordingly
$mM_{1}=nmn^{-1}M_{1}$. ∎

For any $X\subset G$, set $X^{\flat}=M^{\flat}X$. This procedure does not
alter the structure of the object we are interested in, e.g. if
$K_{\mathcal{F}}$ is a parahoric subgroup attached to a facet
$\mathcal{F}\subset\mathcal{A}$, then $K_{\mathcal{F}}^{\flat}$ is still a
group.

When $M^{\flat}=M_{1}$, the resulting objects will sometimes be described as
”parahoric” and the supscript ${\flat}$ will be simply omitted, e.g.
$I^{\flat}=I$ the Iwahori subgroup, $K_{\mathcal{F}}^{\flat}=K_{\mathcal{F}}$
the parahoric subgroup attached to a facet $\mathcal{F}$, and so on … When
$M^{\flat}=M^{1}$, the resulting objects will be described as ”geometric” and
${\flat}$ will be replaced simply by $1$, e.g.
$I^{1}=\widetilde{I},K^{1}=\widetilde{K}$.

Thanks to Lemma 2.11, we have an isomorphism of groups
$W_{\textit{\emph{\text{aff}}}}^{\flat}:=N_{1}^{\flat}/M^{\flat}\simeq
W_{\textit{\emph{\text{aff}}}}$. Recall that
$\mathcal{S}(\mathfrak{a})=\cup_{\mathcal{F}\text{ facet of
}{\mathfrak{a}}}\mathcal{T}_{\mathcal{F}}$ is the set of reflections at the
walls of $\mathfrak{a}$ and it generates $W_{\textit{\emph{\text{aff}}}}$.
Here, $r$ is the number of irreducible components of the Dynkin diagram of
$\Phi$. For any facet $\mathcal{F}$ of $\mathfrak{a}$, let
$\mathcal{T}_{\mathcal{F}}^{\flat}\subset\mathcal{S}^{\flat}\subset
W_{\textit{\emph{\text{aff}}}}^{\flat}$ be the sets corresponding to
$\mathcal{T}_{\mathcal{F}}\subset\mathcal{S}(\mathfrak{a})\subset
W_{\textit{\emph{\text{aff}}}}$. Similarly, we put
$\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}:=M^{\flat}M^{\textit{\emph{\text{aff}}}}/M^{\flat}\simeq\Lambda_{\textit{\emph{\text{aff}}}}$
and we define for any facet $\mathcal{F}\subset\mathcal{A}$, the group
${W}_{\mathcal{F}}^{\flat}:=N_{1,\mathcal{F}}^{\flat}/M^{\flat}\simeq{W}_{\mathcal{F}}$
where $N_{1,\mathcal{F}}^{\flat}:=N\cap
K_{\mathcal{F}}^{\flat}=M^{\flat}N_{1,\mathcal{F}}$. The group
${W}_{\mathcal{F}}^{\flat}$ is generated by
$\mathcal{T}_{\mathcal{F}}^{\flat}$. For example, $W_{a_{\circ}}^{\flat}$ is
canonically isomorphic to $W$. Given Lemma 2.6, we have

$W_{\textit{\emph{\text{aff}}}}^{\flat}=\Lambda_{\textit{\emph{\text{aff}}}}^{\flat}\rtimes
W_{a_{\circ}}^{\flat}.$

######  Corollary 3.2.

The quadruple $(G_{1}^{\flat},I^{\flat},N_{1}^{\flat},\mathcal{S}^{\flat})$ is
a double Tits system whose Weyl group is isomorphic to
$W_{\textit{\emph{\text{aff}}}}$ and the injection
$G_{1}^{\flat}\hookrightarrow G$ is
$I_{\mathfrak{a}}^{\flat}$-$N_{1}^{\flat}$-adapted of connected type.

###### Proof.

This follows readily from Theorem 2.1. ∎

Consequently, the pair
$(W_{\textit{\emph{\text{aff}}}}^{\flat},\mathcal{S}^{\flat})$ is a Coxeter
system [Bou68, §3] subject to the following properties:

  1. $T_{1}.$

The subgroup $N_{1}^{\flat}\cap I^{\flat}=M^{\flat}$ is normal in
$N_{1}^{\flat}$ and $G_{1}^{\flat}$ is generated by $I^{\flat}$ and
$N_{1}^{\flat}$,

  2. $T_{2}.$

the elements of $\mathcal{S}^{\flat}$ have order 2 and generate
$W_{\textit{\emph{\text{aff}}}}^{\flat}$,

  3. $T_{3}.$

For all $s\in\mathcal{S}^{\flat},w\in W_{\textit{\emph{\text{aff}}}}^{\flat}$,
we have $sI^{\flat}w\subset I^{\flat}wI^{\flat}\sqcup I^{\flat}swI^{\flat}$,

  4. $T_{4}.$

For all $s\in\mathcal{S}^{\flat}$, we have $sI^{\flat}s\neq I^{\flat}$.

######  Corollary 3.3.

  1. (1)

$G_{1}^{\flat}=\sqcup_{w\in
W_{\textit{\emph{\text{aff}}}}^{\flat}}I^{\flat}wI^{\flat}$.

  2. (2)

For any facet $\mathcal{F}\subset\overline{\mathfrak{a}}$, the subgroup
$K_{\mathcal{F}}^{\flat}=I^{\flat}W_{\mathcal{F}}^{\flat}I^{\flat}$ is its own
normalizer in $G_{1}^{\flat}$.

  3. (3)

For any two facet
$\mathcal{F},\mathcal{F}^{\prime}\subset\overline{\mathfrak{a}}$, and $w\in
W_{\textit{\emph{\text{aff}}}}^{\flat}$, we have
$K_{\mathcal{F}}^{\flat}wK_{\mathcal{F}^{\prime}}^{\flat}=I^{\flat}W_{\mathcal{F}}^{\flat}wW_{\mathcal{F}^{\prime}}^{\flat}I^{\flat}$
and the map $w\mapsto I^{\flat}wI^{\flat}$ defines a bijection

$W_{\mathcal{F}}^{\flat}\backslash
W_{\textit{\emph{\text{aff}}}}^{\flat}/W_{\mathcal{F}^{\prime}}^{\flat}\cong
K_{\mathcal{F}}^{\flat}\backslash
G_{1}^{\flat}/K_{\mathcal{F}^{\prime}}^{\flat}.$

###### Proof.

These facts are consequences of the axioms $T_{1},\dots,T_{4}$, for which
proofs may be found in [Bou68, Ch. IV, §2 n∘ 5 & 6] since for any
$\mathcal{F}\subset\overline{\mathfrak{a}}$ we know that
$W_{\mathcal{F}}^{\flat}$ is precisely the subgroup of
$W_{\textit{\emph{\text{aff}}}}^{\flat}$ generated by
$\mathcal{T}_{\mathcal{F}}$, the type of $\mathcal{F}$. ∎

######  Corollary 3.4 (Iwahori factorizations).

Let ${\mathfrak{b}}\subset\mathcal{A}$ be an alcove. The map
$U_{\mathfrak{b}}^{+}\times U_{\mathfrak{b}}^{-}\times M^{\flat}\to
I_{\mathfrak{b}}^{\flat}$ is a bijection. In particular,
$I_{\mathfrak{b}}^{\flat}$ admits the Iwahori factorization
$I_{\mathfrak{b}}^{\flat}=U_{\mathfrak{b}}^{+}U_{\mathfrak{b}}^{-}M^{\flat}$
in which the factors commute. For $\mathfrak{b}=\mathfrak{a}$, the product
maps $\prod_{\alpha\in\Phi^{+}\cap\Phi_{red}}U_{\alpha+0}\to
U_{\mathfrak{a}}^{+}=:I^{+}\text{ and
}\prod_{\alpha\in\Phi^{-}\cap\Phi_{red}}U_{\alpha+n_{\alpha}^{-1}}\to
U_{\mathfrak{a}}^{-}=:I^{-}$ are homeomeprisms. Similarly, the product maps
$\prod_{\alpha\in\Phi^{\pm}\cap\Phi_{red}}U_{\alpha+0}\to
U_{a_{\circ}}^{\pm}=K\cap U^{\pm}$ are isomorphisms and
$K^{\flat}=U_{a_{\circ}}^{+}U_{a_{\circ}}^{-}\,U_{a_{\circ}}^{+}M^{\flat}$.

###### Proof.

The corollary is a consequence of Proposition 2.9, Lemma 2.11, d of
Proposition 2.5 together with Example 2.6 for the values of
$f_{\mathfrak{a}}(\alpha)$ for all $\alpha\in\Phi_{red}$. ∎

###  3.2. Dominance in $\Lambda_{M}^{\flat}$

Let $\Lambda_{M}^{\flat}:=M/M^{\flat}$. Recall that we have a canonical
isomorphism (see §2.8.2)

$\Lambda_{M}=M/M_{1}\simeq\kappa_{M}(M)=(X^{*}(Z(\widehat{\mathbf{M}}))_{\text{In}})^{\langle\sigma\rangle}.$

This shows that $\Lambda_{M}$ is a finitely generated abelian group with
finite torsion subgroup $(\Lambda_{M})_{\text{tor}}=M^{1}/M_{1}$, which
injects into the set of permutations of the vertices of $\mathfrak{a}$. So
$\Lambda_{M}^{\flat}$ is finitely generated abelian group with torsion
subgroup $M^{1}/M^{\flat}$. Set $q_{tor}^{\flat}:=|M^{1}/M^{\flat}|$.

There exists a natural injective finite-cokernel homomorphism
$X_{*}(\mathbf{S})\hookrightarrow
X^{*}(Z(\widehat{\mathbf{M}}))_{\text{In}}^{\sigma}$ [Ros15, §2.7], which
yields an isomorphisms

$X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}\simeq
X^{*}(Z(\widehat{\mathbf{M}}))_{\text{In}}^{\sigma}\otimes_{\mathds{Z}}\mathds{R}\simeq\Lambda_{M}\otimes_{\mathds{Z}}\mathds{R}.$

The map $\nu_{M}\colon M\to X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}$
(Lemma 2.1) identifies
$\Lambda_{M}^{1}=M/M^{1}(=\Lambda_{M}/(\Lambda_{M})_{\text{tor}})$ with a
lattice $\underline{\Lambda}_{M}$ in
$X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}$ and
$\text{rank}({\Lambda_{M}/(\Lambda_{M})_{\text{tor}}})=\dim_{\mathds{R}}X_{*}(\mathbf{S})\otimes_{\mathds{Z}}\mathds{R}$.
By the above isomorphisms, the extended apartment $\mathcal{A}_{ext}$ acquires
the structure of an affine space over
$\Lambda_{M}\otimes_{\mathds{Z}}\mathds{R}\simeq\Lambda_{M}^{\flat}\otimes_{\mathds{Z}}\mathds{R}\simeq\underline{\Lambda}_{M}\otimes_{\mathds{Z}}\mathds{R}$,
thus one gets an embedding

$\underline{\Lambda}_{M}\to\mathcal{A}_{ext},\quad
v\mapsto\nu_{M}(v)+(a_{\circ},0_{V_{G}}).$

Define

$\displaystyle M^{\pm}$ $\displaystyle:=\\{m\in
M\colon(\alpha+0)(\nu(m)+a_{\circ})=\langle\nu_{M}(m),\alpha\rangle\geq
0,\,\,\forall\alpha\in\Phi_{red}^{\pm}\\}.$

We call $M^{+}$ (resp. $M^{-}$) the set of dominant (resp. antidominant)
elements of $M$. Let ${\Lambda}_{M}^{\pm,\flat}\subset{\Lambda}_{M}^{\flat}$
and $\underline{\Lambda}_{M}^{\pm}\subset\underline{\Lambda}_{M}$ denote the
images of $M^{\pm,\flat}$ by the natural projections
$M\twoheadrightarrow{\Lambda}_{M}^{\pm,\flat}\twoheadrightarrow\underline{\Lambda}_{M}^{\pm}$.
In other words

$\underline{\Lambda}_{M}^{\pm}:=\left((a_{\circ}+\underline{\Lambda}_{M})\cap\mathcal{C}^{\pm}\right)-a_{\circ}\subset\underline{\Lambda}_{M},$

where $\mathcal{C}^{\pm}$ are the two opposite vectorial chambers

$\mathcal{C}^{\pm}:=\\{a\in\mathcal{A}_{ext}\colon(\alpha+0)(a)=\alpha(a-a_{\circ})=\langle
a-a_{\circ},\alpha\rangle\geq 0,\,\,\forall\alpha\in\Phi_{red}^{\pm}\\}.$

######  Remark 3.1.

An element $m\in M$ is in $M^{+}$ if and only if
$\nu_{N}(m)(a_{\circ})=\nu(m)+a_{\circ}\in\overline{\mathcal{C}}^{+}$
(topological closure) and accordingly also
$\nu_{N}(m)(\mathfrak{a})\subset\mathcal{C}^{+}$ since by definition
$\mathfrak{a}\subset\mathcal{C}^{+}$.

###  3.3. Iwahori–Weyl group

Define the $\flat$-Iwahori–Weyl group for $G$ to be

$\widetilde{W}^{\flat}:=N/M^{\flat}.$

Lemma 3.1 simply says that $\ker\nu/M_{1}\subset Z(\widetilde{W})$ the center
of $\widetilde{W}$. We now present two semidirect product decompositions of
the $\flat$-Iwahori–Weyl group. These decompositions will be useful for
studying the Iwahori–Hecke algebra. Here is the first one:

######  Lemma 3.1.

The ${\flat}$-Weyl–Iwahori subgroup has a natural structure of a quasi-Coxeter
group

$\widetilde{W}^{\flat}=W_{\textit{\emph{\text{aff}}}}^{\flat}\rtimes\widetilde{\Omega}^{\flat}\simeq
W_{\textit{\emph{\text{aff}}}}^{\flat}\rtimes\Lambda_{G}^{\flat},$

where $\widetilde{\Omega}^{\flat}:=N_{\mathfrak{a}}^{\dagger}/M^{\flat}$, that
is the subgroup of stabilizers of $\mathfrak{a}$ in $\widetilde{W}^{\flat}$
and $\Lambda_{G}^{\flat}:=G/G_{1}^{\flat}$.

###### Proof.

It suffices to consider the short exact sequence
${1}$${N_{1}^{\flat}/M^{\flat}}$${N/M^{\flat}}$${N/N_{1}^{\flat}}$${1}$which
splits given that $N_{\mathfrak{a}}^{\dagger}/M^{\flat}\simeq
N/N_{1}^{\flat}\simeq G/G_{1}^{\flat}$ (Corollary 2.7). ∎

######  Lemma 3.2.

The ${\flat}$-Iwahori–Weyl group $\widetilde{W}^{\flat}$ admits the following
decomposition

$\widetilde{W}^{\flat}=\Lambda_{M}^{\flat}\rtimes
W_{a_{\circ}}^{\flat}\simeq\Lambda_{M}^{\flat}\rtimes W.$

If $\mathcal{F}\subset\mathcal{A}$ is a facet, we have a bijection of
$\Lambda_{M}^{\flat}$-sets

$\widetilde{W}^{\flat}/W_{\mathcal{F}}^{\flat}\cong\Lambda_{M}^{\flat}\times(W/\jmath_{W}(W_{\mathcal{F}}))$

where, $\jmath_{W}\colon N\to W=N/M$ is the quotient map.

###### Proof.

Using the canonical isomorphism $W\simeq W_{a_{\circ}}$, proved in Lemma 2.4,
one gets a short exact sequence

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which clearly splits, i.e.
$\widetilde{W}^{\flat}\simeq\Lambda_{M}^{\flat}\rtimes W$.

For the bijection statement, fix a set of representatives
$D_{\mathcal{F}}=\\{n_{{I^{\flat}}}^{\mathcal{F}}\in N\colon 1\leq
i\leq|W/\jmath_{W}(W_{\mathcal{F}}^{\flat})|\\}$ for
$W/\jmath_{W}(W_{\mathcal{F}}^{\flat})$. Note that
$\jmath_{W}(W_{\mathcal{F}}^{\flat})=\jmath_{W}(W_{\mathcal{F}})$. Given that
$M\cap K_{\mathcal{F}}^{\flat}=M^{\flat}$ (by Corollary 2.3), one easily shows
that for any $x\in N$ there exist a unique pair
$(m_{x},n_{x})\in\Lambda_{M}^{\flat}\times D_{\mathcal{F}}$ such that
$xN_{1,\mathcal{F}}^{\flat}=m_{x}n_{x}N_{1,\mathcal{F}}^{\flat}$. This
completes the proof of the Lemma. ∎

######  Remark 3.3.

When $M^{\flat}=M^{1}=\nu_{N,ext}(\mathbf{N}(F))$, ${\widetilde{W}}^{1}$ is
called the extended affine Weyl group and usually denoted by
${\widetilde{W}}_{\textit{\emph{\text{aff}}}}$ and it can be viewed as a group
of affine-linear transformations on $\mathcal{A}_{ext}$ marked with
$(a_{\circ},0_{V_{G}})$ as the ”origin”.

######  Remark 3.4.

(i) For any $x\in\widetilde{W}^{\flat}$, we write $m_{x}$ and $w_{x}$ for its
projection in $\Lambda_{M}^{\flat}$ and $W_{a_{\circ}}^{\flat}$, respectively.

(ii) The natural projection morphism
$\square^{\flat}\colon\widetilde{W}\twoheadrightarrow\widetilde{W}^{\flat}$
induces two isomorphisms;
$W_{\textit{\emph{\text{aff}}}}\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}W_{\textit{\emph{\text{aff}}}}^{\flat}$
and
$W_{a_{\circ}}\xrightarrow{\raisebox{-2.5pt}{\smash{$\sim$}}}W_{a_{\circ}}^{\flat}$
for which the following diagram commutes

${\Lambda_{M}\rtimes
W_{a_{\circ}}}$${\widetilde{W}}$${W_{\textit{\emph{\text{aff}}}}\rtimes\widetilde{\Omega}}$${\Lambda_{M}^{\flat}\rtimes
W_{a_{\circ}}^{\flat}}$${\widetilde{W}^{\flat}}$${W_{\textit{\emph{\text{aff}}}}^{\flat}\rtimes\widetilde{\Omega}^{\flat}}$$\scriptstyle{\square^{\flat}\times\text{Id}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\square^{\flat}}$$\scriptstyle{\text{Id}\times\square^{\flat}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

In particular, the projection $\Lambda_{M}\to\Lambda_{M}^{\flat}$ is
$W$-equivariant.

###  3.4. Bruhat and Iwasawa decompositions

Let us recall Iwasawa and Bruhat decompositions for $G$:

######  Proposition 3.1 (The Bruhat and Iwasawa decompositions).

Let $K_{\mathcal{F}}$ resp. $K_{\mathcal{F}^{\prime}}$ be parahoric subgroups
associated with facets $\mathcal{F}$ resp. $\mathcal{F}^{\prime}$ contained in
the apartment associated with $\mathbf{S}$.

$G=U_{\mathcal{F}}NU_{\mathcal{F}^{\prime}}\quad\text{(The Bruhat
decomposition)},$ $G=U^{\pm}NB=U^{\pm}NU_{\mathcal{F}}\quad\text{(The Iwasawa
decomposition)}.$

Moreover, the maps $n\mapsto BnB$, $n\mapsto
K_{\mathcal{F}}^{\flat}nK_{\mathcal{F}^{\prime}}^{\flat}$ and $n\mapsto
U^{\pm}nK_{\mathcal{F}}^{\flat}$ induce the following bijections

$W\cong B\backslash G/B,\quad\,K_{\mathcal{F}}^{\flat}\backslash
G/K_{\mathcal{F}^{\prime}}^{\flat}\cong{W}_{\mathcal{F}}^{\flat}\backslash\widetilde{W}^{\flat}/{W}_{\mathcal{F}^{\prime}}^{\flat}\quad\text{
and }\quad U^{\pm}\backslash
G/K_{\mathcal{F}}^{\flat}\cong\widetilde{W}^{\flat}/W_{\mathcal{F}}^{\flat}.$

###### Proof.

  * •

By Corollary 2.7 and Corollary 3.3 we have
$G=G_{1}N_{\mathfrak{b}}^{\dagger}=I^{\flat}N_{1}^{\flat}I^{\flat}N_{\mathfrak{b}}^{\dagger}$.
But $N_{\mathfrak{b}}^{\dagger}$ normlaizes $I$ (Lemma 2.5) and hence

$G=I^{\flat}N_{1}^{\flat}N_{\mathfrak{b}}^{\dagger}I^{\flat}=U_{\mathfrak{a}}NU_{\mathfrak{a}}=U_{\mathcal{F}}NU_{\mathcal{F}^{\prime}}.$

Accordingly, the map $\widetilde{W}^{\flat}\mapsto
K_{\mathcal{F}}^{\flat}\backslash G/K_{\mathcal{F}^{\prime}}^{\flat}$ is
surjective and factors through
${W}_{\mathcal{F}}^{\flat}\backslash\widetilde{W}^{\flat}/{W}_{\mathcal{F}^{\prime}}^{\flat}$.
Let $w,w^{\prime}\in\widetilde{W}^{\flat}$ (with representatives
$n,n^{\prime}\in N$) such that
$K_{\mathcal{F}}^{\flat}nK_{\mathcal{F}^{\prime}}^{\flat}=K_{\mathcal{F}}^{\flat}n^{\prime}K_{\mathcal{F}^{\prime}}^{\flat}$.
Now $K_{\mathcal{F}}\subset G_{1}$, hence
$\kappa_{G}(nM^{\flat})=\kappa_{G}(n^{\prime}M^{\flat})$ so that $w=\tau\cdot
w_{a}$ and $\tau\cdot w_{a}^{\prime}$ for a unique
$\tau\in\widetilde{\Upomega}^{\flat}$ and $w_{a},w_{a}^{\prime}\in
W_{\textit{\emph{\text{aff}}}}^{\flat}$, by Lemma 3.1. Consequently,
$K_{\tau(\mathcal{F})}^{\flat}w_{a}K_{\mathcal{F}^{\prime}}^{\flat}=K_{\tau(\mathcal{F})}^{\flat}w_{a}^{\prime}K_{\mathcal{F}^{\prime}}^{\flat}$
with $\tau(\mathcal{F})\subset\overline{\mathfrak{a}}$, by definition of
$\widetilde{\Upomega}^{\flat}$. So by (2) Corollary 3.3,
$w_{a}=w_{a}^{\prime}$ which ends the proof of the Bruhat decomposition and
the bijection $K_{\mathcal{F}}\backslash
G/K_{\mathcal{F}^{\prime}}\simeq{W}_{\mathcal{F}}\backslash\widetilde{W}/{W}_{\mathcal{F}^{\prime}}$.

  * •

Let us prove the remaining statements. Let ${\mathfrak{b}}$ be an alcove that
contains in its closure $\mathcal{F}$. By [BT72, Theorem 5.1.3 (vi)] together
with Theorem 2.1, we have $G_{1}=B_{1}N_{1}I_{\mathfrak{b}}$. Using Corollary
2.7 and Lemma 2.5 again we get

$G=B_{1}N_{1}I_{\mathfrak{b}}N_{\mathfrak{b}}^{\dagger}=U^{+}NI_{\mathfrak{b}}=U^{+}NU_{\mathcal{F}}.$

Assume $n,n^{\prime}\in N$ with $n^{\prime}\in
U^{+}nK_{\mathcal{F}}^{\flat}=U^{+}K_{n\cdot\mathcal{F}}^{\flat}n=U^{+}U_{n\cdot\mathcal{F}}^{-}(N\cap
K_{n\cdot\mathcal{F}}^{\flat})n$, hence $n^{\prime}\in(N\cap
K_{n\cdot\mathcal{F}}^{\flat})n=n(N\cap K_{\mathcal{F}}^{\flat})$ (using
$N\cap U^{+}U^{-}=\\{1\\}$). This proves that $U^{+}\backslash
G/K_{\mathcal{F}}^{\flat}\simeq
N/N_{1,\mathcal{F}}^{\flat}=\widetilde{W}^{\flat}/W_{\mathcal{F}}^{\flat}$.
Finally, using the longest element $w_{0}\in W$, one obtains
$G=U^{-}NU_{\mathcal{F}}$ and this yields the corresponding bijection
$U^{-}\backslash
G/K_{\mathcal{F}}^{\flat}\simeq\widetilde{W}^{\flat}/W_{\mathcal{F}}^{\flat}$.

  * •

For $G=\sqcup_{w\in W}BwB$ we refer to [Mil17, Theorem 21.73]. ∎

######  Corollary 3.2 (Iwasawa decomposition II).

If $\mathbf{P}=\mathbf{L}\ltimes\mathbf{U}_{P}^{+}$ is any standard parabolic
subgroup of $\mathbf{G}$ with Levi factor $\mathbf{L}$ and unipotent radical
$\mathbf{U}_{P}^{+}$, then $G=PK$, i.e. $K$ is a good open compact subgroup of
$G$.

###### Proof.

By Proposition 3.1, one has an Iwasawa decomposition $G=U^{+}NK=\cup_{w\in
N/M}U^{+}MwK$. As observed in Lemma 2.4, the classes of the quotient $W=N/M$
admit representatives in $K$, therefore $G=B\cdot K=P\cdot K$. See the proof
of Proposition 3.2 for the factorization $B\cap K=M_{1}(U^{+}\cap K)$.∎

######  Proposition 3.3 (Cartan decomposition for $K$).

The map $M\to K^{\flat}\backslash G/K^{\flat}$ defined by $m\mapsto
K^{\flat}mK^{\flat}$, induces a bijection

$\Lambda_{M}^{-,{\flat}}\cong W^{\flat}\backslash\Lambda_{M}^{\flat}\cong
W_{a_{\circ}}^{\flat}\backslash\widetilde{W}^{\flat}/W_{a_{\circ}}^{\flat}\cong
K^{\flat}\backslash G/K^{\flat}.$

###### Proof.

It suffices to prove the Lemma for $M^{\flat}=M_{1}$. In which case, the first
bijection is [HV15, §6.3 Lemma], the second is a consequence of the
decomposition of Lemma 3.2 and the third is given by Proposition 3.1 for
$\mathcal{F}=\\{a_{\circ}\\}$. ∎

###  3.5. Iwahori decompositions

We now give an Iwahori decomposition for the special parahoric subgroup $K$.
Note that we are no longer in the unramified case where we could have pulled
up the Bruhat decomposition for the residue field of $F$.

######  Proposition 3.1 (Iwahori decomposition of $K$ \- I).

Let $\mathcal{F}$ be a facet of the alcove $\mathfrak{a}$. We have the
decomposition

$K_{\mathcal{F}}^{\flat}=\bigsqcup_{w\in{W}_{\mathcal{F}}^{\flat}}I^{\flat}wI^{\flat}.$

###### Proof.

This is an immediate consequence of
$G=\bigsqcup_{w\in\widetilde{W}^{\flat}}I^{\flat}wI^{\flat}$ (Proposition 3.1
for $\mathcal{F}=\mathcal{F}^{\prime}=\mathfrak{a}$). ∎

######  Proposition 3.2 (Iwahori decomposition of $K$ \- II).

Let $\mathcal{F}$ be a facet of the alcove $\mathfrak{a}$. We have the
decomposition

$K_{\mathcal{F}}^{\flat}=\bigsqcup_{w\in
W_{\mathcal{F}}^{\flat}}U_{\mathcal{F}}^{+}wI^{\flat}=\bigsqcup_{w\in
W_{\mathcal{F}}^{\flat}}U_{\mathcal{F}}^{-}wI^{\flat}.$

###### Proof.

By the Iwahori decomposition $K_{\mathcal{F}}^{\flat}=U_{\mathcal{F}}^{+}\cdot
U_{\mathcal{F}}^{-}\cdot N_{1,\mathcal{F}}^{\flat}$ one shows $(U^{\pm}\cdot
N)\cap K_{\mathcal{F}}^{\flat}=U_{\mathcal{F}}^{\pm}\cdot
N_{1,\mathcal{F}}^{\flat}$. Combining this equality with the decomposition
$G=\sqcup_{w\in\widetilde{W}^{\flat}}U^{\pm}wI^{\flat}$ (Proposition 3.1), it
becomes clear that

$\displaystyle K_{\mathcal{F}}^{\flat}$
$\displaystyle=U_{\mathcal{F}}^{\pm}\cdot N_{1,\mathcal{F}}^{\flat}\cdot
I^{\flat}=\sqcup_{w\in
W_{\mathcal{F}}^{\flat}}U_{\mathcal{F}}^{\pm}wI^{\flat}.\qed$

###  3.6. Conjugations

######  Lemma 3.1.

Let $\mathcal{F}$ be a facet of $\mathfrak{a}$ that contains in its closure
$a_{\circ}$. For any $n\in N_{1,a_{\circ}}$, we have

$nU_{\mathcal{F}}^{-}n^{-1}\subset K_{\mathcal{F}}\text{ and
}(nK_{\mathcal{F}}n^{-1})\cap U^{+}\subset I^{+}.$

###### Proof.

Let $n\in N_{1,a_{\circ}}$, with image $w$ in $W$. By Lemma 2.2 we have
$nU_{\alpha+f_{\mathcal{F}}(\alpha)}n^{-1}=U_{\beta},$ where,
$\beta\colon\mathcal{A}\to\mathds{R}$ is the affine map
$w(\alpha)+f_{\mathcal{F}}(\alpha)-w(\alpha)(\nu_{N}(n)(a_{\circ})-a_{\circ})$,
thus $\beta=w(\alpha)+f_{\mathcal{F}}(\alpha)$ since $n$ fixes $a_{\circ}$. By
Proposition 2.9, we have for any fixed ordering $\prec$ of
$\Phi^{\pm}\cap\Phi_{red}$

$nU_{\mathcal{F}}^{\pm}n^{-1}=\prod_{\alpha\in\Phi^{\pm}\cap\Phi_{red}}^{\prec}U_{w(\alpha)+f_{\mathcal{F}}(\alpha)}.$

If $w(\alpha)\in\Phi_{red}\cap\Phi^{+}$ then
$U_{w(\alpha)+n_{\alpha}^{-1}}\subset U_{w(\alpha)+0}\subset K_{\mathcal{F}}$
(Example 2.6). If $w(\alpha)\in\Phi_{red}\cap\Phi^{-}$, then
$U_{w(\alpha)+n_{\alpha}^{-1}}=U_{w(\alpha)+n_{w(\alpha)}^{-1}}\subset
K_{\mathcal{F}}$. (Lemma 2.9). Therefore, $nU_{\mathcal{F}}^{-}n^{-1}\subset
K_{\mathcal{F}}.$

For the second equality, note that $(nK_{\mathcal{F}}n^{-1})\cap
U^{+}=U_{\nu_{N}(n)(\mathcal{F})}^{+}\subset U_{a_{\circ}}^{+}=I^{+}$ by
Corollary 3.4. ∎

######  Lemma 3.2.

Let ${\Upomega}\subset\mathcal{A}$ be any bounded subset and $\Psi$ any closed
subset of roots in $\Phi^{+}\cap\Phi_{red}$. For any $m\in M^{-}$, we have

$m(U_{\Psi}\cap K_{\Upomega})m^{-1}\subset U_{\Psi}\cap K_{\Upomega}\text{ and
}m^{-1}(U_{-\Psi}\cap K_{\Upomega})m\subset U_{-\Psi}\cap K_{\Upomega}.$

In particular, if $\Psi=\Phi^{+}\cap\Phi_{red}$ one gets
$m^{-1}U_{\Upomega}^{-}m\subset U_{\Upomega}^{-}\text{ and
}m\,U_{\Upomega}^{+}m^{-1}\subset U_{\Upomega}^{+}$.

###### Proof.

For any $r\in\Gamma_{\alpha}$, we have by Lemma 2.2

$\displaystyle mU_{\alpha+r}m^{-1}$
$\displaystyle=U_{\alpha+r-\alpha(\nu_{N}(m)(a_{\circ})-a_{\circ})}=U_{\alpha+r-\langle\nu(m),\alpha\rangle}$

and so, for all $r\in\Gamma_{\alpha}$

$mU_{\alpha+r}m^{-1}\begin{cases}\subset U_{\alpha+r}&\text{if