Spherical Functions on a Group of p-adic Type

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 13 February 2014

Chapter II: Groups of p-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. 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*. For each non-zero aA* let aA be the image of 2a/a,a under this isomorphism. Let ha denote the kernel of a, and let wa be the reflection in the hyperplane ha: wa(x)=x-a (x)a (xA). We make wa act also on A*, by transposition: wa(b)=bwa.

A root system Σ0 in A* is a subset Σ0 of A* satisfying the following axioms:

(RS 1) Σ0 is finite, spans A* and does not contain 0;
(RS 2) aΣ0wa(Σ0)=Σ0;
(RS 3) a,bΣ0a(b).

We shall assume throughout that Σ0 is reduced:

(RS 4) If a,bΣ0 are proportional, then b=±a;
and irreducible:
(RS 5) If Σ0=Σ1Σ2 such that a1,a2=0 for all a1Σ1 and a2Σ2, then either Σ1 or Σ2 is empty.

The elements of Σ0 are called roots. The group W0 generated by the reflections wa (aΣ0) is a finite group of isometries of A, called the Weyl group of Σ0. The connected components of the set A-aΣ0ha are open simplicial cones called the chambers of Σ0. These chambers are permuted simply transitively by W0, and the closure of each chamber is a fundamental region for W0.

Choose a chamber C0. This choice determines a set Π0 of l roots such that C0 is the intersection of the open half-spaces {xA:a(x)>0} for aΠ0. The elements of Π0 are called the simple roots (relative to the chamber C0). A root b is said to be positive or negative according as it is positive or negative on C0. Let Σ0+ (resp. Σ0-) denote the set of positive (resp. negative) roots.

Each root b can be expressed as a linear combination of the simple roots: b=aΠ0 maa with coefficients ma all of the same sign. The integer ma, the sum of the coefficients, is called the height of b. There is a unique root (the 'highest root') whose height is maximal.

Let C0 denote the 'obtuse cone' C0= { xA:x,y 0for allyC0 } . If xC0 and wW0, then x-wxC0.

A subset Φ0 of Σ0 is said to be closed if a,bΦ0and a+bΣ0 a+b Φ0. Suppose Φ0 is closed and Φ0-Φ0=. Then there exists wW0 such that wΦ0Σ0+.

The Weyl group W0 is generated by the reflections wa for aΠ0. Hence any wW can be written as a product w=w1wr, where wi=wai and aiΠ0 (1ir). If the number of factors wi is as small as possible, then w1wr is a reduced word for w, and r is the length of w, written l(w). The length of w is also equal to the number of positive roots b such that w-1b is negative. More precisely, these roots are bi=w1wi-1ai (1ir). (All this of course is relative to the choice of the chamber C0.)

In particular, there exists a unique element w0 of W0 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 (b1,,bn) such that for each r=0,1,,n the set of roots { -b1,-b2,, -br,br+1, br+2,,bn } is the set of positive roots relative to some chamber Cr.


Let w1w2wn be a reduced word for w0, where wi=wai, aiΠ0, and as before put bi=w1w2wi-1ai (1in). For each r, wr+1wn is a reduced word, and therefore from the facts quoted above { ar+1,wr+1 ar+2,, wr+1wr+2 wn-1an } =Σ0+ ( wr+1wn Σ0- ) . Consequently { br+1, br+2,, bn } = (w1wrΣ0+) (w0Σ0-) = (w1wrΣ0+) Σ0+ and therefore {b1,b2,,br} =(w1wrΣ0-) Σ0+ so that {-b1,-b2,,-br} =(w1wrΣ0+) Σ0- and finally {-b1,,-br,br+1,,bn} =w1wrΣ0+ is the set of positive roots for the chamber Cr=w1wrC0.

Affine roots

We retain the notation of (2.1). For each aΣ0 and k we have an affine-linear function a+k on A (namely, xa(x)+k). Let Σ={a+k:aΣ0,k}. The elements of Σ are called affine roots. We shall denote them by Greek letters α,β,. If α is an affine root, so are -α and α+k for all integers k. If α=a+k then a (Σ0) is the gradient of α (in the usual sense of elementary calculus).

For each αΣ, let hα be the (affine) hyperplane on which α vanishes, and let wα be the reflection in hα. (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 wα as α runs through Σ is an infinite group of displacements of A, called the Weyl group of Σ or the affine Weyl group. W0 is the subgroup of W which fixes the point 0. The translations belonging to W form a free abelian group T of rank l, and W is the semi-direct product of T and W0.

For each aΣ0 let ta=wa wa+1T. The ta (aΠ0) are a basis of T. We have ta(0)=a, and the mapping TA defined by tt(0) maps T isomorphically onto the lattice spanned by the a for aΣ0.

The affine Weyl group acts on Σ by transposition: if wW, αΣ then w(α) is defined to be αw-1.

The connected components of the set A-αΣhα are open rectilinear l-simplexes called the chambers of Σ. The chambers are permuted simply transitively by W, and the closure of each chamber is a fundamental region for W. The (non-empty) faces of the chambers are locally closed simplexes, called facets. Each point xA lies in a unique facet F. The subgroup of W which fixes x also fixes each point of F.

Choose a chamber C. This choice determines a set Π of l+1 affine roots, such that C is the intersection of the open half-spaces {xA:α(x)>0} for αΠ. The affine roots belonging to Π are called the simple affine roots (relative to the chamber C).

The affine Weyl group W is generated by the relations wα for αΠ. As in the case of W0, we define the length l(w) of an element w of W. The length of w is also equal to the number of affine roots α which are positive on C and negative on wC, i.e. to the number of hyperplanes hα which separate C and wC.

If F is a facet of C, then the subgroup of W which fixes a point of F is generated by the wα (αΠ) which fix x.

We shall need the following result later:

Let Φ be a set of affine roots whose gradients are all positive and distinct. Then there exists a translation tT such that each αΦ is positive on t(C0).


Let the elements of Φ be a+ra (aΣ0+) and let tT. We have (a+ra)(t(x)) =a(x+t(0))+ ra=a(x)+a (t(0))+ra. If xC0 then a(x)>0 for all positive roots a. Hence a+ra will be positive on t(C0) provided that a(t(0))+ra0. Hence we have only to choose t so that t(0) satisfies a(t(0))- ra for all aΣ0+ which occur as gradients of elements of Φ, and this is certainly possible.

For later use, it will be convenient to choose C to be the unique chamber of Σ which is contained in the cone C0 and has the origin 0 as one vertex. The following lemma presupposes this choice of C.

For each wW, let Φw be the set of affine roots which are positive on C and negative on wC.

Let w be an element of W such that CwC0. Then for all wW0 we have Φww= ΦwwΦw.


If βΦw is positive on wwC, then α=w-1β is negative on C and positive on wC, hence α(0)=0 (because the origin 0 belongs to the intersection of the closures of C and wC). Consequently α is negative on C0, hence β is negative on wC0 and therefore on C: contradiction. Hence β must be negative on wwC0, i.e. βΦww.

Next, let αΦw and put β=wα. Then β(wC0)>0 and β(wwC0)<0, hence β is positive on C and negative on wwC, i.e. βΦww. Hence ΦwwϕwΦww.

Conversely, let βΦww. If β(wC)<0, then βΦw; and if β(wC)>0, then w-1βΦw. This establishes the opposite inclusion. Finally, the sets Φw and wΦw are disjoint, because β(wC)<0 for all βΦw and β(wC)>0 for all βwΦw.

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

(Root system of type Al).

Let A be the set of all x=(x0,,xl)l+1 such that i=0lxi=0. The scalar product is x,y=xiyi. Let ei be the ith coordinate function on A (ei(x)=xi) and let Σ0= {ei-ej:ij}. Then Σ0 satisfies (RS 1)-(RS 5). The Weyl group W0 acts by permuting the coordinates x0,,xl of xA, hence is isomorphic to the symmetric group Sl+1, of order (l+1)!. We may take the chamber C0 to be C0= { xA:x0>x1> >xl } and the simple roots are then ei-1-ei (1il). The chamber C is the open simplex C= { xA:1+xl>x0 >x1>>xl } .


Let G be a group, B and N subgroups of G, and R a subset of the coset space N/(BN). The pair (B,N) 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=BN is normal in N.
(BN 2) R generates the group W=N/H and each rR has order 2.
(BN 3) rBwBwBBrwB, for all rR and wW.
(BN 4) rBrB, for each rR.

The group W=N/H is the Weyl group of (B,N). 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. If S is a subset of W, then BSB means wSBwB.

One shows that the set R is uniquely determined by B and N: namely an element wW belongs to R if and only if BBwB is a group. The elements of R are called the distinguished generators of W.

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 wBwB is a bijection of W onto the double coset space B\G/B.

2.3.2 For each subset S of R let WS be the subgroup of W generated by S, and let PS=BWSB. Then PS is a subgroup of G.

2.3.3 The mapping SPS is an inclusion-preserving bijection of the set of subsets of R onto the set of subgroups of G containing B.

2.3.4 If S, S are subsets of R, and PS is conjugate to PS, then S=S.

2.3.5 If S, S are subsets of R, and wW, then PSwPS=B WSwWSB.

2.3.6 Each PS is its own normalizer in G.

2.3.7 If rR and wW and l(rw)>l(w), then BrwB=BrB·BwB. (The length l(w) of wW is defined as the length of a reduced word in the generating set R, just as in (2.1) and (2.2).)


Let G be a group with BN-pair (B,N) as in (2.3). We shall assume now that the Weyl group of (B,N) '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 Σ0 in a Euclidean space A, a chamber C for the associated affine root system Σ, and a homomorphism ν of N onto the affine Weyl group W such that

(1) Ker(ν)=H
so that we may identify (via ν) the Weyl group N/H of (B,N) with the Weyl group W of Σ;
(2) under this identification, the distinguished generators of N/H are
the reflections in the walls of the chamber C; in other words, R={wα:αΠ}.

The conjugates of B in G are called Iwahori subgroups of G. 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 PS for some proper subset S of R, and S is uniquely determined by P.

We set up a one-one correspondence SF 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 wαR which fix F. In this correspondence, C; R; and S a vertex of CC is a maximal proper subset of R.

If SF we write PF for PS.

Each parahoric subgroup P determines uniquely a facet F(P) of C: namely F(P)=FP is conjugate to PF.

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

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

The group G acts on by inner automorphisms: g(P,x)=(gPg-1,x).


Consider the following subset 𝒜0 of : 𝒜0=wW (wBw-1). Equivalently, 𝒜0 is the union of the facets n(P) where nN and PB.

Remark. Since we have identified N/H with W by means of the isomorphism induced by ν, expressions such as wPw-1 are meaningful, where wW and P is a subgroup of G containing B. Hence W acts on 𝒜0: w(P,x)=(wPw-1,x)=(nPn-1,x) for any nN such that ν(n)=w.

There exists a unique bisection j:A𝒜0 such that

(1) for each facet F of C and each xF, j(x)=(PF,x);
(2) jw=wj for all wW.


Let yA. Then y is congruent under W to a unique point x of C: say y=wx where wW and xF, and F is a facet of the chamber C. Define j(y) to be the point (wPFw-1,x)𝒜0. We have to check that j is well defined. If also y=wx, then w-1w fixes x and therefore, by a property of reflection groups recalled in (2.2), belongs to the subgroup of W generated by the wα, αΠ, which fix x. In other words, w-1wWS, where SF. Since WSPS=PF it follows that wPFw-1=wPFw-1. Hence j is well defined, and clearly satisfies (1) and (2). The uniqueness of j is obvious.

If g𝒜0=𝒜0 then j-1(g|𝒜0)jW.


Let 𝒷0 be the chamber j(C)=(B). We have g𝒷0=n0𝒷0 for some n0N. Hence g0=n0-1g fixes 𝒜0 as a whole and also fixes the chamber 𝒷0 and all its facets; also it maps each facet to another facet of the same type. Transferring our attention to A, it follows that γ=j-1(g0|𝒜0)j maps A_ bijectively onto A and fixes the chamber C and all its facets. Hence γ also fixes the adjacent chambers wαC (αΠ), and so on. It follows that γ is the identity mapping and hence that j-1(g|𝒜0)j=ν(n0)W.

The subcomplexes g𝒜0 of , as g runs through G, are called the apartments of the building . If 𝒜=g𝒜0 is an apartment, we may transport the Euclidean structure of A to 𝒜 via the bijection (g|𝒜0)j. If also 𝒜=g𝒜0, then it follows from (2.4.2) that the two mappings (g|𝒜0)j and (g|𝒜0)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 d𝒜(x,y) defined for x,y𝒜.

Thus the building may be thought of as obtained by sticking together many copies of the Euclidean space A.

Any two facets of are contained in a single apartment.


Consider two facets (P1) and (P2), where P1, P2 are parahoric subgroups of G: say Pi=giPFigi-1 (i=1,2), where F1, F2 are facets of the chamber C in A. By (2.3.1) we can write g1-1g2=b1nb2 with b1,b2B and nN. If g=g1b1, then P1=gPF1g-1 ,P2=g (nPF2n-1) g-1 and therefore (P1) and (P2) are both in g𝒜0.

G is transitive on the set of pairs (𝒷,𝒜), where 𝒜 is an apartment and 𝒷 is a chamber in 𝒜.


We have 𝒷=(g0Bg0-1)=g0𝒷0 for some g0G, where 𝒷0=(B). Hence we may assume that 𝒷=𝒷0. If 𝒜=g𝒜0 contains 𝒷0, then g-1𝒷0 is a chamber in 𝒜0 and hence g-1𝒷0=n𝒷0 for some nN. So if g1=gn we have 𝒜=g1𝒜0 and 𝒷0=g1𝒷0.

Let 𝒜, 𝒜 be, two apartments and let 𝒷 be a chamber contained in 𝒜𝒜. Then there exists a unique bijecton ρ:𝒜𝒜 such that

(1) There exists gG such that ρx=gx for all x𝒜;
(2) ρx=x for all x𝒷.
Moreover, ρx=x for all x𝒜𝒜, and d𝒜(x,y)=d𝒜(ρx,ρy) for all x,y𝒜.


The existence of ρ follows immediately from (2.4.4). Uniqueness follows from (2.4.2): if ρ1,ρ2:𝒜𝒜 both satisfy (1) and (2), then ρ1ρ2-1 fixes 𝒜 as a whole and fixes the chamber 𝒷, hence is the identity map. That d𝒜(x,y)=D𝒜(ρx,ρy) follows from the definition of the metric on an apartment given after (2.4.2).

It remains to show that ρx=x for all x𝒜𝒜. Suppose =(P) is a facet contained in 𝒜𝒜. By (2.4.4) we may assume that 𝒜=𝒜0, 𝒜=g𝒜0 and 𝒷=𝒷0=(B). Since g𝒷0=𝒷0 it follows that g normalizes B and therefore by (2.3.6) that gB.

Since (P)𝒜0g𝒜0 we have P=n1PFn1-1 =g(n2PFn2-1) g-1 for some facet F of C and n1,n2N. Since PF is its own normalizer (2.3.6), we have n1-1gn2PF and therefore Bn2PF=Bn1PF. By (2.3.5), Bn2WFB=Bn1WFB where WF is the subgroup of W which fixes F. Hence (2.3.1) n2WF=n1WF; since WFPF it follows that n1PFn1-1=n2PFn2-1, hence =g=ρ.

Retraction of the building onto an apartment

Let 𝒜 be an apartment and 𝒷 a chamber in 𝒜. Then there exists a unique mapping ρ:𝒜 such that, for all apartments 𝒜 containing 𝒷, ρ|𝒜 is the bisection 𝒜𝒜 of (2.4.5).


Let x. By (2.4.3) there exists an apartment 𝒜1 containing x and 𝒷. Let ρ1:𝒜1𝒜 be the isomorphism of (2.4.5). If also x and 𝒷 are contained in another apartment 𝒜2, then we have also ρ2:𝒜2𝒜. But then ρ1-1ρ2:𝒜2𝒜1 is the isomorphism of (2.4.5) for the apartments 𝒜1 and 𝒜2 and the chamber 𝒷. Hence ρ1-1ρ2 fixes x, that is to say ρ1(x)=ρ2(x). It follows that ρ as in the theorem is well defined, and the uniqueness is obvious.

The mapping ρ of (2.4.7) is called the retraction of onto 𝒜 with centre 𝒷. It has the following properties:

(1) ρx=x for all x𝒜.
(2) For each facet in , ρ| is an affine isometry of onto ρ.
(3) If x𝒷, then ρ-1(x)={x}.


(1) and (2) are clear. As to (3), let be a facet such that =ρ is a facet of 𝒷. By (2.4.3) there exists an apartment 𝒜 containing 𝒷 and , and ρ|𝒜 is a bijection of 𝒜 onto 𝒜 leaving 𝒷 fixed. It follows that =.

(i) There exists a unique function d:×+ such that d|𝒜×𝒜 is the metric d𝒜, for each apartment 𝒜 in .
(ii) If ρ is a retraction of onto an apartment 𝒜 as in (2.4.6), then d(ρ(x),ρ(y))d(x,y) for all x,y.
(iii) d is a G-invariant metric on .


Let x,y0. By (2.4.3) there is an apartment 𝒜 containing x and y. We wish to define d(x,y) to be the Euclidean distance d𝒜(x,y) between x and y in 𝒜. We must therefore show that, if x, y belong also to another apartment 𝒜, then d𝒜(x,y)=d𝒜(x,y).

Let 𝒷 be a chamber in 𝒜 such that x𝒷 and let 𝒷 be a chamber in 𝒜 such that y𝒷. By (2.4.3) there is an apartment 𝒜 containing 𝒷 and 𝒷. 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-invariant (from the definition of d𝒜).

(ii) Let 𝒜 be an apartment containing x and y. The image under ρ of the straight line segment joining x to y in 𝒜 will be a polygonal line in 𝒜 of the same total length, by (2.4.7)(ii), and endpoints ρ(x) and ρ(y). Hence the result.

(iii) Let x,y,z and let 𝒜 be an apartment containing x and y. Let ρ be a retraction of onto 𝒜. Then d(x,y) d(x,ρ(z))+ d(ρ(z),y) d(x,z)+ d(z,y) by (ii) above, since ρ(x)=x and ρ(y)=y. Hence d satisfies the triangle equality, and we have already observed that d is G-invariant.

Let x,y. Then there is a unique geodesic joining x to y.


Let 𝒜 be an apartment containing x and y and let [xy] denote the straight line segment joining x to y in 𝒜. Suppose z lies on a geodesic from x to y. Then d(x,z)+d(z,y)=d(x,y). Let z0[x,y] be such that d(x,z0)=d(x,z) and hence also d(z0,y)=d(z,y). Let ρ be the retraction of onto 𝒜 with centre a chamber 𝒷 such that z0𝒷. 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,ρ(z) are in the same Euclidean space 𝒜 it follows that ρ(z)=z0. Hence zρ-1(z0) and therefore z=z0 by (2.4.7)(iii).

It follows from (2.4.9) that the straight line segment [xy] lies in all apartments containing x and y.

is complete with respect to the metric d.


Let (xn) be a Cauchy sequence in . If ρ is a retraction of onto an apartment 𝒜, then (ρ(xn)) is a Cauchy sequence in 𝒜, by (2.4.8)(ii), and therefore converges to a point x𝒜. The set of real numbers d(x,g(x)), where gG is such that xgx, has a positive lower bound μ, and we shall have d(ρ(xn),x)< 14μ,d (xn,xn+1)< 14μ for all sufficiently large n.

Now each ρ(xn) is of the form gnxn for some gnG. Put yn=gn-1x. 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. Hence yn=yn+1==y and we have d(xn,y)= d(xn,yn)= d(ρ(xn),x) 0 as n. Hence xny.

Fixed point theorem

A subset X of is convex if x,yX[xy]X.

Let X be a bounded non-empty subset of . Then the group of isometries γ of such that γ(X)X has a fixed point in the closure of the convex hull of X.

For the proof we shall require the following lemma:

Let x,y,z and let m be the midpoint of [xy]. Then d(z,x)2+ d(z,y)2 2d(z,m)2+ 12d(x,y)2.


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 m𝒷. Retract onto 𝒜 with centre 𝒷. 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.

Proof of 2.4.11.

Let δ(X)=supx,yXd(x,y) be the diameter of X. Choose a real number k(0,1) and let fX be the set of all m such that m is the midpoint of [xy] with x,yX and d(x,y)k·δ(X). Clearly fX is non-empty and is contained in the convex hull of X. If mfX and zX, 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 k1<1.

If now mfX and zfX, then again m is the midpoint of a segment [xy] with x,yX and d(x,y)kδ(X), 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 k2<1. From (2) it follows that δ(fnX)0 asn. (3)

For each n, pick xnfnX. Then by (1) and (2) d(xn,xn+1) k1δ(fnX) k1k2nδ(X). Hence (xn) is a Cauchy sequence in , hence by (2.4.10) converges to some x. Clearly x lies in the closure of the convex hull of X.

Now let γ be an isometry of such that γXX. Then γfnXfnX for all n. Let xn=γ(xn). Then (xn) is a Cauchy sequence with xnfnX for all n and converges to γx. But by (3) d(xn,xn)0 as n. Hence γx=x. Hence x is a fixed point of the group of isometries γ of such that γXX.

A subset M of G is said to be bounded if MX is bounded for all bounded subsets X of .

M is bounded M intersects only finitely many double cosets BwB (wW).


Let ρ be the retraction of onto the apartment 𝒜0 with centre 𝒷0=(B). Then X is bounded ρX is bounded ρX is contained in a finite union of closed chambers 𝒷 of 𝒜0. Hence M is bounded M𝒷 is bounded, for each chamber 𝒷𝒜0M𝒷0 is bounded.

If mM, define wmW by the relation mBwmB. Then M𝒷0 is bounded mMwm𝒷0 is bounded the set {wm:mM} is finite M intersects only finitely many BwB.

A subgroup Γ of Γ is bounded Γ is contained in a parahoric subgroup.


Let x. Then X=Γx is bounded. Applying the fixed point theorem (2.4.11), we see that there exists y such that Γy=y. If y lies in the facet (P), then Γ normalizes the parahoric subgroup P and hence ΓP by (2.3.6).

Replacing Γ by a conjugate, we may assume that ΓPS, where S is a proper subset of R={wα:αΠ}. Now PS=BWSB, and WS is finite because SR. Hence Γ is bounded by (2.4.13).

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.

Groups with affine root structure

(a) Universal Chevalley groups

We retain the notation of (2.1) and (2.2). Let L= { uA*:u(a) for allaΣ0 } . Then L is a lattice of rank l (=dimA) in A*, containing Σ0.

Let 𝓀 be any field, and 𝓀+,𝓀× the additive and multiplicative groups of 𝓀. Associated with the pair (Σ0,𝓀) there is a universal Chevalley group G=G(Σ0,𝓀). 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 ua(ξ) (aΣ0,ξ𝓀) and contains elements h(χ) (χHom(L,𝓀×)) with various relations between the u's and the h's. In particular, each ua is an injective homomorphism of 𝓀+ into G, and h is an injective homomorphism of Hom(L,𝓀×) into G.

If Σ0 is the root system of type Al (2.2.3) then the universal Chevalley group G(Σ0,𝓀) is the special linear group SLl+1(𝓀). In this case the ua(ξ) and h(χ) may be taken as follows:

If a=ei-ej (ij) then ua(ξ)=1+ξEij, where 1 is the unit matrix and Eij is the matrix with 1 in the (i,j) place and 0 elsewhere. h(χ) is the diagonal matrix with χ(e0),,χ(el) in order down the diagonal.

Now suppose that the field 𝓀 carries a discrete valuation v, i.e. a surjective homomorphism v:𝓀× such that v(ξ+η)min(v(ξ),v(η)) for all ξ,η𝓀 (conventionally v(0)=+). We consider certain subgroups of the Chevalley group G=G(Σ0,𝓀).

First, let Z be the subgroup consisting of the h(χ) and let N be the normalizer of Z. Then there is a canonical isomorphism N/ZW0, and one can show that this lifts uniquely to a homomorphism ν of N onto the affine Weyl group W. The restriction of ν to is described as follows. If χHom(L,𝓀×) then vχ is a homomorphism L, which defines a linear form A*, hence a vector in A. Then ν(h(χ)) is translation by this vector, and this translation belongs to W. We have ν(Z)=T and Z=ν-1(T), where T is as before the translation subgroup of W. The kernel H of ν is the set of all h(χ) such that χ(L) is contained in the group of units of 𝓀.

Next, for each αΣ we have a subgroup Uα of G, defined as follows. If α=a+k, then Ua+k= { ua(χ):ξ𝓀 andv(ξ)k } . If Φ is any subset of Σ, we denote by UΦ the subgroup of G generated by the Uα such that αΦ, and by PΦ the subgroup generated by UΦ and H. Then the triple (N,ν,(Uα)αΣ) has the following properties, for all nN and α,βΣ:

(I) nUαn-1= Uν(n)α.
From this it follows that H normalizes each Uα, hence each UΦ. Consequently PΦ=H· UΦ=UΦ ·H.
(II) If k is a positive integer, then Uα+k is a proper subgroup of Uα, and k0Uα+k={1}.
From this it follows that for each aΣ0, the union of the Ua+r, r is a group, which we denote by U(a). (In the present situation U(a)=ua(𝓀+).)

The next three properties relate to a pair of subgroups Uα and Uβ in the following situations: (III) β+α is a positive constant; (IV) β+α=0; (V) neither β+α nor β-α is constant (i.e. the hyperplanes hα, hβ are not parallel).

(III) If β=-α+k with k>0, then P{α,β}=UαHUβ.
(IV) P{α,-α} (Uα·ν-1(wα)·Uα) (UαHU-α+1) .

For α,βΣ let [α,β] denote the set of all γΣ of the form mα+nβ with m, n positive integers, and let [Uα,Uβ] be the subgroup of G generated by all commutators (u,v)=uvu-1v-1 with uUα and vUβ. Then from Chevalley's commutator relations we have

(V) If hα and hβ are not parallel, then [Uα,Uβ]U[α,β].

Let U+=U(a):aΣ0+ be the subgroup generated by the U(a) with a positive, and likewise U-=U(a):aΣ0-. Then

(VI) U+ZU-={1}.


(VII) G is generated by N and the Uα (αΣ).

(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. Let 𝒮 be a maximal 𝓀-split torus in 𝒢, and let 𝒵,𝒩 be the centralizer and normalizer respectively of 𝒮 in 𝒢. Let G=𝒢(𝓀) be the group of 𝓀-rational points of 𝒢. Then one of the main results of the structure theory of Bruhat and Tits ([BTi1966], [BTi1967]) is:

There exists a reduced irreducible root system Σ0, subgroups N and Uα (αΣ) of G, and a surjective homomorphism ν:NW such that the triple (N,ν,(Uα)αΣ) satisfies (I)-(VII) above.

In fact one takes N=𝒩(𝓀), Z=𝒵(𝓀), and H=Ker(ν) is the set of all z𝒵(𝓀) such that v(χ(z))=0 for all characters χ of 𝒵. W0 is the relative Weyl group of 𝒢 with respect to 𝒮, but Σ0 is not in general the relative root system, even if the latter is reduced. But every aΣ0 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) then we are back in the situation of (a) above.

If 𝓀 is a p-adic field (i.e. if the residue field of 𝓀 is finite) then G=𝒢(𝓀) 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 (G,K) 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 Σ0 be a reduced irreducible root system and let Σ, W be the associated affine root system and affine Weyl group, as in (2.2). An affine root structure of type Σ0 on a group G is a triple (N,ν,(Uα)αΣ), where N and the Uα are subgroups of G, and ν 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 (N,ν,(Uα)αΣ) of type Σ0 (2.5.3). We shall develop some consequences of the axioms (I)-(VII).

Let Φ be a subset of Σ such that

(1) if α,βΦ and α+βΣ, then α+β-kΦ for some k0;
(2) if α,βΦ and αβ, then the hyperplanes hα,hβ are not parallel.
Let UΦ be the subgroup of G generated by the Uα, αΦ. Then every element of UΦ can be written uniquely in the form u1ur, where uiUαi (1ir) and α1,,αr are the elements of Φ in any fixed order.


Let al be the gradient of αi. Then a1,,ar are all distinct by (2) and Φ0={a1,,ar} is a closed subset of Σ0 such that Φ0-Φ0= (by (1) and (2)). Hence (2.1.2) there exists wW0 such that wΦ0Σ0+. Choosing nN such that ν(n)=w, we have nUΦn-1=UwΦ by (I); hence we may assume without loss of generality that Φ0Σ0+.

Assume first that (a1,,ar) in this order is a subsequence of the sequence (b1,,bn) of (2.1.1). Let Φ1={α2,,αr}. If m,n are positive integers and i>1, then ma1+naia1, hence U[α1,αi]UΦ1. Now use the commutator axiom (V): if u1Uα1 and uiUαi (i>1) we have u1uiu1-1ui-1U[α1,αi] and therefore u1uiu1-1UΦ1. This shows that UΦ1 is normal in UΦ, and hence that UΦ=Uα1UΦ1. By induction on r=Card(Φ) we conclude that UΦ=Uα1 Uα2Uαr.

Next we shall show that every uUΦ is uniquely of the form u=u1ur with uiUαi (1ir). Again by induction, it is enough to show that u1 is uniquely determined by u, and for this it is enough to show that Uα1UΦ1={1}. By (2.1.1) we can assume that a1 is negative and a2,,ar ar are positive. Then Uα1U- and UΦ1U+, hence the result follows from (VI).

This proves (2.6.1) for the particular ordering chosen. For an arbitrary ordering of Φ it follows from (2.6.2) below, whose proof we leave to the reader.

Let X be a group with subgroups X1,,Xr such that

(1) X=X1X2Xr with uniqueness of expression;
(2) XiXi+1Xr is a normal subgroup of X, for i=1,2,,r.
If p is any permutation of {1,2,,r}, then X=Xp(1)Xp(2)Xp(r) with uniqueness of expression.

U+=aΣ0+U(a) with uniqueness of expression, the positive roots being taken in any fixed order. Likewise U-=aΣ0-U(a).


Arrange the positive roots in a sequence (b1,,bn) satisfying (2.1.1). Follow the proof of (2.6.1) to show that U+=i=1nUbi with uniqueness of expression, and then use (2.6.2).

If S is any subset of A, let (S) denote the set of all αΣ which are 0 on S. Then the subgroups U(S), P(S) are defined.

Let aΣ0. If there is an affine root with gradient a belonging to (S), there is a least such affine root, say α: α(S) and α-1(S). Let U(S),a a denote the corresponding subgroup Uα of G. If there is no affine root with gradient a belonging to (S), define U(S),a to be {1}. Let U(S)+ (resp. U(S)-) be the group generated by the U(S),a with aΣ0+ (resp. aΣ0-).

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


By (2.6.1), U(S)±=aΣ0±U(S),a, the product being taken in any order. Since P(S) is generated by the U(S),a and H it is enough to show that U(S)+HU(S)- is a group. For this it is enough to show that U(S)+HU(S)-=U(S)-HU(S)+.

Arrange the positive roots in a sequence (b1,,bn) satisfying (2.1.1) Let U±r=U(S),±br (1rn), and let Vr+ (resp. Vr-) be the subgroup of G generated by U-1,,U-r,Ur+1,,Un (resp. by U1,,Ur,U-r-1,,U-n). Since S has non-empty interior it follows from (III) that UrHU-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,,n. Consequently U(S)+HU(S)- =V0+HV0-= Vn+HVn-= U(S)-HU(S)+.

If αΣ then UαP(S)α(S).


Suppose UαP(S). Let uUα, then u=u+hu- with u±U(S)± and hH. Suppose, to fix the ideas, that the gradient a of α is a positive root. Then (u+)-1u=hu-U+ZU-={1} by (VI). Hence u=u+, i.e. uU(S)+. By (2.6.1) and (2.6.3) it follows that uU(S),a. Hence UαU(S),α and therefore by (II) we have α(S). The reverse implication is obvious.

We shall now show that the group G carries two BN-pair structures, one with Weyl group W and the other with Weyl group W0 (2.6.8) and (2,6.9). To establish these results we need a couple of preliminary lemmas.

(1) If αΣ, then U-α-U-α+1Uα·ν-1(wα)·Uα.
(2) If aΣ0, then U-aUa·ν0-1(wa)·Ua.

(Here ν0 is the epimorphism NW0 obtained by composing ν:NW with the epimorphism WW0 defined by the semi-direct decomposition W=T·W0.)


(1) Let uU-α, then by (IV) we have uUα·ν-1 (wα)·Uα UαHU-α+1. Suppose that uUαHU-α+1. Then u=u1hu2 with u1Uα, u2U-α+1, hH. Hence uu2-1=u1hUαHU-α={1} by (VI). Consequently u=u2U-α+1, whence the result.

(2) Let uU-a, u1. Then by (II) there exists an affine root α with gradient a such that uU-α-U-α+1. Hence by (1) just proved we have uUα·ν-1 (wα)·Uα Ua·ν0-1 (wa)·Ua.

Let B=P(C) and let B0±=ZU±. Since Z normalizes U+, B0+ is a subgroup of G, and is the semidirect product of Z and U+ by axiom (VI). Similarly for B0-.

(1) If αΠ and nαν-1(wα), then BnαB=BnαUα.
(2) If aΠ0 and naν0-1(wa), then B0+naB0+=B0+naUa.


(1) Since α(C)>0 we have UαB, hence BnαUαBnαB. Conversely, if β(C) and βα, then since hα is the only hyperplane separating C and wαC, it follows that β is positive on wαC, hence that UβP(wαC)=nα-1Bnα. Consequently BnαUβ=Bnα for all such β, and therefore BnαB=BnαUα.

(2) Let Ua be the group generated by the Ub with bΣ0+ and ba. By (2.6.3) we have U+=UaUa. Hence, since wa permutes Σ0+-{a}, B0+naB0+= B0+naZUa Ua=B0+ZUa naUa=B0+ naUa.

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


(1) Let nBN. Then nP(C)n-1=P(C), so that P(wC)=P(C) where w=ν(n). Suppose that w1, so that wCC. Then there exists a hyperplane hα (αΣ) separating C and wC, hence an affine root α which is positive on wC and negative on C. Then UαP(wC)=P(C), and since α(C) this contradicts (2.6.5). Hence w=1 and therefore nH. Hence BN=H.

(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 {ν-1(wα):αΠ}. So is (BN 4), because we have already shown that P(wC)P(C) if w1. As to (BN 3), let wW and αΠ, and let nν-1(w), nαν-1(wα). Then nαBnBnαBn= BnαUαn by (2.6.7). Suppose first that α(wC)>0. Then Uw-1αB and so BnαUαn=B nαnUw-1α BnαnB. If on the other hand α(wC)<0, then BnαUαn=B U-αnαn and by axiom (IV) we have U-αUαnα HUαUαH U-α+1 so that BU-αnαn BnαUαnαn BU-α+1nα n = BU-αnB nαn,since -α+1(C) = BnU-w-1α Bnαn BnBBnαnB, since -w-1α(C). Hence nαBnBnBBnαnB in all cases.

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

Let B0+=ZU+. Then (B0+,N) is a BN-pair in G. We have B0+N=Z, so that ν0 induces an isomorphism of N/(B0+N) onto the Weyl group W0. The distinguished generators of N/(B0+N) correspond under this isomorphism to the reflections in the walls of the cone C0.


(1) Let nB0+N. Then n=z0u+ (z0Z,uU+) and therefore n0=z0-1nNU+. So we have n0=αΣ0+ua with uaUa by (2.6.3). For each aΣ0+, pick an affine root α with gradient a such that uaUα. Applying (2.2.1) to this set of affine roots, we see that n0U(tC0) for some translation tT, and hence n1=z-1n0zU(C0) for some zZ. We have n1NU(C0)NB=H by (2.6.8). Hence n0HU+={1} by axiom (VI). So n=z0Z and therefore B0+N=Z.

(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 ν0-1(wa), aΠ0. As to (BN 4), let aΠ0 and let naν0-1(wa), then if B0+=naB0+na-1 we have B0=B0+ naB0+ na-1 = B0+na Uana-1 by (2.6.7) = B0+U-a so that U-aB0U-=ZU+U-={1} by axiom (VI). Since U-a{1} by axiom (II), we have a contradiction.

Finally, (BN 3). Let wW0, aΠ0, nν0-1(w), naν0-1(wa). Then naB0+nB0+naB0+n=B0+naUan by (2.6.7).

Suppose first that w-1a is positive, then B0+naUan= B0+nan Uw-1a B0+nanB0+. If on the other hand w-1a is negative, then B0+naUan = B0+U-a nan B0+naUa nanby (2.6.6) = B0+U-an= B0+nU-w-1a B0+nB0+.

Remark. Of course there is a corresponding result for B0-=ZU-,



We remark first that G is generated by B0+ and the U-a (aΠ0). For let G=B0+,(U-a)aΠ0, and let α be an affine root with gradient aΠ0. Then G contains the group Uα,U-α,H and hence by axiom (IV) contains ν-1(wα). Since G contains Z it follows that NG, hence that UβG for all βΣ. Hence G=G.

Next, let aΠ0 and let C be any chamber in A. Then U-a P(C)N Ua. (1) For let α be an affine root with gradient a. If -α(C) we have U-αP(C). If on the other hand α(C), 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-αnBNUa for all nN, so that NU-aBNUa. (2)

Now to show that G=BNB0+, it is enough to show that BNB0+gBNB0+ for all g in a set of generators of G. Hence it is enough to show that BNB0+U-aBNB0+ for αΠ0. 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+.

Let T+ (resp. T++) be the set of all translations tT such that t(0)C0 (resp. t(0)C0). Let Z+=ν-1(T+), Z++=ν-1(T++), so that Z+ and Z++ are semigroups contained in Z.

Also let K denote the maximal parahoric subgroup P(0).

(1) (Cartan decomposition) G=KZ++K, and the mapping tK·ν-1(t)·K is a bijection of T++ onto K\G/K.
(2) (Iwasawa decomposition) G=KZU-, and the mapping tK·ν-1(t)·U- is a bijection of T onto K\G/U-.
(3) If z1Z++ and z2Z are such that Kz1KKz2U-=, then z1z2-1Z+.
(4) If zZ++, then KzKKzU-=Kz.


As in (2.4) we shall identify N/H with W by means of the isomorphism induced by ν. Thus KtK means K·ν-1(t)·K, and so on. In particular, K=BW0B by (2.3.2).

(1) If wW, then KwK=BW0wW0B by (2.6.8) and (2.3.6). Now W is the disjoint union of the double cosets W0tW0 as t runs through T++. Hence G=BWB is the disjoint union of the cosets KtK with tT++.

(2) We shall prove (2) with U- replaced by U+: this merely amounts to replacing the chamber C0 by its opposite, -C0. From (2.6.10) we have G=BWB0+=BW0 B0+BW0B· B0+=KB0+=K ZU+. To show that Kt1U+ and Kt2U+ are disjoint if t1t2, it is enough to show that KB0+HU+. So let z0u+K, where z0Z and u+U+.

By (2.2.1) there exists tT such that u+U(tC0)=zU(C0)z-1zBz-1, where zν-1(t). Hence z0u+z0zBz-1K, hence z0zB meets Kz: or, putting t0=v(z0)T, t0tB meets Kt=BW0Bt, and hence t0tB meets BwBt for some wW0. Consequently Bt0tBBwBtB.

Now by expressing wW0 as a reduced word in the generators wa (aΠ0), it follows easily from the axiom (BN 3) that BwBtBBW0tB. Hence Bt0tBBW0tB and therefore t0tW0t by (2.3.1). Hence t0W0T, so that t0=1 and z0H, as required.

The proofs of (3) and (4) are best expressed in the language of buildings (2.4).

(3) Suppose that t1T++ and t2T are such that Kt1K meets Kt2U-. Then
 Kt1-1 contains an element belonging to u-t2-1K for some u-U-. As above, by (2.2.1) we have u-U(-tC0)P(C) for some chamber C. Hence Kt1-1
 meets P(C)t2-1K. Let x1=t1(0) and C=t1(C). Then P(x1)=t1Kt1-1 meets t1P(C)t2-1K=P(C)t1t2-1K. Let t0=t1t2-1 and put x0=t0(0).

Let be the building associated with (B,N) as in (2.4). Identify the affine space A with the apartment 𝒜0 by means of the bijection j defined in (2.4.1). Let ρ be the retraction of onto 𝒜0 with centre C (2.4.6) and let gP(x1)P(C)t0K. Since gP(x1), the image under g of the segment [0,x1]𝒜0 is the segment [g(0),x1]. Since gpt0K for some pP(C), we have g(0)=pt0(0) =p(x0)p𝒜0 and therefore ρ(g(0))=x0, ρ(x1)=x1 (because x1𝒜0). Hence the image of the segment [g(0),x1] under the retraction ρ is a polygonal line γ in 𝒜0 joining x0 to x1. Consequently there exist real numbers λi with 0=λ0<λ1<<λn=1 such that, if yi=ρ(g(λix1)), then γ is the union of the segments [yi-1,yi] (1in). From (2.4.2) and the definition of ρ it follows that there exist elements wiW such that [yi-1,yi]= wi[λi-1x1,λix1]. Writing wi=wi,0τi with wi,0W0 and τiT, we have yi-yi-1= (λi-λi-1) wi,0(x1). Now x1C0, hence x1-wx1C0 for all wW0. So we have yi-yi-1= (λi-λi-1) x1-yi with yiC0. Summing over i, we get x1-x0=yn- y0=x1-yi and therefore x0=yiC0. Since x0=t1t2-1(0), we have proved that t1t2-1T+; as required.

(4) Let tT++ and let gKtKKtU-. Then by (2.2.1) as before we have g-1P(C0)t-1KKt-1K for some translate C0 of the cone -C0, and hence g1=g-1tP(C0)P(x)P(0)P(x), where x=t-1(0). Now x-C0, and -C0 intersects C0. Hence there exists yC0 such that x lies in the segment [0y].

Let g1(x)=x. Since g1pP(x) for some pP(0)=K, we have x=p(x) and p(0)=0. Hence d(0,x)=d (p(0),p(x))= d(0,x). Next, since g1pP(x) for some pP(C0), we have x=p(x) and y=p(y). Consequently d(x,y)= d(p(x),p(y))= d(x,y). It follows that w[0y] by the uniqueness of geodesies in (2.4.9) and hence that x=x. Consequently g1 fixes x, i.e. normalizes P(x) and therefore (2.3.6) g1P(x)=t-1Kt. Hence gKt.

Groups of p-adic type

Let G be an 'abstract' group with affine root structure (N,ν,(uα)αΣ) of type Σ0. Now suppose in addition that G carries a Hausdorff topology compatible with its group structure such that

(VIII) N and the Uα are closed subgroups of G;
(IX) B is a compact open subgroup of G.

A Hausdorff topological group carrying an affine root structure satisfying axioms (I)-(IX) inclusive will be called a simply-connected group of p-adic type. If 𝓀 is a p-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(Σ0,𝓀), and more generally the group of 𝓀-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.


H=BN (2.6.8), so that H is an open subgroup of N, hence closed in N, hence closed in G. Since H is contained in B it follows that H is compact.

A closed subset M of G is compact if and only if M meets only finitely
many cosets BwB.


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

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 (l+1) 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 αΣ, the index (Uα:Uα+1) is finite, and Uα is compact.


There exists a chamber C on which α is positive and for which hα is one of the walls. Transforming by an element of W, we may assume that C=C, i.e. that αΠ.

Since B is open and compact, B/(BwαBwα) is compact and discrete, hence finite. But B/(BwαBwα) (wαBwαB)/ (wαBwα) = (wαBwαUα)/ (wαBwα) by (2.6.7) Uα/(UαwαBwα) = Uα/Uα+1 because UαwαBwα=UαP(wαC)=Uα+1 by (2.6.5). Hence (Uα:Uα+1) is finite. Also Uα is a closed subgroup of B, hence is compact.

(1) For each aΣ0, the group U(a) is closed in G, and for each affine
root α with gradient a, Uα is an open compact subgroup of U(a).
(2) U+ is a closed subgroup of G and U(C0) is an open compact subgroup of U+.


(1) By (2.6.5) U(a)B is of the form Uα, where α is some affine root with gradient a. Hence this Uα is open in U(a). Conjugating Uα by elements of Z, each Uα is open in U(a). Since Uα is compact (2.7.4) it follows that U(a) is locally compact and therefore a closed subgroup of G.

(2) BU+=U(C0), hence U(C0) is open in U+; also by (2.6.1) U(C0) is a product of Uα's, hence is compact. Hence U+ is locally compact, hence closed in G.

B0+=ZU+ is locally compact.


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 B0+.

Consider again the group G=SLl+1(𝓀) (2.5.1) where 𝓀 is now a p-adic field. Let be the ring of integers, 𝓅 the maximal ideal of 
and 𝓀=/𝓅 the (finite) residue field. is a compact open subring of 𝓀.

G is a closed subset of 𝓀(l+1)2, hence is locally compact (and totally disconnected). The maximal compact subgroup K may be taken to be SLl+1(). Let G=SLl+1(k), then the projection of onto k induces an epimorphism π:KG. Let B be the group of upper triangular matrices in G, then B=π-1(B) is an Iwahori subgroup of G. B0+ 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-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.

Published by the Ramanujan Institute, University of Madras, and printed at the Baptist Mission Press, 41A Acharyya Jagadish Bose Road, Calcutta 17, India.

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