Before the Concept · I of II

Groups and the affine Weyl group

The local field, the group $\GF$, its two compact open subgroups $K\supset I$, and the group $\Wt$ that indexes the double cosets of $I$. Each object is defined, then shown for the running example $\SL_2$.

Table 1. The objects of this page and their types.
object	type	defined in
$F$, $\cO$, $t$, $q$	a local field, its integers, a uniformizer, the size of the residue field	§1
$G\supset B\supset T$	a split reductive group, a Borel subgroup, a maximal torus, all over $\cO$	§1
$\ev:K\to G(\Fq)$	reduction modulo $t$ on $K=G(\cO)$	§1
$I\subset K$	the Iwahori subgroup, a compact open subgroup of $\GF$	§2
$\Wt$, $\Waff$, $\Om$	groups: $\Xs\rtimes W$, its Coxeter subgroup, the stabilizer of an alcove	§3, §5
$\len:\Wt\to\N$	the length function	§5
$\GF=\bigsqcup_{w\in\Wt}IwI$	the Iwahori–Bruhat decomposition	§6

The local field and the groups Definition

(the setting)

Let $q$ be a prime power and

$$F=\Fq((t)),\qquad\cO=\Fq[[t]],\qquad\cO/t\cO=\Fq .$$

Let $G$ be a split connected reductive group over $\cO$ with a split maximal torus $T$ and a Borel subgroup $B=TU$ containing it, $U$ its unipotent radical. Write $W=N_G(T)/T$ for the finite Weyl group and $\Xs=\Hom(\mathbb G_m,T)$ for the cocharacter lattice. Put

$$K=G(\cO)\subset\GF,\qquad\ev:K\longrightarrow G(\Fq)\ \text{ the reduction modulo }t .$$

$F$ is a local field: complete for the $t$-adic absolute value, with finite residue field. Only two properties of it are used on these pages: $\cO$ is compact and open in $F$, and $\cO/t^n\cO$ is finite with $q^n$ elements. So $\GF$ is a locally compact totally disconnected group, and $K$ is a compact open subgroup. The theory is the same for the $p$-adic fields of the sources [HKP10, p. 1].

In the running example $G=\SL_2$, $T$ the diagonal matrices, $B$ the upper triangular ones, $U$ the unipotent upper triangular ones. $W=\{e,s_1\}$ with $s_1$ represented by $\dot s_1=\sm{0}{1}{-1}{0}$, and $\Xs=\Z$ with $\mu\mapsto\sm{t^{\mu}}{0}{0}{t^{-\mu}}$. $K=\SL_2(\cO)$. Its own page.

The Iwahori subgroup Definition

(Iwahori subgroup) [HKP10, p. 1], [IM65, §1.1]

$$I=\ev^{-1}\big(B(\Fq)\big)=\{g\in G(\cO):\ \ev(g)\in B(\Fq)\}\ \subset\ K .$$

$I$ is open in $K$ because it contains the kernel of $\ev$, and it has finite index: $K/I\simeq G(\Fq)/B(\Fq)$, the flag variety over the residue field. Iwahori and Matsumoto write $B$ for it and define it by generators [IM65, §1.1]; the two definitions agree by their Theorem 2.5, which is the Iwahori factorization $I=U^-(t\cO)\,T(\cO)\,U(\cO)$ in the notation of SL₂ I §4.

(index of $I$ in $K$) Proved here

$$[K:I]=|G(\Fq)/B(\Fq)|=\sum_{w\in W}q^{\len(w)} .$$

Proof

$\ev$ is surjective because $G$ is smooth over $\cO$ (Hensel's lemma), so $K/I\to G(\Fq)/B(\Fq)$ is a bijection. The Bruhat decomposition $G(\Fq)=\bigsqcup_{w\in W}B(\Fq)wB(\Fq)$ and the count $|B(\Fq)wB(\Fq)/B(\Fq)|=q^{\len(w)}$ (the big cell of $wB/B$ is an affine space of dimension $\len(w)$) give the sum. The same number appears as $W(q)$ in [HKP10, proof of Thm 5.6.1, p. 15].

In the running example $I=\Big\{\sm{a}{b}{c}{d}\in\SL_2(\cO):\ c\in t\cO\Big\}$, and $[K:I]=|\mathbb P^1(\Fq)|=q+1=1+q^{\len(s_1)}$.

The extended affine Weyl group Definition

(extended affine Weyl group) [HKP10, §1.1]

$$\Wt=N_G(T)(F)\,/\,T(\cO)\ \simeq\ \Xs\rtimes W,\qquad \mu\longmapsto t^{\mu}:=\mu(t)\in T(F).$$

The map $\mu\mapsto t^\mu$ identifies $\Xs$ with $T(F)/T(\cO)$, the translation subgroup; $W=N_G(T)(\cO)/T(\cO)$ is a complement.

Elements of $\Wt$ are not elements of $\GF$, but every $w\in\Wt$ has representatives $\dot w\in N_G(T)(F)$, and any two differ by an element of $T(\cO)\subset I$. So the double coset $I\dot wI$ depends only on $w$; it is written $IwI$.

In the running example $\Wt=\Z\rtimes\{e,s_1\}$, the infinite dihedral group, with $s_1$ acting on $\Z$ by $\mu\mapsto-\mu$. The translation $t^{\mu}$ is $\sm{t^{\mu}}{0}{0}{t^{-\mu}}$ and $s_1t^{\mu}s_1=t^{-\mu}$.

Alcoves and the affine Weyl group Definition

$\Wt$ acts on the real vector space $V=\Xs\otimes\R$ by affine maps: $W$ linearly and $\mu\in\Xs$ by translation $x\mapsto x+\mu$. For each root $\alpha$ and integer $k$, the affine root hyperplane is

$$H_{\alpha,k}=\{x\in V:\ \langle\alpha,x\rangle=k\},$$

and the reflection in it is $s_{\alpha,k}(x)=x-(\langle\alpha,x\rangle-k)\alpha^\vee=t^{k\alpha^\vee}s_\alpha$ [IM65, §1.1].

(alcoves, $\Waff$) [IM65, §1.3–1.4], [HKP10, §7.1]

An alcove is a connected component of $V\smallsetminus\bigcup_{\alpha,k}H_{\alpha,k}$. The fundamental alcove and the affine Weyl group are

$$A_0=\{x\in V:\ -1<\langle\alpha,x\rangle<0\ \text{ for every positive root }\alpha\},\qquad \Waff=\Qv\rtimes W\subset\Wt,$$

where $\Qv\subset\Xs$ is the coroot lattice. $\Waff$ is generated by the reflections in the walls of $A_0$: the simple reflections $S$ of $W$ together with $s_0=s_{\tilde\alpha,-1}=t^{-\tilde\alpha^\vee}s_{\tilde\alpha}$ for the highest root $\tilde\alpha$ (one such reflection per simple factor of $G$). Write $\Saff=S\cup\{s_0\}$.

$A_0$ is taken in the negative Weyl chamber, with $0$ in its closure, because $I$ is the inverse image of $B$ and not of the opposite Borel subgroup: with this choice the walls of $A_0$ are the reflections $s$ whose double cosets $IsI$ are the minimal ones outside $I$, as the Concept shows [HKP10, footnote 2, p. 3, and §7.1]. Defining $I$ with the opposite Borel subgroup would replace $A_0$ by $-A_0$.

(simple transitivity) [IM65, Prop 1.2, Cor 1.8, Cor 1.16] Proved (cited)

$\Waff$ is generated by $\Saff$ and acts simply transitively on the set of alcoves; it is a Coxeter group with the relations $s^2=1$ and the braid relations $(st)^{m_{st}}=1$ determined by the angles between the walls of $A_0$.

Proof

Proof idea

Transitivity: an alcove is moved to $A_0$ by successively reflecting in a wall of $A_0$ that separates it from $A_0$, which strictly decreases the number of hyperplanes between them [IM65, proof of Prop 1.1]. Freeness: an element fixing $A_0$ separates it from itself by no hyperplane, and a non-identity reduced word always crosses at least one [IM65, Lemma 1.7, Cor 1.8]. The presentation is [IM65, Cor 1.16].

In the running example $V=\R$ and it is convenient to use the coordinate $y=\langle\alpha,x\rangle$, so $\alpha^\vee$ translates by $2$. The hyperplanes are the integers, the alcoves are the open unit intervals, $A_0=(-1,0)$, and $$s_1(y)=-y,\qquad s_0(y)=-2-y=t^{-\alpha^\vee}s_1(y),$$ $$s_1s_0=t^{\alpha^\vee},\qquad \Waff=\langle s_0,s_1\rangle=\Wt .$$

Figure 1. The apartment of $\SL_2$ in the coordinate $y=\langle\alpha,x\rangle$: walls at the integers, alcoves between them, $A_0=(-1,0)$ shaded. The reflections in its two walls generate $\Waff$, and their product is a translation by $\pm2$, that is by $\pm\alpha^\vee$.

Length, reduced words, and the length-zero subgroup $\Om$ Definition

(length, $\Om$) [IM65, §1.4, §1.7]

$$\len(w)=\#\{H_{\alpha,k}:\ H_{\alpha,k}\ \text{separates}\ A_0\ \text{from}\ wA_0\},\qquad \Om=\{w\in\Wt:\ wA_0=A_0\}=\{w\in\Wt:\len(w)=0\}.$$

A reduced expression of $w\in\Waff$ is a word $w=s_{i_1}\cdots s_{i_r}$ in $\Saff$ of minimal length $r$.

(length is word length; $\Wt=\Waff\rtimes\Om$) [IM65, Prop 1.10, §1.7] Proved (cited)

(i)For $w\in\Waff$, $\len(w)$ is the length of a reduced expression of $w$.

(ii)For $s\in\Saff$ and $w\in\Wt$, $\len(sw)=\len(w)\pm1$.

(iii)$\Wt=\Waff\rtimes\Om$, $\Om$ normalizes $\Saff$ (it permutes the walls of $A_0$), $\len(\sigma w\sigma')=\len(w)$ for $\sigma,\sigma'\in\Om$, and $\Om\simeq\Xs/\Qv$, a finite abelian group.

Proof

Proof idea

(i) Each letter of a word crosses one wall, so $\len(w)\le$ word length; conversely a word of length $\len(w)$ is produced by the reflection procedure of [IM65, Cor 1.6, Prop 1.10]. (ii) The wall $H_s$ of $A_0$ either separates $A_0$ from $wA_0$ or not, and $s$ changes exactly that one count [IM65, Lemma 1.5]. (iii) Simple transitivity gives $\Wt=\Om\cdot\Waff$ with $\Om\cap\Waff=1$; the quotient $\Wt/\Waff=\Xs/\Qv$ [IM65, §1.7].

$\Om$ is the fundamental group of $G$ in the sense of root data: trivial when $G$ is simply connected, $\Z/(n)$ for $\PGL_n$, $\Z$ for $\GL_n$. The extended group $\Wt$ is not a Coxeter group when $\Om\neq1$, but it is one up to the finite group $\Om$, which is why the Hecke algebra is a twisted tensor product (the Concept).

In the running example Every element of $\Wt=\langle s_0,s_1\rangle$ is an alternating word, and the word is its own reduced expression: $\len(s_1s_0s_1)=3$, $\len(t^{n\alpha^\vee})=\len((s_0s_1)^n)=2|n|$. $\Om=1$ because $\Xs=\Z\alpha^\vee=\Qv$. The grown example $\PGL_2$ has $\Om=\Z/2$ (SL₂ III).

The Iwahori–Bruhat decomposition Proved (cited)

(Iwahori–Bruhat decomposition) [IM65, Thm 2.16] Proved (cited)

$$\GF=\bigsqcup_{w\in\Wt}I\,w\,I .$$

Moreover, for $s\in\Saff$ and $w\in\Wt$ [IM65, Prop 2.8]:

(a)if $\len(sw)=\len(w)+1$ then $IsI\cdot IwI=IswI$;

(b)if $\len(sw)=\len(w)-1$ then $IsI\cdot IwI=IwI\sqcup IswI$;

(c)for $\sigma\in\Om$, $\sigma$ normalizes $I$, so $I\sigma I=\sigma I=I\sigma$ [IM65, Prop 2.10, Cor 2.19].

Proof

Proof idea, and where it is proved in full

Iwahori and Matsumoto prove it for Chevalley groups. Every $g\in\GF$ lies in some $K\,t^{\mu}K$ (Cartan decomposition), and $K=\bigsqcup_{w\in W}IwI$ by the Bruhat decomposition of $G(\Fq)$ pulled back along $\ev$; the products of double cosets are controlled by the rank-one computation in $\SL_2$ applied to each simple affine root, which is (a) and (b) [IM65, Prop 2.6, Prop 2.8]. Disjointness is an induction on length using (a) and (b) [IM65, Prop 2.13]. For $\SL_2$ the whole statement is checked by hand on SL₂ I §2–3. For a general split reductive group it is part of Bruhat–Tits theory, and [HKP10, §1.2] takes it as known.

History: 1965

How these objects meet the next page Context

The decomposition says that the functions on $\GF$ invariant under $I$ on both sides have a basis of indicator functions $T_w=\one_{IwI}$, one per element of $\Wt$. The next page makes these functions into an algebra by convolution, and the Concept computes the products $T_sT_w$ from (a)–(c) and the count $|IwI/I|=q^{\len(w)}$.