Spherical Functions on a Group of $p\text{-adic}$ Type

Last update: 13 February 2014

Chapter II: Groups of $p\text{-adic}$ type

Root systems

Let $A$ be a real vector space of finite dimension $l>0,$ and $⟨x,y⟩$ a positive definite scalar product on $A\text{.}$ Let ${A}^{*}$ be the dual vector space. Then the scalar product defines an isomorphism of $A$ onto ${A}^{*},$ and hence a scalar product $⟨a,b⟩$ on ${A}^{*}\text{.}$ For each non-zero $a\in {A}^{*}$ let ${a}^{\vee }\in A$ be the image of $2a/⟨a,a⟩$ under this isomorphism. Let ${h}_{a}$ denote the kernel of $a,$ and let ${w}_{a}$ be the reflection in the hyperplane ${h}_{a}\text{:}$ $wa(x)=x-a (x)a∨ (x∈A).$ We make ${w}_{a}$ act also on ${A}^{*},$ by transposition: ${w}_{a}\left(b\right)=b\circ {w}_{a}\text{.}$

A root system ${\Sigma }_{0}$ in ${A}^{*}$ is a subset ${\Sigma }_{0}$ of ${A}^{*}$ satisfying the following axioms:

 (RS 1) ${\Sigma }_{0}$ is finite, spans ${A}^{*}$ and does not contain $0\text{;}$ (RS 2) $a\in {\Sigma }_{0}⇒{w}_{a}\left({\Sigma }_{0}\right)={\Sigma }_{0}\text{;}$ (RS 3) $a,b\in {\Sigma }_{0}⇒a\left({b}^{\vee }\right)\in ℤ\text{.}$

We shall assume throughout that ${\Sigma }_{0}$ is reduced:

 (RS 4) If $a,b\in {\Sigma }_{0}$ are proportional, then $b=±a\text{;}$
and irreducible:
 (RS 5) If ${\Sigma }_{0}={\Sigma }_{1}\cup {\Sigma }_{2}$ such that $⟨{a}_{1},{a}_{2}⟩=0$ for all ${a}_{1}\in {\Sigma }_{1}$ and ${a}_{2}\in {\Sigma }_{2},$ then either ${\Sigma }_{1}$ or ${\Sigma }_{2}$ is empty.

The elements of ${\Sigma }_{0}$ are called roots. The group ${W}_{0}$ generated by the reflections ${w}_{a}$ $\left(a\in {\Sigma }_{0}\right)$ is a finite group of isometries of $A,$ called the Weyl group of ${\Sigma }_{0}\text{.}$ The connected components of the set $A-\bigcup _{a\in {\Sigma }_{0}}{h}_{a}$ are open simplicial cones called the chambers of ${\Sigma }_{0}\text{.}$ These chambers are permuted simply transitively by ${W}_{0},$ and the closure of each chamber is a fundamental region for ${W}_{0}\text{.}$

Choose a chamber ${C}_{0}\text{.}$ This choice determines a set ${\Pi }_{0}$ of $l$ roots such that ${C}_{0}$ is the intersection of the open half-spaces $\left\{x\in A:a\left(x\right)>0\right\}$ for $a\in {\Pi }_{0}\text{.}$ The elements of ${\Pi }_{0}$ are called the simple roots (relative to the chamber ${C}_{0}\text{).}$ A root $b$ is said to be positive or negative according as it is positive or negative on ${C}_{0}\text{.}$ Let ${\Sigma }_{0}^{+}$ (resp. ${\Sigma }_{0}^{-}\text{)}$ denote the set of positive (resp. negative) roots.

Each root $b$ can be expressed as a linear combination of the simple roots: $b=∑a∈Π0 maa$ with coefficients ${m}_{a}\in ℤ$ all of the same sign. The integer $\sum {m}_{a},$ the sum of the coefficients, is called the height of $b\text{.}$ There is a unique root (the 'highest root') whose height is maximal.

Let ${C}_{0}^{\perp }$ denote the 'obtuse cone' $C0⊥= { x∈A:⟨x,y⟩ ≥0 for all y∈C‾0 } .$ If $x\in {\stackrel{‾}{C}}_{0}$ and $w\in {W}_{0},$ then $x-wx\in {C}_{0}^{\perp }\text{.}$

A subset ${\Phi }_{0}$ of ${\Sigma }_{0}$ is said to be closed if $a,b∈Φ0 and a+b∈Σ0 ⇒a+b∈ Φ0.$ Suppose ${\Phi }_{0}$ is closed and ${\Phi }_{0}\cap -{\Phi }_{0}=\varnothing \text{.}$ Then there exists $w\in {W}_{0}$ such that $w{\Phi }_{0}\subseteq {\Sigma }_{0}^{+}\text{.}$

The Weyl group ${W}_{0}$ is generated by the reflections ${w}_{a}$ for $a\in {\Pi }_{0}\text{.}$ Hence any $w\in W$ can be written as a product $w={w}_{1}\cdots {w}_{r},$ where ${w}_{i}={w}_{{a}_{i}}$ and ${a}_{i}\in {\Pi }_{0}$ $\left(1\le i\le r\right)\text{.}$ If the number of factors ${w}_{i}$ is as small as possible, then ${w}_{1}\cdots {w}_{r}$ is a reduced word for $w,$ and $r$ is the length of $w,$ written $l\left(w\right)\text{.}$ The length of $w$ is also equal to the number of positive roots $b$ such that ${w}^{-1}b$ is negative. More precisely, these roots are ${b}_{i}={w}_{1}\cdots {w}_{i-1}{a}_{i}$ $\left(1\le i\le r\right)\text{.}$ (All this of course is relative to the choice of the chamber ${C}_{0}\text{.)}$

In particular, there exists a unique element ${w}_{0}$ of ${W}_{0}$ which sends every positive root to a negative root. This leads to the following result, which will be useful later:

The positive roots can be arranged in a sequence $\left({b}_{1},\dots ,{b}_{n}\right)$ such that for each $r=0,1,\dots ,n$ the set of roots ${ -b1,-b2,…, -br,br+1, br+2,…,bn }$ is the set of positive roots relative to some chamber ${C}_{r}\text{.}$

 Proof. Let ${w}_{1}{w}_{2}\cdots {w}_{n}$ be a reduced word for ${w}_{0},$ where ${w}_{i}={w}_{{a}_{i}},$ ${a}_{i}\in {\Pi }_{0},$ and as before put ${b}_{i}={w}_{1}{w}_{2}\cdots {w}_{i-1}{a}_{i}$ $\left(1\le i\le n\right)\text{.}$ For each $r,$ ${w}_{r+1}\cdots {w}_{n}$ is a reduced word, and therefore from the facts quoted above ${ ar+1,wr+1 ar+2,…, wr+1wr+2… wn-1an } =Σ0+∩ ( wr+1…wn Σ0- ) .$ Consequently ${ br+1, br+2,…, bn } = (w1⋯wrΣ0+) ∩(w0Σ0-) = (w1⋯wrΣ0+) ∩Σ0+$ and therefore ${b1,b2,…,br} =(w1⋯wrΣ0-) ∩Σ0+$ so that ${-b1,-b2,…,-br} =(w1⋯wrΣ0+) ∩Σ0-$ and finally ${-b1,…,-br,br+1,…,bn} =w1⋯wrΣ0+$ is the set of positive roots for the chamber ${C}_{r}={w}_{1}\cdots {w}_{r}{C}_{0}\text{.}$ $\square$

Affine roots

We retain the notation of (2.1). For each $a\in {\Sigma }_{0}$ and $k\in ℤ$ we have an affine-linear function $a+k$ on $A$ (namely, $x↦a\left(x\right)+k\text{).}$ Let $Σ={a+k:a∈Σ0,k∈ℤ}.$ The elements of $\Sigma$ are called affine roots. We shall denote them by Greek letters $\alpha ,\beta ,\dots \text{.}$ If $\alpha$ is an affine root, so are $-\alpha$ and $\alpha +k$ for all integers $k\text{.}$ If $\alpha =a+k$ then $a$ $\left(\in {\Sigma }_{0}\right)$ is the gradient of $\alpha$ (in the usual sense of elementary calculus).

For each $\alpha \in \Sigma ,$ let ${h}_{\alpha }$ be the (affine) hyperplane on which $\alpha$ vanishes, and let ${w}_{\alpha }$ be the reflection in ${h}_{\alpha }\text{.}$ (From now on, only the affine structure and not the vector-space structure of $A$ will come into play). The group $W$ generated by the ${w}_{\alpha }$ as $\alpha$ runs through $\Sigma$ is an infinite group of displacements of $A,$ called the Weyl group of $\Sigma$ or the affine Weyl group. ${W}_{0}$ is the subgroup of $W$ which fixes the point $0\text{.}$ The translations belonging to $W$ form a free abelian group $T$ of rank $l,$ and $W$ is the semi-direct product of $T$ and ${W}_{0}\text{.}$

For each $a\in {\Sigma }_{0}$ let $ta=wa∘ wa+1∈T.$ The ${t}_{a}$ $\left(a\in {\Pi }_{0}\right)$ are a basis of $T\text{.}$ We have ${t}_{a}\left(0\right)={a}^{\vee },$ and the mapping $T\to A$ defined by $t↦t\left(0\right)$ maps $T$ isomorphically onto the lattice spanned by the ${a}^{\vee }$ for $a\in {\Sigma }_{0}\text{.}$

The affine Weyl group acts on $\Sigma$ by transposition: if $w\in W,$ $\alpha \in \Sigma$ then $w\left(\alpha \right)$ is defined to be $\alpha \circ {w}^{-1}\text{.}$

The connected components of the set $A-\bigcup _{\alpha \in \Sigma }{h}_{\alpha }$ are open rectilinear $l\text{-simplexes}$ called the chambers of $\Sigma \text{.}$ The chambers are permuted simply transitively by $W,$ and the closure of each chamber is a fundamental region for $W\text{.}$ The (non-empty) faces of the chambers are locally closed simplexes, called facets. Each point $x\in A$ lies in a unique facet $F\text{.}$ The subgroup of $W$ which fixes $x$ also fixes each point of $F\text{.}$

Choose a chamber $C\text{.}$ This choice determines a set $\Pi$ of $l+1$ affine roots, such that $C$ is the intersection of the open half-spaces $\left\{x\in A:\alpha \left(x\right)>0\right\}$ for $\alpha \in \Pi \text{.}$ The affine roots belonging to $\Pi$ are called the simple affine roots (relative to the chamber $C\text{).}$

The affine Weyl group $W$ is generated by the relations ${w}_{\alpha }$ for $\alpha \in \Pi \text{.}$ As in the case of ${W}_{0},$ we define the length $l\left(w\right)$ of an element $w$ of $W\text{.}$ The length of $w$ is also equal to the number of affine roots $\alpha$ which are positive on $C$ and negative on $wC,$ i.e. to the number of hyperplanes ${h}_{\alpha }$ which separate $C$ and $wC\text{.}$

If $F$ is a facet of $C,$ then the subgroup of $W$ which fixes a point of $F$ is generated by the ${w}_{\alpha }$ $\left(\alpha \in \Pi \right)$ which fix $x\text{.}$

We shall need the following result later:

Let $\Phi$ be a set of affine roots whose gradients are all positive and distinct. Then there exists a translation $t\in T$ such that each $\alpha \in \Phi$ is positive on $t\left({C}_{0}\right)\text{.}$

 Proof. Let the elements of $\Phi$ be $a+{r}_{a}$ $\left(a\in {\Sigma }_{0}^{+}\right)$ and let $t\in T\text{.}$ We have $(a+ra)(t(x)) =a(x+t(0))+ ra=a(x)+a (t(0))+ra.$ If $x\in {C}_{0}$ then $a\left(x\right)>0$ for all positive roots $a\text{.}$ Hence $a+{r}_{a}$ will be positive on $t\left({C}_{0}\right)$ provided that $a\left(t\left(0\right)\right)+{r}_{a}\ge 0\text{.}$ Hence we have only to choose $t$ so that $t\left(0\right)$ satisfies $a(t(0))≥- ra$ for all $a\in {\Sigma }_{0}^{+}$ which occur as gradients of elements of $\Phi ,$ and this is certainly possible. $\square$

For later use, it will be convenient to choose $C$ to be the unique chamber of $\Sigma$ which is contained in the cone ${C}_{0}$ and has the origin $0$ as one vertex. The following lemma presupposes this choice of $C\text{.}$

For each $w\in W,$ let ${\Phi }_{w}$ be the set of affine roots which are positive on $C$ and negative on $wC\text{.}$

Let $w$ be an element of $W$ such that $C\subset w{C}_{0}\text{.}$ Then for all $w\prime \in {W}_{0}$ we have $Φww′= Φw⊔wΦw′.$

 Proof. If $\beta \in {\Phi }_{w}$ is positive on $ww\prime C,$ then $\alpha ={w}^{-1}\beta$ is negative on $C$ and positive on $w\prime C,$ hence $\alpha \left(0\right)=0$ (because the origin $0$ belongs to the intersection of the closures of $C$ and $w\prime C\text{).}$ Consequently $\alpha$ is negative on ${C}_{0},$ hence $\beta$ is negative on $w{C}_{0}$ and therefore on $C\text{:}$ contradiction. Hence $\beta$ must be negative on $ww\prime {C}_{0},$ i.e. $\beta \in {\Phi }_{ww\prime }\text{.}$ Next, let $\alpha \in {\Phi }_{w\prime }$ and put $\beta =w\alpha \text{.}$ Then $\beta \left(w{C}_{0}\right)>0$ and $\beta \left(ww\prime {C}_{0}\right)<0,$ hence $\beta$ is positive on $C$ and negative on $ww\prime C,$ i.e. $\beta \in {\Phi }_{ww\prime }\text{.}$ Hence ${\Phi }_{w}\cup w{\varphi }_{w\prime }\subset {\Phi }_{ww\prime }\text{.}$ Conversely, let $\beta \in {\Phi }_{ww\prime }\text{.}$ If $\beta \left(wC\right)<0,$ then $\beta \in {\Phi }_{w}\text{;}$ and if $\beta \left(wC\right)>0,$ then ${w}^{-1}\beta \in {\Phi }_{w}\text{.}$ This establishes the opposite inclusion. Finally, the sets ${\Phi }_{w}$ and $w{\Phi }_{w\prime }$ are disjoint, because $\beta \left(wC\right)<0$ for all $\beta \in {\Phi }_{w}$ and $\beta \left(wC\right)>0$ for all $\beta \in w{\Phi }_{w\prime }\text{.}$ $\square$

Before going any further, it may be helpful to have a concrete example of the set-up so far:

(Root system of type ${A}_{l}\text{).}$

Let $A$ be the set of all $x=\left({x}_{0},\dots ,{x}_{l}\right)\in {ℝ}^{l+1}$ such that $\sum _{i=0}^{l}{x}_{i}=0\text{.}$ The scalar product is $⟨x,y⟩=\sum {x}_{i}{y}_{i}\text{.}$ Let ${e}_{i}$ be the $i\text{th}$ coordinate function on $A$ $\left({e}_{i}\left(x\right)={x}_{i}\right)$ and let $Σ0= {ei-ej:i≠j}.$ Then ${\Sigma }_{0}$ satisfies (RS 1)-(RS 5). The Weyl group ${W}_{0}$ acts by permuting the coordinates ${x}_{0},\dots ,{x}_{l}$ of $x\in A,$ hence is isomorphic to the symmetric group ${S}_{l+1},$ of order $\left(l+1\right)!\text{.}$ We may take the chamber ${C}_{0}$ to be $C0= { x∈A:x0>x1> ⋯>xl }$ and the simple roots are then ${e}_{i-1}-{e}_{i}$ $\left(1\le i\le l\right)\text{.}$ The chamber $C$ is the open simplex $C= { x∈A:1+xl>x0 >x1>⋯>xl } .$

BN-pairs

Let $G$ be a group, $B$ and $N$ subgroups of $G,$ and $R$ a subset of the coset space $N/\left(B\cap N\right)\text{.}$ The pair $\left(B,N\right)$ is a BN-pair in $G$ (or a Tits system) if it satisfies the following four axioms:

 (BN 1) $G$ is generated by $B$ and $N,$ and $H=B\cap N$ is normal in $N\text{.}$ (BN 2) $R$ generates the group $W=N/H$ and each $r\in R$ has order $2\text{.}$ (BN 3) $rBw\subset BwB\cup BrwB,$ for all $r\in R$ and $w\in W\text{.}$ (BN 4) $rBr\not\subset B,$ for each $r\in R\text{.}$

The group $W=N/H$ is the Weyl group of $\left(B,N\right)\text{.}$ The elements of $W$ are cosets of $H,$ and an expression such as $BwB$ is to be read in the usual sense as a product of subsets of $G\text{.}$ If $S$ is a subset of $W,$ then $BSB$ means $\bigcup _{w\in S}BwB\text{.}$

One shows that the set $R$ is uniquely determined by $B$ and $N\text{:}$ namely an element $w\in W$ belongs to $R$ if and only if $B\cup BwB$ is a group. The elements of $R$ are called the distinguished generators of $W\text{.}$

The axioms (BN 1)-(BN 4) have the following consequences. The proofs may be found in [Bou1968].

2.3.1 $G=BWB,$ and the mapping $w↦BwB$ is a bijection of $W$ onto the double coset space $B\G/B\text{.}$

2.3.2 For each subset $S$ of $R$ let ${W}_{S}$ be the subgroup of $W$ generated by $S,$ and let ${P}_{S}=B{W}_{S}B\text{.}$ Then ${P}_{S}$ is a subgroup of $G\text{.}$

2.3.3 The mapping $S↦{P}_{S}$ is an inclusion-preserving bijection of the set of subsets of $R$ onto the set of subgroups of $G$ containing $B\text{.}$

2.3.4 If $S,$ $S\prime$ are subsets of $R,$ and ${P}_{S}$ is conjugate to ${P}_{S\prime },$ then $S=S\prime \text{.}$

2.3.5 If $S,$ $S\prime$ are subsets of $R,$ and $w\in W,$ then $PSwPS′=B WSwWS′B.$

2.3.6 Each ${P}_{S}$ is its own normalizer in $G\text{.}$

2.3.7 If $r\in R$ and $w\in W$ and $l\left(rw\right)>l\left(w\right),$ then $BrwB=BrB·BwB\text{.}$ (The length $l\left(w\right)$ of $w\in W$ is defined as the length of a reduced word in the generating set $R,$ just as in (2.1) and (2.2).)

Buildings

Let $G$ be a group with BN-pair $\left(B,N\right)$ as in (2.3). We shall assume now that the Weyl group of $\left(B,N\right)$ 'is' an affine Weyl group $W$ in the sense of (2.2). More precisely, we shall suppose we are given a reduced irreducible root system ${\Sigma }_{0}$ in a Euclidean space $A,$ a chamber $C$ for the associated affine root system $\Sigma ,$ and a homomorphism $\nu$ of $N$ onto the affine Weyl group $W$ such that

 (1) $\text{Ker}\left(\nu \right)=H$
so that we may identify (via $\nu \text{)}$ the Weyl group $N/H$ of $\left(B,N\right)$ with the Weyl group $W$ of $\Sigma \text{;}$
 (2) under this identification, the distinguished generators of $N/H$ are the reflections in the walls of the chamber $C\text{;}$ in other words, $R=\left\{{w}_{\alpha }:\alpha \in \Pi \right\}\text{.}$

The conjugates of $B$ in $G$ are called Iwahori subgroups of $G\text{.}$ A parahoric subgroup of $G$ is a proper subgroup containing an Iwahori subgroup. From the properties of BN-pairs recalled in (2.3), each parahoric subgroup $P$ is conjugate to ${P}_{S}$ for some proper subset $S$ of $R,$ and $S$ is uniquely determined by $P\text{.}$

We set up a one-one correspondence $S⟷F$ between the subsets $S$ of $R$ and the facets $F$ of the chamber $C$ as follows. To a facet $F$ corresponds the set of all ${w}_{\alpha }\in R$ which fix $F\text{.}$ In this correspondence, $\varnothing ⟷C\text{;}$ $R⟷\varnothing \text{;}$ and $S⟷$ a vertex of $C⇔C$ is a maximal proper subset of $R\text{.}$

If $S⟷F$ we write ${P}_{F}$ for ${P}_{S}\text{.}$

Each parahoric subgroup $P$ determines uniquely a facet $F\left(P\right)$ of $C\text{:}$ namely $F\left(P\right)=F⇔P$ is conjugate to ${P}_{F}\text{.}$

The building associated with the given BN-pair structure on $G$ is the set $ℐ$ of all pairs $\left(P,x\right)$ where $P$ is a parahoric subgroup of $G,$ and $x$ is a point in the facet $F\left(P\right)\text{.}$

With each parahoric subgroup $P$ we associate the subset $ℱ\left(P\right)$ of $ℐ,$ where $ℱ(P)= { (P,x):x∈F(P) } .$ $ℱ\left(P\right)$ is called a facet of $ℐ,$ of type $F\left(P\right),$ In particular, if $P$ is an Iwahori subgroup, $ℱ\left(P\right)$ is a chamber of $ℐ\text{.}$ If $P$ is parahoric, there are only finitely many parahoric subgroups $Q$ containing $P,$ and we define $ℱ(P)‾= ⋃Q⊇Pℱ(Q).$ In this way the building $ℐ$ may be regarded as a geometrical simplicial complex, the $ℱ\left(P\right)$ being the 'open' simplexes and the $\stackrel{‾}{ℱ\left(P\right)}$ the 'closed' simplexes.

The group $G$ acts on $ℐ$ by inner automorphisms: $g\left(P,x\right)=\left(gP{g}^{-1},x\right)\text{.}$

Apartments

Consider the following subset ${𝒜}_{0}$ of $ℐ\text{:}$ $𝒜0=⋃w∈W ℱ(wBw-1)‾.$ Equivalently, ${𝒜}_{0}$ is the union of the facets $nℱ\left(P\right)$ where $n\in N$ and $P\supset B\text{.}$

Remark. Since we have identified $N/H$ with $W$ by means of the isomorphism induced by $\nu ,$ expressions such as $wP{w}^{-1}$ are meaningful, where $w\in W$ and $P$ is a subgroup of $G$ containing $B\text{.}$ Hence $W$ acts on ${𝒜}_{0}\text{:}$ $w\left(P,x\right)=\left(wP{w}^{-1},x\right)=\left(nP{n}^{-1},x\right)$ for any $n\in N$ such that $\nu \left(n\right)=w\text{.}$

There exists a unique bisection $j:A\to {𝒜}_{0}$ such that

 (1) for each facet $F$ of $C$ and each $x\in F,$ $j(x)=(PF,x);$ (2) $j\circ w=w\circ j$ for all $w\in W\text{.}$

 Proof. Let $y\in A\text{.}$ Then $y$ is congruent under $W$ to a unique point $x$ of $\stackrel{‾}{C}\text{:}$ say $y=wx$ where $w\in W$ and $x\in F,$ and $F$ is a facet of the chamber $C\text{.}$ Define $j\left(y\right)$ to be the point $\left(w{P}_{F}{w}^{-1},x\right)\in {𝒜}_{0}\text{.}$ We have to check that $j$ is well defined. If also $y=w\prime x,$ then ${w}^{-1}w\prime$ fixes $x$ and therefore, by a property of reflection groups recalled in (2.2), belongs to the subgroup of $W$ generated by the ${w}_{\alpha },$ $\alpha \in \Pi ,$ which fix $x\text{.}$ In other words, ${w}^{-1}w\prime \in {W}_{S},$ where $S⟷F\text{.}$ Since ${W}_{S}\subset {P}_{S}={P}_{F}$ it follows that $w{P}_{F}{w}^{-1}=w\prime {P}_{F}{w\prime }^{-1}\text{.}$ Hence $j$ is well defined, and clearly satisfies (1) and (2). The uniqueness of $j$ is obvious. $\square$

If $g{𝒜}_{0}={𝒜}_{0}$ then ${j}^{-1}\circ \left(g|{𝒜}_{0}\right)\circ j\in W\text{.}$

 Proof. Let ${𝒷}_{0}$ be the chamber $j\left(C\right)=ℱ\left(B\right)\text{.}$ We have $g{𝒷}_{0}={n}_{0}{𝒷}_{0}$ for some ${n}_{0}\in N\text{.}$ Hence ${g}_{0}={n}_{0}^{-1}g$ fixes ${𝒜}_{0}$ as a whole and also fixes the chamber ${𝒷}_{0}$ and all its facets; also it maps each facet to another facet of the same type. Transferring our attention to $A,$ it follows that $\gamma ={j}^{-1}\circ \left({g}_{0}|{𝒜}_{0}\right)\circ j$ maps $\underset{_}{A}$ bijectively onto $A$ and fixes the chamber $C$ and all its facets. Hence $\gamma$ also fixes the adjacent chambers ${w}_{\alpha }C$ $\left(\alpha \in \Pi \right),$ and so on. It follows that $\gamma$ is the identity mapping and hence that ${j}^{-1}\circ \left(g|{𝒜}_{0}\right)\circ j=\nu \left({n}_{0}\right)\in W\text{.}$ $\square$

The subcomplexes $g{𝒜}_{0}$ of $ℐ,$ as $g$ runs through $G,$ are called the apartments of the building $ℐ\text{.}$ If $𝒜=g{𝒜}_{0}$ is an apartment, we may transport the Euclidean structure of $A$ to $𝒜$ via the bijection $\left(g|{𝒜}_{0}\right)\circ j\text{.}$ If also $𝒜=g\prime {𝒜}_{0},$ then it follows from (2.4.2) that the two mappings $\left(g|{𝒜}_{0}\right)\circ j$ and $\left(g\prime |{𝒜}_{0}\right)\circ j$ differ by an element of $W,$ and hence we have a well-defined structure of Euclidean space on each apartment $𝒜,$ and in particular a distance ${d}_{𝒜}\left(x,y\right)$ defined for $x,y\in 𝒜\text{.}$

Thus the building $ℐ$ may be thought of as obtained by sticking together many copies of the Euclidean space $A\text{.}$

Any two facets of $ℐ$ are contained in a single apartment.

 Proof. Consider two facets $ℱ\left({P}_{1}\right)$ and $ℱ\left({P}_{2}\right),$ where ${P}_{1},$ ${P}_{2}$ are parahoric subgroups of $G\text{:}$ say ${P}_{i}={g}_{i}{P}_{{F}_{i}}{g}_{i}^{-1}$ $\left(i=1,2\right),$ where ${F}_{1},$ ${F}_{2}$ are facets of the chamber $C$ in $A\text{.}$ By (2.3.1) we can write ${g}_{1}^{-1}{g}_{2}={b}_{1}n{b}_{2}$ with ${b}_{1},{b}_{2}\in B$ and $n\in N\text{.}$ If $g={g}_{1}{b}_{1},$ then $P1=gPF1g-1 ,P2=g (nPF2n-1) g-1$ and therefore $ℱ\left({P}_{1}\right)$ and $ℱ\left({P}_{2}\right)$ are both in $g{𝒜}_{0}\text{.}$ $\square$

$G$ is transitive on the set of pairs $\left(𝒷,𝒜\right),$ where $𝒜$ is an apartment and $𝒷$ is a chamber in $𝒜\text{.}$

 Proof. We have $𝒷=ℱ\left({g}_{0}B{g}_{0}^{-1}\right)={g}_{0}{𝒷}_{0}$ for some ${g}_{0}\in G,$ where ${𝒷}_{0}=ℱ\left(B\right)\text{.}$ Hence we may assume that $𝒷={𝒷}_{0}\text{.}$ If $𝒜=g{𝒜}_{0}$ contains ${𝒷}_{0},$ then ${g}^{-1}{𝒷}_{0}$ is a chamber in ${𝒜}_{0}$ and hence ${g}^{-1}{𝒷}_{0}=n{𝒷}_{0}$ for some $n\in N\text{.}$ So if ${g}_{1}=gn$ we have $𝒜={g}_{1}{𝒜}_{0}$ and ${𝒷}_{0}={g}_{1}{𝒷}_{0}\text{.}$ $\square$

Let $𝒜,$ $𝒜\prime$ be, two apartments and let $𝒷$ be a chamber contained in $𝒜\cap 𝒜\prime \text{.}$ Then there exists a unique bijecton $\rho :𝒜\prime \to 𝒜$ such that

 (1) There exists $g\in G$ such that $\rho x=gx$ for all $x\in 𝒜\prime \text{;}$ (2) $\rho x=x$ for all $x\in 𝒷\text{.}$
Moreover, $\rho x=x$ for all $x\in 𝒜\cap 𝒜\prime ,$ and ${d}_{𝒜\prime }\left(x,y\right)={d}_{𝒜}\left(\rho x,\rho y\right)$ for all $x,y\in 𝒜\prime \text{.}$

 Proof. The existence of $\rho$ follows immediately from (2.4.4). Uniqueness follows from (2.4.2): if ${\rho }_{1},{\rho }_{2}:𝒜\prime \to 𝒜$ both satisfy (1) and (2), then ${\rho }_{1}\circ {\rho }_{2}^{-1}$ fixes $𝒜$ as a whole and fixes the chamber $𝒷,$ hence is the identity map. That ${d}_{𝒜\prime }\left(x,y\right)={D}_{𝒜}\left(\rho x,\rho y\right)$ follows from the definition of the metric on an apartment given after (2.4.2). It remains to show that $\rho x=x$ for all $x\in 𝒜\cap 𝒜\prime \text{.}$ Suppose $ℱ=ℱ\left(P\right)$ is a facet contained in $𝒜\cap 𝒜\prime \text{.}$ By (2.4.4) we may assume that $𝒜\prime ={𝒜}_{0},$ $𝒜=g{𝒜}_{0}$ and $𝒷={𝒷}_{0}=ℱ\left(B\right)\text{.}$ Since $g{𝒷}_{0}={𝒷}_{0}$ it follows that $g$ normalizes $B$ and therefore by (2.3.6) that $g\in B\text{.}$ Since $ℱ\left(P\right)\subseteq {𝒜}_{0}\cap g{𝒜}_{0}$ we have $P=n1PFn1-1 =g(n2PFn2-1) g-1$ for some facet $F$ of $C$ and ${n}_{1},{n}_{2}\in N\text{.}$ Since ${P}_{F}$ is its own normalizer (2.3.6), we have ${n}_{1}^{-1}g{n}_{2}\in {P}_{F}$ and therefore $B{n}_{2}{P}_{F}=B{n}_{1}{P}_{F}\text{.}$ By (2.3.5), $B{n}_{2}{W}_{F}B=B{n}_{1}{W}_{F}B$ where ${W}_{F}$ is the subgroup of $W$ which fixes $F\text{.}$ Hence (2.3.1) ${n}_{2}{W}_{F}={n}_{1}{W}_{F}\text{;}$ since ${W}_{F}\subset {P}_{F}$ it follows that ${n}_{1}{P}_{F}{n}_{1}^{-1}={n}_{2}{P}_{F}{n}_{2}^{-1},$ hence $ℱ=gℱ=\rho ℱ\text{.}$ $\square$

Retraction of the building onto an apartment

Let $𝒜$ be an apartment and $𝒷$ a chamber in $𝒜\text{.}$ Then there exists a unique mapping $\rho :ℐ\to 𝒜$ such that, for all apartments $𝒜\prime$ containing $𝒷,$ $\rho |𝒜\prime$ is the bisection $𝒜\prime \to 𝒜$ of (2.4.5).

 Proof. Let $x\in ℐ\text{.}$ By (2.4.3) there exists an apartment ${𝒜}_{1}$ containing $x$ and $𝒷\text{.}$ Let ${\rho }_{1}:{𝒜}_{1}\to 𝒜$ be the isomorphism of (2.4.5). If also $x$ and $𝒷$ are contained in another apartment ${𝒜}_{2},$ then we have also ${\rho }_{2}:{𝒜}_{2}\to 𝒜\text{.}$ But then ${\rho }_{1}^{-1}\circ {\rho }_{2}:{𝒜}_{2}\to {𝒜}_{1}$ is the isomorphism of (2.4.5) for the apartments ${𝒜}_{1}$ and ${𝒜}_{2}$ and the chamber $𝒷\text{.}$ Hence ${\rho }_{1}^{-1}\circ {\rho }_{2}$ fixes $x,$ that is to say ${\rho }_{1}\left(x\right)={\rho }_{2}\left(x\right)\text{.}$ It follows that $\rho$ as in the theorem is well defined, and the uniqueness is obvious. $\square$

The mapping $\rho$ of (2.4.7) is called the retraction of $ℐ$ onto $𝒜$ with centre $𝒷\text{.}$ It has the following properties:

 (1) $\rho x=x$ for all $x\in 𝒜\text{.}$ (2) For each facet $ℱ$ in $ℐ,$ $\rho |\stackrel{‾}{ℱ}$ is an affine isometry of $\stackrel{‾}{ℱ}$ onto $\stackrel{‾}{\rho ℱ}\text{.}$ (3) If $x\in \stackrel{‾}{𝒷},$ then ${\rho }^{-1}\left(x\right)=\left\{x\right\}\text{.}$

 Proof. (1) and (2) are clear. As to (3), let $ℱ\prime$ be a facet such that $ℱ=\rho ℱ\prime$ is a facet of $𝒷\text{.}$ By (2.4.3) there exists an apartment $𝒜\prime$ containing $𝒷$ and $ℱ\prime ,$ and $\rho |𝒜\prime$ is a bijection of $𝒜\prime$ onto $𝒜$ leaving $𝒷$ fixed. It follows that $ℱ\prime =ℱ\text{.}$ $\square$

 (i) There exists a unique function $d:ℐ×ℐ\to {ℝ}_{+}$ such that $d|𝒜×𝒜$ is the metric ${d}_{𝒜},$ for each apartment $𝒜$ in $ℐ\text{.}$ (ii) If $\rho$ is a retraction of $ℐ$ onto an apartment $𝒜$ as in (2.4.6), then $d\left(\rho \left(x\right),\rho \left(y\right)\right)\le d\left(x,y\right)$ for all $x,y\in ℐ\text{.}$ (iii) $d$ is a $G\text{-invariant}$ metric on $ℐ\text{.}$

 Proof. Let $x,y\in {ℐ}_{0}\text{.}$ By (2.4.3) there is an apartment $𝒜$ containing $x$ and $y\text{.}$ We wish to define $d\left(x,y\right)$ to be the Euclidean distance ${d}_{𝒜}\left(x,y\right)$ between $x$ and $y$ in $𝒜\text{.}$ We must therefore show that, if $x,$ $y$ belong also to another apartment $𝒜\prime ,$ then ${d}_{𝒜}\left(x,y\right)={d}_{𝒜\prime }\left(x,y\right)\text{.}$ Let $𝒷$ be a chamber in $𝒜$ such that $x\in \stackrel{‾}{𝒷}$ and let $𝒷\prime$ be a chamber in $𝒜\prime$ such that $y\in \stackrel{‾}{𝒷}\prime \text{.}$ By (2.4.3) there is an apartment $𝒜″$ containing $𝒷$ and $𝒷\prime \text{.}$ From (2.4.5) we have $d𝒜(x,y)= d𝒜″(x,y)$ because $𝒜$ and $𝒜″$ have a chamber in common; and for the same reason $d𝒜′(x,y)= d𝒜″(x,y).$ Hence $d$ is well defined and is $G\text{-invariant}$ (from the definition of ${d}_{𝒜}\text{).}$ (ii) Let $𝒜\prime$ be an apartment containing $x$ and $y\text{.}$ The image under $\rho$ of the straight line segment joining $x$ to $y$ in $𝒜\prime$ will be a polygonal line in $𝒜$ of the same total length, by (2.4.7)(ii), and endpoints $\rho \left(x\right)$ and $\rho \left(y\right)\text{.}$ Hence the result. (iii) Let $x,y,z\in ℐ$ and let $𝒜$ be an apartment containing $x$ and $y\text{.}$ Let $\rho$ be a retraction of $ℐ$ onto $𝒜\text{.}$ Then $d(x,y) ≤ d(x,ρ(z))+ d(ρ(z),y) ≤ d(x,z)+ d(z,y)$ by (ii) above, since $\rho \left(x\right)=x$ and $\rho \left(y\right)=y\text{.}$ Hence $d$ satisfies the triangle equality, and we have already observed that $d$ is $G\text{-invariant.}$ $\square$

Let $x,y\in ℐ\text{.}$ Then there is a unique geodesic joining $x$ to $y\text{.}$

 Proof. Let $𝒜$ be an apartment containing $x$ and $y$ and let $\left[xy\right]$ denote the straight line segment joining $x$ to $y$ in $𝒜\text{.}$ Suppose $z$ lies on a geodesic from $x$ to $y\text{.}$ Then $d\left(x,z\right)+d\left(z,y\right)=d\left(x,y\right)\text{.}$ Let ${z}_{0}\in \left[x,y\right]$ be such that $d\left(x,{z}_{0}\right)=d\left(x,z\right)$ and hence also $d\left({z}_{0},y\right)=d\left(z,y\right)\text{.}$ Let $\rho$ be the retraction of $ℐ$ onto $𝒜$ with centre a chamber $𝒷$ such that ${z}_{0}\in \stackrel{‾}{𝒷}\text{.}$ Then we have $d(x,y) ≤ d(x,ρ(z))+ d(ρ(z),y) ≤ d(x,z)+ d(z,y) by (2.4.8)(ii) = d(x,y).$ Consequently $d(x,y)= d(x,ρ(z))+ d(ρ(z),y) and d(x,ρ(z))= d(x,z).$ Since $x,y,\rho \left(z\right)$ are in the same Euclidean space $𝒜$ it follows that $\rho \left(z\right)={z}_{0}\text{.}$ Hence $z\in {\rho }^{-1}\left({z}_{0}\right)$ and therefore $z={z}_{0}$ by (2.4.7)(iii). $\square$

It follows from (2.4.9) that the straight line segment $\left[xy\right]$ lies in all apartments containing $x$ and $y\text{.}$

$ℐ$ is complete with respect to the metric $d\text{.}$

 Proof. Let $\left({x}_{n}\right)$ be a Cauchy sequence in $ℐ\text{.}$ If $\rho$ is a retraction of $ℐ$ onto an apartment $𝒜,$ then $\left(\rho \left({x}_{n}\right)\right)$ is a Cauchy sequence in $𝒜,$ by (2.4.8)(ii), and therefore converges to a point $x\in 𝒜\text{.}$ The set of real numbers $d\left(x,g\left(x\right)\right),$ where $g\in G$ is such that $x\ne gx,$ has a positive lower bound $\mu ,$ and we shall have $d(ρ(xn),x)< 14μ,d (xn,xn+1)< 14μ$ for all sufficiently large $n\text{.}$ Now each $\rho \left({x}_{n}\right)$ is of the form ${g}_{n}{x}_{n}$ for some ${g}_{n}\in G\text{.}$ Put ${y}_{n}={g}_{n}^{-1}x\text{.}$ Then $d(yn,yn+1) ≤ d(yn,xn)+ d(xn,xn+1)+ d(xn+1,yn+1) = d(x,ρ(xn))+ d(xn,xn+1)+ d(ρ(xn+1),x) < 34μ<μ$ for all large $n\text{.}$ Hence ${y}_{n}={y}_{n+1}=\dots =y$ and we have $d(xn,y)= d(xn,yn)= d(ρ(xn),x) →0$ as $n\to \infty \text{.}$ Hence ${x}_{n}\to y\text{.}$ $\square$

Fixed point theorem

A subset $X$ of $ℐ$ is convex if $x,y\in X⇒\left[xy\right]\subset X\text{.}$

Let $X$ be a bounded non-empty subset of $ℐ\text{.}$ Then the group of isometries $\gamma$ of $ℐ$ such that $\gamma \left(X\right)\subseteq X$ has a fixed point in the closure of the convex hull of $X\text{.}$

For the proof we shall require the following lemma:

Let $x,y,z\in ℐ$ and let $m$ be the midpoint of $\left[xy\right]\text{.}$ Then $d(z,x)2+ d(z,y)2≥ 2d(z,m)2+ 12d(x,y)2.$

 Proof. If $x,y,z$ are all in the same apartment $𝒜,$ this is true with equality, by elementary geometry (take $z$ as origin). In the general case, let $𝒜$ be an apartment containing $x$ and $y$ and choose a chamber $𝒷$ in $𝒜$ such that $m\in \stackrel{‾}{𝒷}\text{.}$ Retract $ℐ$ onto $𝒜$ with centre $𝒷\text{.}$ Then $d(z,x)2+ d(z,y)2 ≥ d(ρ(z),x)2+ d(ρ(z),y)2 = 2d(ρ(z),m)2+ 12d(x,y)2 = 2d(z,m)2+12d (x,y)2.$ $\square$

 Proof of 2.4.11. Let $\delta \left(X\right)=\underset{x,y\in X}{\text{sup}}d\left(x,y\right)$ be the diameter of $X\text{.}$ Choose a real number $k\in \left(0,1\right)$ and let $fX$ be the set of all $m\in ℐ$ such that $m$ is the midpoint of $\left[xy\right]$ with $x,y\in X$ and $d\left(x,y\right)\ge k·\delta \left(X\right)\text{.}$ Clearly $fX$ is non-empty and is contained in the convex hull of $X\text{.}$ If $m\in fX$ and $z\in X,$ then by (2.4.12) $d(z,m)2 = 12d(z,x)2+ 12d(z,y)2- 14d(x,y)2 ≤ 12δ(X)2+ 12δ(X)2- 14k2δ(X)2 = (1-14k2)δ (X)2$ so that $d(z,m)≤k1 δ(X) (1)$ for some ${k}_{1}<1\text{.}$ If now $m\in fX$ and $z\in fX,$ then again $m$ is the midpoint of a segment $\left[xy\right]$ with $x,y\in X$ and $d\left(x,y\right)\ge k\delta \left(X\right),$ so that $d(z,m)2 ≤ 12d(z,x)2+ 12d(z,y)2- 14d(x,y)2 ≤ (1-14k2)δ (X)2-14 k2δ(X)2 = (1-12k2) δ(X)2$ so that $δ(fX)≤k2 δ(X) (2)$ for some ${k}_{2}<1\text{.}$ From (2) it follows that $δ(fnX)→0 as n→∞. (3)$ For each $n,$ pick ${x}_{n}\in {f}^{n}X\text{.}$ Then by (1) and (2) $d(xn,xn+1)≤ k1δ(fnX)≤ k1k2nδ(X).$ Hence $\left({x}_{n}\right)$ is a Cauchy sequence in $ℐ,$ hence by (2.4.10) converges to some $x\in ℐ\text{.}$ Clearly $x$ lies in the closure of the convex hull of $X\text{.}$ Now let $\gamma$ be an isometry of $ℐ$ such that $\gamma X\subseteq X\text{.}$ Then $\gamma {f}^{n}X\subseteq {f}^{n}X$ for all $n\text{.}$ Let ${x}_{n}^{\prime }=\gamma \left({x}_{n}\right)\text{.}$ Then $\left({x}_{n}^{\prime }\right)$ is a Cauchy sequence with ${x}_{n}^{\prime }\in {f}^{n}X$ for all $n$ and converges to $\gamma x\text{.}$ But by (3) $d\left({x}_{n},{x}_{n}^{\prime }\right)\to 0$ as $n\to \infty \text{.}$ Hence $\gamma x=x\text{.}$ Hence $x$ is a fixed point of the group of isometries $\gamma$ of $ℐ$ such that $\gamma X\subseteq X\text{.}$ $\square$

A subset $M$ of $G$ is said to be bounded if $MX$ is bounded for all bounded subsets $X$ of $ℐ\text{.}$

$M$ is bounded $⇔$ $M$ intersects only finitely many double cosets $BwB$ $\left(w\in W\right)\text{.}$

 Proof. Let $\rho$ be the retraction of $ℐ$ onto the apartment ${𝒜}_{0}$ with centre ${𝒷}_{0}=ℱ\left(B\right)\text{.}$ Then $X\subset ℐ$ is bounded $⇔$ $\rho X$ is bounded $⇔$ $\rho X$ is contained in a finite union of closed chambers $\stackrel{‾}{𝒷}$ of ${𝒜}_{0}\text{.}$ Hence $M$ is bounded $⇔$ $M𝒷$ is bounded, for each chamber $𝒷\subset {𝒜}_{0}⇔M{𝒷}_{0}$ is bounded. If $m\in M,$ define ${w}_{m}\in W$ by the relation $m\in B{w}_{m}B\text{.}$ Then $M{𝒷}_{0}$ is bounded $⇔\bigcup _{m\in M}{w}_{m}{𝒷}_{0}$ is bounded $⇔$ the set $\left\{{w}_{m}:m\in M\right\}$ is finite $⇔M$ intersects only finitely many $BwB\text{.}$ $\square$

A subgroup $\Gamma$ of $\Gamma$ is bounded $⇔$ $\Gamma$ is contained in a parahoric subgroup.

 Proof. $⇒$ Let $x\in ℐ\text{.}$ Then $X=\Gamma x$ is bounded. Applying the fixed point theorem (2.4.11), we see that there exists $y\in ℐ$ such that $\Gamma y=y\text{.}$ If $y$ lies in the facet $ℱ\left(P\right),$ then $\Gamma$ normalizes the parahoric subgroup $P$ and hence $\Gamma \subseteq P$ by (2.3.6). $⇐$ Replacing $\Gamma$ by a conjugate, we may assume that $\Gamma \subset {P}_{S},$ where $S$ is a proper subset of $R=\left\{{w}_{\alpha }:\alpha \in \Pi \right\}\text{.}$ Now ${P}_{S}=B{W}_{S}B,$ and ${W}_{S}$ is finite because $S\ne R\text{.}$ Hence $\Gamma$ is bounded by (2.4.13). $\square$

Every bounded subgroup of $G$ is contained in a maximal bounded subgroup. The maximal bounded subgroups are precisely the maximal parahoric subgroups of $G,$ and form $l+1$ conjugacy classes, corresponding to the vertices of the chamber $C\text{.}$

Groups with affine root structure

(a) Universal Chevalley groups

We retain the notation of (2.1) and (2.2). Let $L= { u∈A*:u(a∨) ∈ℤ for all a∈Σ0 } .$ Then $L$ is a lattice of rank $l$ $\left(=\text{dim} A\right)$ in ${A}^{*},$ containing ${\Sigma }_{0}\text{.}$

Let $𝓀$ be any field, and ${𝓀}^{+},{𝓀}^{×}$ the additive and multiplicative groups of $𝓀\text{.}$ Associated with the pair $\left({\Sigma }_{0},𝓀\right)$ there is a universal Chevalley group $G=G\left({\Sigma }_{0},𝓀\right)\text{.}$ We shall not go into the details of the construction of $G,$ for which we refer to Steinberg [Ste1967]. All we shall say here is that $G$ is generated by elements ${u}_{a}\left(\xi \right)$ $\left(a\in {\Sigma }_{0},\xi \in 𝓀\right)$ and contains elements $h\left(\chi \right)$ $\left(\chi \in \text{Hom}\left(L,{𝓀}^{×}\right)\right)$ with various relations between the $u\text{'s}$ and the $h\text{'s.}$ In particular, each ${u}_{a}$ is an injective homomorphism of ${𝓀}^{+}$ into $G,$ and $h$ is an injective homomorphism of $\text{Hom}\left(L,{𝓀}^{×}\right)$ into $G\text{.}$

If ${\Sigma }_{0}$ is the root system of type ${A}_{l}$ (2.2.3) then the universal Chevalley group $G\left({\Sigma }_{0},𝓀\right)$ is the special linear group ${SL}_{l+1}\left(𝓀\right)\text{.}$ In this case the ${u}_{a}\left(\xi \right)$ and $h\left(\chi \right)$ may be taken as follows:

If $a={e}_{i}-{e}_{j}$ $\left(i\ne j\right)$ then ${u}_{a}\left(\xi \right)=1+\xi {E}_{ij},$ where $1$ is the unit matrix and ${E}_{ij}$ is the matrix with $1$ in the $\left(i,j\right)$ place and $0$ elsewhere. $h\left(\chi \right)$ is the diagonal matrix with $\chi \left({e}_{0}\right),\dots ,\chi \left({e}_{l}\right)$ in order down the diagonal.

Now suppose that the field $𝓀$ carries a discrete valuation $v,$ i.e. a surjective homomorphism $v:{𝓀}^{×}\to ℤ$ such that $v\left(\xi +\eta \right)\ge \text{min}\left(v\left(\xi \right),v\left(\eta \right)\right)$ for all $\xi ,\eta \in 𝓀$ (conventionally $v\left(0\right)=+\infty \text{).}$ We consider certain subgroups of the Chevalley group $G=G\left({\Sigma }_{0},𝓀\right)\text{.}$

First, let $Z$ be the subgroup consisting of the $h\left(\chi \right)$ and let $N$ be the normalizer of $Z\text{.}$ Then there is a canonical isomorphism $N/Z\to {W}_{0},$ and one can show that this lifts uniquely to a homomorphism $\nu$ of $N$ onto the affine Weyl group $W\text{.}$ The restriction of $\nu$ to $ℤ$ is described as follows. If $\chi \in \text{Hom}\left(L,{𝓀}^{×}\right)$ then $v\circ \chi$ is a homomorphism $L\to ℤ,$ which defines a linear form ${A}^{*}\to ℝ,$ hence a vector in $A\text{.}$ Then $\nu \left(h\left(\chi \right)\right)$ is translation by this vector, and this translation belongs to $W\text{.}$ We have $\nu \left(Z\right)=T$ and $Z={\nu }^{-1}\left(T\right),$ where $T$ is as before the translation subgroup of $W\text{.}$ The kernel $H$ of $\nu$ is the set of all $h\left(\chi \right)$ such that $\chi \left(L\right)$ is contained in the group of units of $𝓀\text{.}$

Next, for each $\alpha \in \Sigma$ we have a subgroup ${U}_{\alpha }$ of $G,$ defined as follows. If $\alpha =a+k,$ then $Ua+k= { ua(χ):ξ∈𝓀 and v(ξ)≥k } .$ If $\Phi$ is any subset of $\Sigma ,$ we denote by ${U}_{\Phi }$ the subgroup of $G$ generated by the ${U}_{\alpha }$ such that $\alpha \in \Phi ,$ and by ${P}_{\Phi }$ the subgroup generated by ${U}_{\Phi }$ and $H\text{.}$ Then the triple $\left(N,\nu ,{\left({U}_{\alpha }\right)}_{\alpha \in \Sigma }\right)$ has the following properties, for all $n\in N$ and $\alpha ,\beta \in \Sigma \text{:}$

 (I) $n{U}_{\alpha }{n}^{-1}={U}_{\nu \left(n\right)\alpha }\text{.}$
From this it follows that $H$ normalizes each ${U}_{\alpha },$ hence each ${U}_{\Phi }\text{.}$ Consequently $PΦ=H· UΦ=UΦ ·H.$
 (II) If $k$ is a positive integer, then ${U}_{\alpha +k}$ is a proper subgroup of ${U}_{\alpha },$ and $\bigcap _{k\ge 0}{U}_{\alpha +k}=\left\{1\right\}\text{.}$
From this it follows that for each $a\in {\Sigma }_{0},$ the union of the ${U}_{a+r},$ $r\in ℤ$ is a group, which we denote by ${U}_{\left(a\right)}\text{.}$ (In the present situation ${U}_{\left(a\right)}={u}_{a}\left({𝓀}^{+}\right)\text{.)}$

The next three properties relate to a pair of subgroups ${U}_{\alpha }$ and ${U}_{\beta }$ in the following situations: (III) $\beta +\alpha$ is a positive constant; (IV) $\beta +\alpha =0\text{;}$ (V) neither $\beta +\alpha$ nor $\beta -\alpha$ is constant (i.e. the hyperplanes ${h}_{\alpha },$ ${h}_{\beta }$ are not parallel).

 (III) If $\beta =-\alpha +k$ with $k>0,$ then ${P}_{\left\{\alpha ,\beta \right\}}={U}_{\alpha }H{U}_{\beta }\text{.}$ (IV) ${P}_{\left\{\alpha ,-\alpha \right\}}\left({U}_{\alpha }·{\nu }^{-1}\left({w}_{\alpha }\right)·{U}_{\alpha }\right)\cup \left({U}_{\alpha }H{U}_{-\alpha +1}\right)\text{.}$

For $\alpha ,\beta \in \Sigma$ let $\left[\alpha ,\beta \right]$ denote the set of all $\gamma \in \Sigma$ of the form $m\alpha +n\beta$ with $m,$ $n$ positive integers, and let $\left[{U}_{\alpha },{U}_{\beta }\right]$ be the subgroup of $G$ generated by all commutators $\left(u,v\right)=uv{u}^{-1}{v}^{-1}$ with $u\in {U}_{\alpha }$ and $v\in {U}_{\beta }\text{.}$ Then from Chevalley's commutator relations we have

 (V) If ${h}_{\alpha }$ and ${h}_{\beta }$ are not parallel, then $\left[{U}_{\alpha },{U}_{\beta }\right]\subseteq {U}_{\left[\alpha ,\beta \right]}\text{.}$

Let ${U}^{+}=⟨{U}_{\left(a\right)}:a\in {\Sigma }_{0}^{+}⟩$ be the subgroup generated by the ${U}_{\left(a\right)}$ with $a$ positive, and likewise ${U}^{-}=⟨{U}_{\left(a\right)}:a\in {\Sigma }_{0}^{-}⟩\text{.}$ Then

 (VI) ${U}^{+}\cap Z{U}^{-}=\left\{1\right\}\text{.}$

Finally,

 (VII) $G$ is generated by $N$ and the ${U}_{\alpha }$ $\left(\alpha \in \Sigma \right)\text{.}$

(b) Simply-connected simple algebraic groups

As before let $𝓀$ be a field with a discrete valuation $v,$ and assume now that $𝓀$ is complete with respect to $v,$ with perfect residue field.

Let $𝒢$ be a simply-connected simple linear algebraic group defined over $𝓀\text{.}$ Let $𝒮$ be a maximal $𝓀\text{-split}$ torus in $𝒢,$ and let $𝒵,𝒩$ be the centralizer and normalizer respectively of $𝒮$ in $𝒢\text{.}$ Let $G=𝒢\left(𝓀\right)$ be the group of $𝓀\text{-rational}$ points of $𝒢\text{.}$ Then one of the main results of the structure theory of Bruhat and Tits ([BTi1966], [BTi1967]) is:

There exists a reduced irreducible root system ${\Sigma }_{0},$ subgroups $N$ and ${U}_{\alpha }$ $\left(\alpha \in \Sigma \right)$ of $G,$ and a surjective homomorphism $\nu :N\to W$ such that the triple $\left(N,\nu ,{\left({U}_{\alpha }\right)}_{\alpha \in \Sigma }\right)$ satisfies (I)-(VII) above.

In fact one takes $N=𝒩\left(𝓀\right),$ $Z=𝒵\left(𝓀\right),$ and $H=\text{Ker}\left(\nu \right)$ is the set of all $z\in 𝒵\left(𝓀\right)$ such that $v\left(\chi \left(z\right)\right)=0$ for all characters $\chi$ of $𝒵\text{.}$ ${W}_{0}$ is the relative Weyl group of $𝒢$ with respect to $𝒮,$ but ${\Sigma }_{0}$ is not in general the relative root system, even if the latter is reduced. But every $a\in {\Sigma }_{0}$ is proportional to a root of $G$ relative to $𝒮,$ and conversely.

If $𝒢$ is split over $𝓀$ (i.e. if $𝒮$ is a maximal torus of $𝒢\text{)}$ then we are back in the situation of (a) above.

If $𝓀$ is a $p\text{-adic}$ field (i.e. if the residue field of $𝓀$ is finite) then $G=𝒢\left(𝓀\right)$ inherits a topology from $𝓀$ for which $G$ is a locally compact topological group. If $K$ is a suitably chosen maximal compact subgroup of $G,$ then as we shall see later (3.3.7) the algebra $ℒ\left(G,K\right)$ is commutative, and the theory of zonal spherical functions on $G$ relative to $K$ is almost entirely a consequence of the properties (I)-(VII) listed above. We shall therefore take these properties as a set of axioms:

Let ${\Sigma }_{0}$ be a reduced irreducible root system and let $\Sigma ,$ $W$ be the associated affine root system and affine Weyl group, as in (2.2). An affine root structure of type ${\Sigma }_{0}$ on a group $G$ is a triple $\left(N,\nu ,{\left({U}_{\alpha }\right)}_{\alpha \in \Sigma }\right),$ where $N$ and the ${U}_{\alpha }$ are subgroups of $G,$ and $\nu$ is a homomorphism of $N$ onto $W,$ satisfying (I)-(VII) above.

Cartan and Iwasawa decompositions

In this section $G$ is a group endowed with an affine root structure $\left(N,\nu ,{\left({U}_{\alpha }\right)}_{\alpha \in \Sigma }\right)$ of type ${\Sigma }_{0}$ (2.5.3). We shall develop some consequences of the axioms (I)-(VII).

Let $\Phi$ be a subset of $\Sigma$ such that

 (1) if $\alpha ,\beta \in \Phi$ and $\alpha +\beta \in \Sigma ,$ then $\alpha +\beta -k\in \Phi$ for some $k\ge 0\text{;}$ (2) if $\alpha ,\beta \in \Phi$ and $\alpha \ne \beta ,$ then the hyperplanes ${h}_{\alpha },{h}_{\beta }$ are not parallel.
Let ${U}_{\Phi }$ be the subgroup of $G$ generated by the ${U}_{\alpha },$ $\alpha \in \Phi \text{.}$ Then every element of ${U}_{\Phi }$ can be written uniquely in the form ${u}_{1}\cdots {u}_{r},$ where ${u}_{i}\in {U}_{{\alpha }_{i}}$ $\left(1\le i\le r\right)$ and ${\alpha }_{1},\dots ,{\alpha }_{r}$ are the elements of $\Phi$ in any fixed order.

 Proof. Let ${a}_{l}$ be the gradient of ${\alpha }_{i}\text{.}$ Then ${a}_{1},\dots ,{a}_{r}$ are all distinct by (2) and ${\Phi }_{0}=\left\{{a}_{1},\dots ,{a}_{r}\right\}$ is a closed subset of ${\Sigma }_{0}$ such that ${\Phi }_{0}\cap -{\Phi }_{0}=\varnothing$ (by (1) and (2)). Hence (2.1.2) there exists $w\in {W}_{0}$ such that $w{\Phi }_{0}\subseteq {\Sigma }_{0}^{+}\text{.}$ Choosing $n\in N$ such that $\nu \left(n\right)=w,$ we have $n{U}_{\Phi }{n}^{-1}={U}_{w\Phi }$ by (I); hence we may assume without loss of generality that ${\Phi }_{0}\subseteq {\Sigma }_{0}^{+}\text{.}$ Assume first that $\left({a}_{1},\dots ,{a}_{r}\right)$ in this order is a subsequence of the sequence $\left({b}_{1},\dots ,{b}_{n}\right)$ of (2.1.1). Let ${\Phi }_{1}=\left\{{\alpha }_{2},\dots ,{\alpha }_{r}\right\}\text{.}$ If $m,n$ are positive integers and $i>1,$ then $m{a}_{1}+n{a}_{i}\ne {a}_{1},$ hence ${U}_{\left[{\alpha }_{1},{\alpha }_{i}\right]}\subseteq {U}_{{\Phi }_{1}}\text{.}$ Now use the commutator axiom (V): if ${u}_{1}\in {U}_{{\alpha }_{1}}$ and ${u}_{i}\in {U}_{{\alpha }_{i}}$ $\left(i>1\right)$ we have ${u}_{1}{u}_{i}{u}_{1}^{-1}{u}_{i}^{-1}\in {U}_{\left[{\alpha }_{1},{\alpha }_{i}\right]}$ and therefore ${u}_{1}{u}_{i}{u}_{1}^{-1}\in {U}_{{\Phi }_{1}}\text{.}$ This shows that ${U}_{{\Phi }_{1}}$ is normal in ${U}_{\Phi },$ and hence that ${U}_{\Phi }={U}_{{\alpha }_{1}}{U}_{{\Phi }_{1}}\text{.}$ By induction on $r=\text{Card}\left(\Phi \right)$ we conclude that $UΦ=Uα1 Uα2⋯Uαr.$ Next we shall show that every $u\in {U}_{\Phi }$ is uniquely of the form $u={u}_{1}\cdots {u}_{r}$ with ${u}_{i}\in {U}_{{\alpha }_{i}}$ $\left(1\le i\le r\right)\text{.}$ Again by induction, it is enough to show that ${u}_{1}$ is uniquely determined by $u,$ and for this it is enough to show that ${U}_{{\alpha }_{1}}\cap {U}_{{\Phi }_{1}}=\left\{1\right\}\text{.}$ By (2.1.1) we can assume that ${a}_{1}$ is negative and ${a}_{2},\dots ,{a}_{r}$ ar are positive. Then ${U}_{{\alpha }_{1}}\subset {U}^{-}$ and ${U}_{{\Phi }_{1}}\subset {U}^{+},$ hence the result follows from (VI). This proves (2.6.1) for the particular ordering chosen. For an arbitrary ordering of $\Phi$ it follows from (2.6.2) below, whose proof we leave to the reader. $\square$

Let $X$ be a group with subgroups ${X}_{1},\dots ,{X}_{r}$ such that

 (1) $X={X}_{1}{X}_{2}\cdots {X}_{r}$ with uniqueness of expression; (2) ${X}_{i}{X}_{i+1}\cdots {X}_{r}$ is a normal subgroup of $X,$ for $i=1,2,\dots ,r\text{.}$
If $p$ is any permutation of $\left\{1,2,\dots ,r\right\},$ then $X={X}_{p\left(1\right)}{X}_{p\left(2\right)}\cdots {X}_{p\left(r\right)}$ with uniqueness of expression.

${U}^{+}=\prod _{a\in {\Sigma }_{0}^{+}}{U}_{\left(a\right)}$ with uniqueness of expression, the positive roots being taken in any fixed order. Likewise ${U}^{-}=\prod _{a\in {\Sigma }_{0}^{-}}{U}_{\left(a\right)}\text{.}$

 Proof. Arrange the positive roots in a sequence $\left({b}_{1},\dots ,{b}_{n}\right)$ satisfying (2.1.1). Follow the proof of (2.6.1) to show that ${U}^{+}=\prod _{i=1}^{n}{U}_{{b}_{i}}$ with uniqueness of expression, and then use (2.6.2). $\square$

If $S$ is any subset of $A,$ let $\left(S\right)$ denote the set of all $\alpha \in \Sigma$ which are $\ge 0$ on $S\text{.}$ Then the subgroups ${U}_{\left(S\right)},$ ${P}_{\left(S\right)}$ are defined.

Let $a\in {\Sigma }_{0}\text{.}$ If there is an affine root with gradient $a$ belonging to $\left(S\right),$ there is a least such affine root, say $\alpha :$ $\alpha \in \left(S\right)$ and $\alpha -1\notin \left(S\right)\text{.}$ Let ${U}_{\left(S\right),a}$ a denote the corresponding subgroup ${U}_{\alpha }$ of $G\text{.}$ If there is no affine root with gradient $a$ belonging to $\left(S\right),$ define ${U}_{\left(S\right),a}$ to be $\left\{1\right\}\text{.}$ Let ${U}_{\left(S\right)}^{+}$ (resp. ${U}_{\left(S\right)}^{-}\text{)}$ be the group generated by the ${U}_{\left(S\right),a}$ with $a\in {\Sigma }_{0}^{+}$ (resp. $a\in {\Sigma }_{0}^{-}\text{).}$

Suppose $S$ has non-empty interior. Then $P(S)= U(S)+H U(S)-.$

 Proof. By (2.6.1), ${U}_{\left(S\right)}^{±}=\prod _{a\in {\Sigma }_{0}^{±}}{U}_{\left(S\right),a},$ the product being taken in any order. Since ${P}_{\left(S\right)}$ is generated by the ${U}_{\left(S\right),a}$ and $H$ it is enough to show that ${U}_{\left(S\right)}^{+}H{U}_{\left(S\right)}^{-}$ is a group. For this it is enough to show that ${U}_{\left(S\right)}^{+}H{U}_{\left(S\right)}^{-}={U}_{\left(S\right)}^{-}H{U}_{\left(S\right)}^{+}\text{.}$ Arrange the positive roots in a sequence $\left({b}_{1},\dots ,{b}_{n}\right)$ satisfying (2.1.1) Let ${U}_{±r}={U}_{\left(S\right),±{b}_{r}}$ $\left(1\le r\le n\right),$ and let ${V}_{r}^{+}$ (resp. ${V}_{r}^{-}\text{)}$ be the subgroup of $G$ generated by ${U}_{-1},\dots ,{U}_{-r},{U}_{r+1},\dots ,{U}_{n}$ (resp. by ${U}_{1},\dots ,{U}_{r},{U}_{-r-1},\dots ,{U}_{-n}\text{).}$ Since $S$ has non-empty interior it follows from (III) that ${U}_{r}H{U}_{-r}$ is a group and therefore $UrHU-r= U-rHUr.$ Hence, by (2.6.1), we have $Vr-1+H Vr-1-= Vr+HVr-$ for $r=1,2,\dots ,n\text{.}$ Consequently $U(S)+HU(S)- =V0+HV0-= Vn+HVn-= U(S)-HU(S)+.$ $\square$

If $\alpha \in \Sigma$ then ${U}_{\alpha }\subseteq {P}_{\left(S\right)}⇒\alpha \in \left(S\right)\text{.}$

 Proof. Suppose ${U}_{\alpha }\subseteq {P}_{\left(S\right)}\text{.}$ Let $u\in {U}_{\alpha },$ then $u={u}^{+}h{u}^{-}$ with ${u}^{±}\in {U}_{\left(S\right)}^{±}$ and $h\in H\text{.}$ Suppose, to fix the ideas, that the gradient $a$ of $\alpha$ is a positive root. Then ${\left({u}^{+}\right)}^{-1}u=h{u}^{-}\in {U}^{+}\cap Z{U}^{-}=\left\{1\right\}$ by (VI). Hence $u={u}^{+},$ i.e. $u\in {U}_{\left(S\right)}^{+}\text{.}$ By (2.6.1) and (2.6.3) it follows that $u\in {U}_{\left(S\right),a}\text{.}$ Hence ${U}_{\alpha }\subseteq {U}_{\left(S\right),\alpha }$ and therefore by (II) we have $\alpha \in \left(S\right)\text{.}$ The reverse implication is obvious. $\square$

We shall now show that the group $G$ carries two BN-pair structures, one with Weyl group $W$ and the other with Weyl group ${W}_{0}$ (2.6.8) and (2,6.9). To establish these results we need a couple of preliminary lemmas.

 (1) If $\alpha \in \Sigma ,$ then ${U}_{-\alpha }-{U}_{-\alpha +1}\subset {U}_{\alpha }·{\nu }^{-1}\left({w}_{\alpha }\right)·{U}_{\alpha }\text{.}$ (2) If $a\in {\Sigma }_{0},$ then ${U}_{-a}\subset {U}_{a}·{\nu }_{0}^{-1}\left({w}_{a}\right)·{U}_{a}\text{.}$

(Here ${\nu }_{0}$ is the epimorphism $N\to {W}_{0}$ obtained by composing $\nu :N\to W$ with the epimorphism $W\to {W}_{0}$ defined by the semi-direct decomposition $W=T·{W}_{0}\text{.)}$

 Proof. (1) Let $u\in {U}_{-\alpha },$ then by (IV) we have $u∈Uα·ν-1 (wα)·Uα ∪UαHU-α+1.$ Suppose that $u\in {U}_{\alpha }H{U}_{-\alpha +1}\text{.}$ Then $u={u}_{1}h{u}_{2}$ with ${u}_{1}\in {U}_{\alpha },$ ${u}_{2}\in {U}_{-\alpha +1},$ $h\in H\text{.}$ Hence $u{u}_{2}^{-1}={u}_{1}h\in {U}_{\alpha }H\cap {U}_{-\alpha }=\left\{1\right\}$ by (VI). Consequently $u={u}_{2}\in {U}_{-\alpha +1},$ whence the result. (2) Let $u\in {U}_{-a},$ $u\ne 1\text{.}$ Then by (II) there exists an affine root $\alpha$ with gradient $a$ such that $u\in {U}_{-\alpha }-{U}_{-\alpha +1}\text{.}$ Hence by (1) just proved we have $u∈Uα·ν-1 (wα)·Uα⊂ Ua·ν0-1 (wa)·Ua.$ $\square$

Let $B={P}_{\left(C\right)}$ and let ${B}_{0}^{±}=Z{U}^{±}\text{.}$ Since $Z$ normalizes ${U}^{+},$ ${B}_{0}^{+}$ is a subgroup of $G,$ and is the semidirect product of $Z$ and ${U}^{+}$ by axiom (VI). Similarly for ${B}_{0}^{-}\text{.}$

 (1) If $\alpha \in \Pi$ and ${n}_{\alpha }\in {\nu }^{-1}\left({w}_{\alpha }\right),$ then $B{n}_{\alpha }B=B{n}_{\alpha }{U}_{\alpha }\text{.}$ (2) If $a\in {\Pi }_{0}$ and ${n}_{a}\in {\nu }_{0}^{-1}\left({w}_{a}\right),$ then ${B}_{0}^{+}{n}_{a}{B}_{0}^{+}={B}_{0}^{+}{n}_{a}{U}_{a}\text{.}$

 Proof. (1) Since $\alpha \left(C\right)>0$ we have ${U}_{\alpha }\subset B,$ hence $B{n}_{\alpha }{U}_{\alpha }\subseteq B{n}_{\alpha }B\text{.}$ Conversely, if $\beta \in \left(C\right)$ and $\beta \ne \alpha ,$ then since ${h}_{\alpha }$ is the only hyperplane separating $C$ and ${w}_{\alpha }C,$ it follows that $\beta$ is positive on ${w}_{\alpha }C,$ hence that ${U}_{\beta }\subset {P}_{\left({w}_{\alpha }C\right)}={n}_{\alpha }^{-1}B{n}_{\alpha }\text{.}$ Consequently $B{n}_{\alpha }{U}_{\beta }=B{n}_{\alpha }$ for all such $\beta ,$ and therefore $B{n}_{\alpha }B=B{n}_{\alpha }{U}_{\alpha }\text{.}$ (2) Let ${U}^{a}$ be the group generated by the ${U}_{b}$ with $b\in {\Sigma }_{0}^{+}$ and $b\ne a\text{.}$ By (2.6.3) we have ${U}^{+}={U}^{a}{U}_{a}\text{.}$ Hence, since ${w}_{a}$ permutes ${\Sigma }_{0}^{+}-\left\{a\right\},$ $B0+naB0+= B0+naZUa Ua=B0+ZUa naUa=B0+ naUa.$ $\square$

Let $B={P}_{\left(C\right)}\text{.}$ Then $\left(B,N\right)$ is a BN-pair in $G\text{.}$ We have $B\cap N=H,$ so that $\nu$ induces an isomorphism of $N/\left(B\cap N\right)$ onto the affine Weyl group $W\text{.}$ The distinguished generators of $N/\left(B\cap N\right)$ correspond under this isomorphism to the reflections in the walls of the chamber $C\text{.}$

 Proof. (1) Let $n\in B\cap N\text{.}$ Then $n{P}_{\left(C\right)}{n}^{-1}={P}_{\left(C\right)},$ so that ${P}_{\left(wC\right)}={P}_{\left(C\right)}$ where $w=\nu \left(n\right)\text{.}$ Suppose that $w\ne 1,$ so that $wC\ne C\text{.}$ Then there exists a hyperplane ${h}_{\alpha }$ $\left(\alpha \in \Sigma \right)$ separating $C$ and $wC,$ hence an affine root $\alpha$ which is positive on $wC$ and negative on $C\text{.}$ Then ${U}_{\alpha }\subset {P}_{\left(wC\right)}={P}_{\left(C\right)},$ and since $\alpha \notin \left(C\right)$ this contradicts (2.6.5). Hence $w=1$ and therefore $n\in H\text{.}$ Hence $B\cap N=H\text{.}$ (2) We have to verify the axioms (BN 1)-(BN 4) of (2.3). Of these, (BN 1) follows immediately from (I) and (VII). (BN 2) is clear: we define $R$ to be the set $\left\{{\nu }^{-1}\left({w}_{\alpha }\right):\alpha \in \Pi \right\}\text{.}$ So is (BN 4), because we have already shown that ${P}_{\left(wC\right)}\ne {P}_{\left(C\right)}$ if $w\ne 1\text{.}$ As to (BN 3), let $w\in W$ and $\alpha \in \Pi ,$ and let $n\in {\nu }^{-1}\left(w\right),$ ${n}_{\alpha }\in {\nu }^{-1}\left({w}_{\alpha }\right)\text{.}$ Then $nαBn⊂BnαBn= BnαUαn$ by (2.6.7). Suppose first that $\alpha \left(wC\right)>0\text{.}$ Then ${U}_{{w}^{-1}\alpha }\subset B$ and so $BnαUαn=B nαnUw-1α ⊂BnαnB.$ If on the other hand $\alpha \left(wC\right)<0,$ then $BnαUαn=B U-αnαn$ and by axiom (IV) we have $U-α⊂Uαnα HUα∪UαH U-α+1$ so that $BU-αnαn ⊂ BnαUαnαn ∪BU-α+1nα n = BU-αn∪B nαn,since -α+1∈(C) = BnU-w-1α ∪Bnαn ⊂ BnB∪BnαnB,$ since $-{w}^{-1}\alpha \in \left(C\right)\text{.}$ Hence ${n}_{\alpha }Bn\subset BnB\cup B{n}_{\alpha }nB$ in all cases. $\square$

It follows that all the results of (2.4) are applicable in the present situation. The group $B={P}_{\left(C\right)}$ and its conjugates are the Iwahori subgroups of $G\text{;}$ the groups ${P}_{\left(F\right)}$ and their conjugates, where $F$ is a facet of $C,$ are the parahoric subgroups.

Let ${B}_{0}^{+}=Z{U}^{+}\text{.}$ Then $\left({B}_{0}^{+},N\right)$ is a BN-pair in $G\text{.}$ We have ${B}_{0}^{+}\cap N=Z,$ so that ${\nu }_{0}$ induces an isomorphism of $N/\left({B}_{0}^{+}\cap N\right)$ onto the Weyl group ${W}_{0}\text{.}$ The distinguished generators of $N/\left({B}_{0}^{+}\cap N\right)$ correspond under this isomorphism to the reflections in the walls of the cone ${C}_{0}\text{.}$

 Proof. (1) Let $n\in {B}_{0}^{+}\cap N\text{.}$ Then $n={z}_{0}{u}^{+}$ $\left({z}_{0}\in Z,u\in {U}^{+}\right)$ and therefore ${n}_{0}={z}_{0}^{-1}n\in N\cap {U}^{+}\text{.}$ So we have ${n}_{0}=\prod _{\alpha \in {\Sigma }_{0}^{+}}{u}_{a}$ with ${u}_{a}\in {U}_{a}$ by (2.6.3). For each $a\in {\Sigma }_{0}^{+},$ pick an affine root $\alpha$ with gradient $a$ such that ${u}_{a}\in {U}_{\alpha }\text{.}$ Applying (2.2.1) to this set of affine roots, we see that ${n}_{0}\in {U}_{\left(t{C}_{0}\right)}$ for some translation $t\in T,$ and hence ${n}_{1}={z}^{-1}{n}_{0}z\in {U}_{\left({C}_{0}\right)}$ for some $z\in Z\text{.}$ We have ${n}_{1}\in N\cap {U}_{\left({C}_{0}\right)}\subset N\cap B=H$ by (2.6.8). Hence ${n}_{0}\in H\cap {U}^{+}=\left\{1\right\}$ by axiom (VI). So $n={z}_{0}\in Z$ and therefore ${B}_{0}^{+}\cap N=Z\text{.}$ (2) We have to verify the BN-axioms. As before, (BN 1) follows immediately from (I) and (VII). (BN 2): take the distinguished generators of $N/Z$ to be ${\nu }_{0}^{-1}\left({w}_{a}\right),$ $a\in {\Pi }_{0}\text{.}$ As to (BN 4), let $a\in {\Pi }_{0}$ and let ${n}_{a}\in {\nu }_{0}^{-1}\left({w}_{a}\right),$ then if ${B}_{0}^{+}={n}_{a}{B}_{0}^{+}{n}_{a}^{-1}$ we have $B0=B0+ naB0+ na-1 = B0+na Uana-1 by (2.6.7) = B0+U-a$ so that ${U}_{-a}\subset {B}_{0}\cap {U}^{-}=Z{U}^{+}\cap {U}^{-}=\left\{1\right\}$ by axiom (VI). Since ${U}_{-a}\ne \left\{1\right\}$ by axiom (II), we have a contradiction. Finally, (BN 3). Let $w\in {W}_{0},$ $a\in {\Pi }_{0},$ $n\in {\nu }_{0}^{-1}\left(w\right),$ ${n}_{a}\in {\nu }_{0}^{-1}\left({w}_{a}\right)\text{.}$ Then ${n}_{a}{B}_{0}^{+}n\subset {B}_{0}^{+}{n}_{a}{B}_{0}^{+}n={B}_{0}^{+}{n}_{a}{U}_{a}n$ by (2.6.7). Suppose first that ${w}^{-1}a$ is positive, then $B0+naUan= B0+nan Uw-1a⊂ B0+nanB0+.$ If on the other hand ${w}^{-1}a$ is negative, then $B0+naUan = B0+U-a nan ⊂ B0+naUa nanby (2.6.6) = B0+U-an= B0+nU-w-1a ⊂B0+nB0+.$ $\square$

Remark. Of course there is a corresponding result for ${B}_{0}^{-}=Z{U}^{-},$

$G=Bn{B}_{0}^{+}\text{.}$

 Proof. We remark first that $G$ is generated by ${B}_{0}^{+}$ and the ${U}_{-a}$ $\left(a\in {\Pi }_{0}\right)\text{.}$ For let $G\prime =⟨{B}_{0}^{+},{\left({U}_{-a}\right)}_{a\in {\Pi }_{0}}⟩,$ and let $\alpha$ be an affine root with gradient $a\in {\Pi }_{0}\text{.}$ Then $G\prime$ contains the group $⟨{U}_{\alpha },{U}_{-\alpha },H⟩$ and hence by axiom (IV) contains ${\nu }^{-1}\left({w}_{\alpha }\right)\text{.}$ Since $G\prime$ contains $Z$ it follows that $N\subset G\prime ,$ hence that ${U}_{\beta }\subset G\prime$ for all $\beta \in \Sigma \text{.}$ Hence $G\prime =G\text{.}$ Next, let $a\in {\Pi }_{0}$ and let $C\prime$ be any chamber in $A\text{.}$ Then $U-a⊂ P(C′)N Ua. (1)$ For let $\alpha$ be an affine root with gradient $a\text{.}$ If $-\alpha \in \left(C\prime \right)$ we have ${U}_{-\alpha }\subset {P}_{\left(C\prime \right)}\text{.}$ If on the other hand $\alpha \in \left(C\prime \right),$ then $U-α- U-α+1 ⊂ Uα·ν-1 (wα)· Uαby (2.6.6) ⊂ P(C′)N Ua.$ This proves (1), from which it follows that ${U}_{-\alpha }\subset nBN{U}_{a}$ for all $n\in N,$ so that $NU-a⊂BNUa. (2)$ Now to show that $G=BN{B}_{0}^{+},$ it is enough to show that $BN{B}_{0}^{+}g\subseteq BN{B}_{0}^{+}$ for all $g$ in a set of generators of $G\text{.}$ Hence it is enough to show that $BN{B}_{0}^{+}{U}_{-a}\subseteq BN{B}_{0}^{+}$ for $\alpha \in {\Pi }_{0}\text{.}$ But $BNB0+U-a = BNUaUa U-a=BNUa U-aUa ⊂ BNUaν0-1 (wa)UaUa by (2.6.6) = BNν0-1(wa) U-aU+ = BNU-aU+ ⊂ BNUaU+ by (2) above = BNU+=BNB0+.$ $\square$

Let ${T}^{+}$ (resp. ${T}^{++}\text{)}$ be the set of all translations $t\in T$ such that $t\left(0\right)\in {\stackrel{‾}{C}}_{0}$ (resp. $t\left(0\right)\in {C}_{0}^{\perp }\text{).}$ Let ${Z}^{+}={\nu }^{-1}\left({T}^{+}\right),$ ${Z}^{++}={\nu }^{-1}\left({T}^{++}\right),$ so that ${Z}^{+}$ and ${Z}^{++}$ are semigroups contained in $Z\text{.}$

Also let $K$ denote the maximal parahoric subgroup ${P}_{\left(0\right)}\text{.}$

 (1) (Cartan decomposition) $G=K{Z}^{++}K,$ and the mapping $t↦K·{\nu }^{-1}\left(t\right)·K$ is a bijection of ${T}^{++}$ onto $K\G/K\text{.}$ (2) (Iwasawa decomposition) $G=KZ{U}^{-},$ and the mapping $t↦K·{\nu }^{-1}\left(t\right)·{U}^{-}$ is a bijection of $T$ onto $K\G/{U}^{-}\text{.}$ (3) If ${z}_{1}\in {Z}^{++}$ and ${z}_{2}\in Z$ are such that $K{z}_{1}K\cap K{z}_{2}{U}^{-}=\varnothing ,$ then ${z}_{1}{z}_{2}^{-1}\in {Z}^{+}\text{.}$ (4) If $z\in {Z}^{++},$ then $KzK\cap Kz{U}^{-}=Kz\text{.}$

 Proof. As in (2.4) we shall identify $N/H$ with $W$ by means of the isomorphism induced by $\nu \text{.}$ Thus $KtK$ means $K·{\nu }^{-1}\left(t\right)·K,$ and so on. In particular, $K=B{W}_{0}B$ by (2.3.2). (1) If $w\in W,$ then $KwK=B{W}_{0}w{W}_{0}B$ by (2.6.8) and (2.3.6). Now $W$ is the disjoint union of the double cosets ${W}_{0}t{W}_{0}$ as $t$ runs through ${T}^{++}\text{.}$ Hence $G=BWB$ is the disjoint union of the cosets $KtK$ with $t\in {T}^{++}\text{.}$ (2) We shall prove (2) with ${U}^{-}$ replaced by ${U}^{+}\text{:}$ this merely amounts to replacing the chamber ${C}_{0}$ by its opposite, $-{C}_{0}\text{.}$ From (2.6.10) we have $G=BWB0+=BW0 B0+⊂BW0B· B0+=KB0+=K ZU+.$ To show that $K{t}_{1}{U}^{+}$ and $K{t}_{2}{U}^{+}$ are disjoint if ${t}_{1}\ne {t}_{2},$ it is enough to show that $K\cap {B}_{0}^{+}\subset H{U}^{+}\text{.}$ So let ${z}_{0}{u}^{+}\in K,$ where ${z}_{0}\in Z$ and ${u}^{+}\in {U}^{+}\text{.}$ By (2.2.1) there exists $t\in T$ such that ${u}^{+}\in {U}_{\left(t{C}_{0}\right)}=z{U}_{\left({C}_{0}\right)}{z}^{-1}\subset zB{z}^{-1},$ where $z\in {\nu }^{-1}\left(t\right)\text{.}$ Hence ${z}_{0}{u}^{+}\in {z}_{0}zB{z}^{-1}\cap K,$ hence ${z}_{0}zB$ meets $Kz\text{:}$ or, putting ${t}_{0}=v\left({z}_{0}\right)\in T,$ ${t}_{0}tB$ meets $Kt=B{W}_{0}Bt,$ and hence ${t}_{0}tB$ meets $BwBt$ for some $w\in {W}_{0}\text{.}$ Consequently $B{t}_{0}tB\subset BwBtB\text{.}$ Now by expressing $w\in {W}_{0}$ as a reduced word in the generators ${w}_{a}$ $\left(a\in {\Pi }_{0}\right),$ it follows easily from the axiom (BN 3) that $BwBtB\subset B{W}_{0}tB\text{.}$ Hence $B{t}_{0}tB\subset B{W}_{0}tB$ and therefore ${t}_{0}t\in {W}_{0}t$ by (2.3.1). Hence ${t}_{0}\in {W}_{0}\cap T,$ so that ${t}_{0}=1$ and ${z}_{0}\in H,$ as required. The proofs of (3) and (4) are best expressed in the language of buildings (2.4). (3) Suppose that ${t}_{1}\in {T}^{++}$ and ${t}_{2}\in T$ are such that $K{t}_{1}K$ meets $K{t}_{2}{U}^{-}\text{.}$ Then  $K{t}_{1}^{-1}$ contains an element belonging to ${u}^{-}{t}_{2}^{-1}K$ for some ${u}^{-}\in {U}^{-}\text{.}$ As above, by (2.2.1) we have ${u}^{-}\in {U}_{\left(-t{C}_{0}\right)}\subset {P}_{\left(C\prime \right)}$ for some chamber $C\prime \text{.}$ Hence $K{t}_{1}^{-1}$  meets ${P}_{\left(C\prime \right)}{t}_{2}^{-1}K\text{.}$ Let ${x}_{1}={t}_{1}\left(0\right)$ and $C″={t}_{1}\left(C\prime \right)\text{.}$ Then ${P}_{\left({x}_{1}\right)}={t}_{1}K{t}_{1}^{-1}$ meets ${t}_{1}{P}_{\left(C\prime \right)}{t}_{2}^{-1}K={P}_{\left(C″\right)}{t}_{1}{t}_{2}^{-1}K\text{.}$ Let ${t}_{0}={t}_{1}{t}_{2}^{-1}$ and put ${x}_{0}={t}_{0}\left(0\right)\text{.}$ Let $ℐ$ be the building associated with $\left(B,N\right)$ as in (2.4). Identify the affine space $A$ with the apartment ${𝒜}_{0}$ by means of the bijection $j$ defined in (2.4.1). Let $\rho$ be the retraction of $ℐ$ onto ${𝒜}_{0}$ with centre $C″$ (2.4.6) and let $g\in {P}_{\left({x}_{1}\right)}\cap {P}_{\left(C″\right)}{t}_{0}K\text{.}$ Since $g\in {P}_{\left({x}_{1}\right)},$ the image under $g$ of the segment $\left[0,{x}_{1}\right]\subset {𝒜}_{0}$ is the segment $\left[g\left(0\right),{x}_{1}\right]\text{.}$ Since $g\in p{t}_{0}K$ for some $p\in {P}_{\left(C″\right)},$ we have $g(0)=pt0(0) =p(x0)∈p𝒜0$ and therefore $\rho \left(g\left(0\right)\right)={x}_{0},$ $\rho \left({x}_{1}\right)={x}_{1}$ (because ${x}_{1}\in {𝒜}_{0}\text{).}$ Hence the image of the segment $\left[g\left(0\right),{x}_{1}\right]$ under the retraction $\rho$ is a polygonal line $\gamma$ in ${𝒜}_{0}$ joining ${x}_{0}$ to ${x}_{1}\text{.}$ Consequently there exist real numbers ${\lambda }_{i}$ with $0={\lambda }_{0}<{\lambda }_{1}<\cdots <{\lambda }_{n}=1$ such that, if ${y}_{i}=\rho \left(g\left({\lambda }_{i}{x}_{1}\right)\right),$ then $\gamma$ is the union of the segments $\left[{y}_{i-1},{y}_{i}\right]$ $\left(1\le i\le n\right)\text{.}$ From (2.4.2) and the definition of $\rho$ it follows that there exist elements ${w}_{i}\in W$ such that $[yi-1,yi]= wi[λi-1x1,λix1].$ Writing ${w}_{i}={w}_{i,0}{\tau }_{i}$ with ${w}_{i,0}\in {W}_{0}$ and ${\tau }_{i}\in T,$ we have $yi-yi-1= (λi-λi-1) wi,0(x1).$ Now ${x}_{1}\in {\stackrel{‾}{C}}_{0},$ hence ${x}_{1}-w{x}_{1}\in {C}_{0}^{\perp }$ for all $w\in {W}_{0}\text{.}$ So we have $yi-yi-1= (λi-λi-1) x1-yi′$ with ${y}_{i}^{\prime }\in {C}_{0}^{\perp }\text{.}$ Summing over $i,$ we get $x1-x0=yn- y0=x1-∑yi′$ and therefore ${x}_{0}=\sum {y}_{i}^{\prime }\in {C}_{0}^{\perp }\text{.}$ Since ${x}_{0}={t}_{1}{t}_{2}^{-1}\left(0\right),$ we have proved that ${t}_{1}{t}_{2}^{-1}\in {T}^{+}\text{;}$ as required. (4) Let $t\in {T}^{++}$ and let $g\in KtK\cap Kt{U}^{-}\text{.}$ Then by (2.2.1) as before we have ${g}^{-1}\in {P}_{\left({C}_{0}^{\prime }\right)}{t}^{-1}K\cap K{t}^{-1}K$ for some translate ${C}_{0}^{\prime }$ of the cone $-{C}_{0},$ and hence ${g}_{1}={g}^{-1}t\in {P}_{\left({C}_{0}^{\prime }\right)}{P}_{\left(x\right)}\cap {P}_{\left(0\right)}{P}_{\left(x\right)},$ where $x={t}^{-1}\left(0\right)\text{.}$ Now $x\in -{\stackrel{‾}{C}}_{0},$ and $-{\stackrel{‾}{C}}_{0}$ intersects ${C}_{0}^{\prime }\text{.}$ Hence there exists $y\in {C}_{0}^{\prime }$ such that $x$ lies in the segment $\left[0y\right]\text{.}$ Let ${g}_{1}\left(x\right)=x\prime \in ℐ\text{.}$ Since ${g}_{1}\in p{P}_{\left(x\right)}$ for some $p\in {P}_{\left(0\right)}=K,$ we have $x\prime =p\left(x\right)$ and $p\left(0\right)=0\text{.}$ Hence $d(0,x′)=d (p(0),p(x))= d(0,x).$ Next, since ${g}_{1}\in p\prime {P}_{\left(x\right)}$ for some $p\prime \in {P}_{\left({C}_{\prime }^{0}\right)},$ we have $x\prime =p\prime \left(x\right)$ and $y=p\prime \left(y\right)\text{.}$ Consequently $d(x′,y)= d(p′(x),p′(y))= d(x,y).$ It follows that $w\prime \in \left[0y\right]$ by the uniqueness of geodesies in $ℐ$ (2.4.9) and hence that $x\prime =x\text{.}$ Consequently ${g}_{1}$ fixes $x,$ i.e. normalizes ${P}_{\left(x\right)}$ and therefore (2.3.6) ${g}_{1}\in {P}_{\left(x\right)}={t}^{-1}Kt\text{.}$ Hence $g\in Kt\text{.}$ $\square$

Groups of $p\text{-adic}$ type

Let $G$ be an 'abstract' group with affine root structure $\left(N,\nu ,{\left({u}_{\alpha }\right)}_{\alpha \in \Sigma }\right)$ of type ${\Sigma }_{0}\text{.}$ Now suppose in addition that $G$ carries a Hausdorff topology compatible with its group structure such that

 (VIII) $N$ and the ${U}_{\alpha }$ are closed subgroups of $G\text{;}$ (IX) $B$ is a compact open subgroup of $G\text{.}$

A Hausdorff topological group carrying an affine root structure satisfying axioms (I)-(IX) inclusive will be called a simply-connected group of $p\text{-adic}$ type. If $𝓀$ is a $p\text{-adic}$ field (i.e. a locally compact field which is neither discrete nor connected) then $𝓀$ is complete with respect to a discrete valuation, and the residue field is finite. The universal Chevalley group $G\left({\Sigma }_{0},𝓀\right),$ and more generally the group of $𝓀\text{-rational}$ points of a simply-connected simple linear algebraic group defined over $𝓀,$ then inherits a topology from $𝓀$ which satisfies (VIII) and (IX).

The two new axioms have the following consequences:

$H$ is open in $N$ and compact.

 Proof. $H=B\cap N$ (2.6.8), so that $H$ is an open subgroup of $N,$ hence closed in $N,$ hence closed in $G\text{.}$ Since $H$ is contained in $B$ it follows that $H$ is compact. $\square$

A closed subset $M$ of $G$ is compact if and only if $M$ meets only finitely many cosets $BwB\text{.}$

 Proof. The cosets $BwB$ are mutually disjoint, open and compact. The assertion follows immediately from this. $\square$

Comparing (2.7.2) with (2.4.13), we see that a subset $M$ of $G$ is bounded in the sense of (2.4) if and only if $M$ has compact closure. Hence, from (2.4.15):

Every compact subgroup of $G$ is contained in a maximal compact subgroup. The maximal compact subgroups are precisely the maximal parahoric  subgroups, and form $\left(l+1\right)$ conjugacy classes. All parahoric subgroups are open and compact. $G$ itself is locally compact, but not compact.

The last two assertions follow from the facts that a parahoric subgroup is a finite union of double cosets $BwB,$ and that $G$ itself is not.

For each $\alpha \in \Sigma ,$ the index $\left({U}_{\alpha }:{U}_{\alpha +1}\right)$ is finite, and ${U}_{\alpha }$ is compact.

 Proof. There exists a chamber $C\prime$ on which $\alpha$ is positive and for which ${h}_{\alpha }$ is one of the walls. Transforming by an element of $W,$ we may assume that $C\prime =C,$ i.e. that $\alpha \in \Pi \text{.}$ Since $B$ is open and compact, $B/\left(B\cap {w}_{\alpha }B{w}_{\alpha }\right)$ is compact and discrete, hence finite. But $B/(B∩wαBwα) ≅ (wαBwαB)/ (wαBwα) = (wαBwαUα)/ (wαBwα) by (2.6.7) ≅ Uα/(Uα∩wαBwα) = Uα/Uα+1$ because ${U}_{\alpha }\cap {w}_{\alpha }B{w}_{\alpha }={U}_{\alpha }\cap {P}_{\left({w}_{\alpha }C\right)}={U}_{\alpha +1}$ by (2.6.5). Hence $\left({U}_{\alpha }:{U}_{\alpha +1}\right)$ is finite. Also ${U}_{\alpha }$ is a closed subgroup of $B,$ hence is compact. $\square$

 (1) For each $a\in {\Sigma }_{0},$ the group ${U}_{\left(a\right)}$ is closed in $G,$ and for each affine root $\alpha$ with gradient $a,$ ${U}_{\alpha }$ is an open compact subgroup of ${U}_{\left(a\right)}\text{.}$ (2) ${U}^{+}$ is a closed subgroup of $G$ and ${U}_{\left({C}_{0}\right)}$ is an open compact subgroup of ${U}^{+}\text{.}$

 Proof. (1) By (2.6.5) ${U}_{\left(a\right)}\cap B$ is of the form ${U}_{\alpha },$ where $\alpha$ is some affine root with gradient $a\text{.}$ Hence this ${U}_{\alpha }$ is open in ${U}_{\left(a\right)}\text{.}$ Conjugating ${U}_{\alpha }$ by elements of $Z,$ each ${U}_{\alpha }$ is open in ${U}_{\left(a\right)}\text{.}$ Since ${U}_{\alpha }$ is compact (2.7.4) it follows that ${U}_{\left(a\right)}$ is locally compact and therefore a closed subgroup of $G\text{.}$ (2) $B\cap {U}^{+}={U}_{\left({C}_{0}\right)},$ hence ${U}_{\left({C}_{0}\right)}$ is open in ${U}^{+}\text{;}$ also by (2.6.1) ${U}_{\left({C}_{0}\right)}$ is a product of ${U}_{\alpha }\text{'s,}$ hence is compact. Hence ${U}^{+}$ is locally compact, hence closed in $G\text{.}$ $\square$

${B}_{0}^{+}=Z{U}^{+}$ is locally compact.

 Proof. $Z$ is locally compact, because it has $H$ as a compact open subgroup by (2.7.1). Also ${U}^{+}$ is locally compact by (2.7.5), hence so is ${B}_{0}^{+}\text{.}$ $\square$

Consider again the group $G={SL}_{l+1}\left(𝓀\right)$ (2.5.1) where $𝓀$ is now a $p\text{-adic}$ field. Let $ℴ$ be the ring of integers, $𝓅$ the maximal ideal of $ℴ$ and $𝓀=ℴ/𝓅$ the (finite) residue field. $ℴ$ is a compact open subring of $𝓀\text{.}$

$G$ is a closed subset of ${𝓀}^{{\left(l+1\right)}^{2}},$ hence is locally compact (and totally disconnected). The maximal compact subgroup $K$ may be taken to be ${SL}_{l+1}\left(ℴ\right)\text{.}$ Let $\stackrel{‾}{G}={SL}_{l+1}\left(k\right),$ then the projection of $ℴ$ onto $k$ induces an epimorphism $\pi :K\to \stackrel{‾}{G}\text{.}$ Let $\stackrel{‾}{B}$ be the group of upper triangular matrices in $\stackrel{‾}{G},$ then $B={\pi }^{-1}\left(\stackrel{‾}{B}\right)$ is an Iwahori subgroup of $G\text{.}$ ${B}_{0}^{+}$ is the group of upper triangular matrices in $G,$ and $Z$ is the diagonal matrices.

Notes and references

This is a typed version of the book Spherical Functions on a Group of $p\text{-adic}$ Type by I. G. Macdonald, Magdalen College, University of Oxford. This book is copyright the University of Madras, Madras 5, India and was first published November 1971.