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 III: Spherical functions on a group of p-adic type

Throughout this chapter and the succeeding ones, G is a simply-connected group of p-adic type, as defined in (2.7), and the notation and hypotheses of Chapter II remain in force.

The root system Σ1

For each affine root α, the group Uα is of finite index in Uα-1, by (2.7.4). We define qα= (Uα-1:Uα). Then it follows from axiom (I) that (3.1.1) qwα=qα for all wW. In particular, since wα+1wα sends α+r to α+r+2 for all r, we have (3.1.2) qα=qα+2. Hence, if aΣ0 and r, qa+r is equal to either qa or qa+1. But it can happen that qaqa+1.

As another particular case of (3.1.1) we have (3.1.3) qa+r=qwa+r for all wW0.

We define a set Σ1 of vectors in A* as follows:

(1) Σ0Σ1Σ012Σ0;
(2) If aΣ0, then 12aΣ1qaqa+1.

Σ1 is an irreducible root system in A*, with Weyl group W0.


We have to show that Σ1 satisfies the axioms of (2.1), except for (RS 4). (RS 1) and (RS 5) follow immediately from the corresponding facts for Σ0, and (RS 2) is an immediate consequence of (3.1.3). So it remains to show that b(c) for all b,cΣ1.

If b,c are both in Σ0 there is nothing to prove. If c12Σ0, say c=12a, then c=2a, and b(2a) is an even integer if bΣ0, an integer if b12Σ0. The only remaining possibility is that cΣ0 and b=12a12Σ0. Then we have to show that a(c) is an even integer.

Suppose that a(c) is odd. The translation tc (defined by tc(0)=c) is in the affine Weyl group, and tc(a)=a-a(c). Hence by (3.1.1) we have qa=qa-a(c), and therefore qa=qa+1 by (3.1.2), because a(c) is odd. But then b=12aΣ1.

For each aΣ0 define qa/2=qa+1 /qa. Thus if bΣ012Σ0 we have qb1bΣ1. From (3.1.3), qwb=qb for all wW0.

If G1,G2 are subgroups of a group and G1G2, we write (G1:G2) to mean (G2:G1)-1. With this convention,

(Ua-r:Ua)= qa/2[r/2] qar where [r2] is the integral part of r2.


The proof is a straightforward calculation, using (3.1.2).

Let aΠ0. Then from (2.7.4) we have (B:BwaBwa)=(Ua:Ua+1)=qa+1=qa/2qa, so that (3.1.6) (BwaB:B)= qa/2qa (aΠ0).

For each wW, let q(w)= (BwB:B).

Let w=w1wr be a reduced word for wW. Then q(w)=q(w1) q(wr).


If αΠ and wW are such that l(wαw)>l(w) then (2.3.7) BwαwB=BwαB·BwB. Since wαw, the cosets BwαB and BwB are disjoint, and
therefore q(wαw)=q(wα)q(w). (3.1.7) now follows by induction on the length.

If wW0, then q(w)=bqb, the product being taken over the bΣ1 which are positive on C0 and negative on wC0.


Let w=w1wr be a reduced word for w in W0, where wi=wai, aiΠ0. Then the roots aΣ0 which are positive on C0 and negative on wC0 are congruent under W0 to a1,,ar. The result now follows from (3.1.7), (3.1.6) and the fact that qwb=qb for all bΣ1, wW0.

For each bΣ1 let ξb be an indeterminate over such that ξwb=ξb for all wW0. Define ξ(w)=b ξb(wW0) where as above the product is over the bΣ1 which are positive on C0 and negative on wC0, and put Q(ξ)=wW0 ξ(w).

Q(ξ) is the Poincaré polynomial of the root system Σ1. It is a polynomial in one variable if all the roots of Σ1 are of the same length (types Al, Dl, E6, E7, E8), two variables if Σ1 is of type Bl, Cl, F4 or G2, and three variables in the case of BCl.

Since K=wW0BwB, we have (3.1.9) (K:B)=wW0 q(w)=Q(q)say.

Haar measures

If Γ is any locally compact topological group, the modulus function ΔΓ (see e.g. [Die1967], Chapter XIV) is a continuous homomorphism of Γ into the multiplicative group of positive real numbers, and therefore is equal to 1 on any compact subgroup of Γ.

In the case of our group G, it follows (using (2.7.3)) from this remark that the kernel of ΔG contains all parahoric subgroups and hence is the whole of G: in other words, G is unimodular. So are K and Uα, because they are compact ((2.7.3) and (2.7.4)); so also are U(a) and U±, because they are unions of compact subgroups. Finally, the group Z is unimodular, for it has H as a compact normal subgroup with Z/H discrete, by (2.7.1).

On the other hand, the groups B0±=ZU± are not in general unimodular.

We fix Haar measures dg, dk, dz, dua on G, K, Z, U(a) (aΣ0) respectively, such that Hdg=Kdk= Hdz=Ua dua=1.

Next, consider the group U+. Arrange the positive roots in a sequence a1,,an such that height (ai) height (aj) if i<j. Then, as we ought to have proved in (2.6), the group Uj=ij U(ai) is a normal subgroup of U+ for j=1,2,,n, and Uj is the semi-direct product of U(aj) and Uj+1. Hence if duj+1 is a Haar measure on Uj+1, it follows that duj=duajduj+1 is a Haar measure on Uj. Consequently (3.2.1) du+=i=1n duai, where u+=i=1nuai, is a Haar measure on U1=U+, for which the subgroup U(C0)=i=1nUai has measure 1.

The left and right Haar measures on B0+=ZU+ are then dl(zu+) = dzdu+, dr(zu+) = d(zu+z-1) dz=Δ(z)dz du+ where Δ is a continuous homomorphism of Z into the multiplicative group +* of positive real numbers, defined by Δ(z)=d (zu+z-1) /du+.

For any integrable function on G we have Gf(g)dg= KZU+ f(zku+)dk dr(zu+) or symbolically (3.2.2) dg=Δ(z)dk dzdu+.

Since Z normalizes each U(a) we have homomorphisms Δ(a):Z+* defined by Δa(z)=d (zuaz-1) /dua. Δ and Δa are trivial on H, hence induce homomorphisms δ, δa:T+* such that Δ=δν, Δa=δaν.

From (3.2.1) we have immediately (3.2.3) Δ(z)= aΣ0+ Δa(z).

Also the value of Δa(z) is given by

Let aΣ0, zZ, t=ν(z)T. Then Δa(z)= δa(t)= qa/212a(x) qaa(x) where x=t(0). If a(x)0 then δa(t) is a positive integer.


Since t(a)=at-1=a-a(x) we have zUaz-1=Ua-a(x) and hence Δa(z) = (zUaz-1:Ua) = (Ua-a(x):Ua) = qa/2[r/2] qar by (3.1.5), where r=a(x). Now x belongs to the lattice spanned by the b (bΣ0). If qa/21, that is if 12aΣ1, then 12a(x) by (3.1.4) and therefore r is an even integer, i.e. [r/2]=12a(x).

Δ(z)=δ(t)= bΣ1+ qbb(x).


(3.2.3), (3.2.4).

If wW0 and nν-1(w), then Δwa(nzn-1) =Δa(z), δwa(wtw-1) =δa(t) (zZ, tT).

For qwb=qb (bΣ1).

Let aΠ0. Then δ(ta)=δa (ta)=qa/2 qa2.


Consider the product bδb(ta) extended over the roots bΣ0+, ba. Since wa permutes this set of roots, we have bδb(ta)= bδwab(ta) =bδb(ta-1) by (3.2.6), so that bδb(ta)=1. Hence δ(ta)=δa(ta), and δa(ta)=qa/2qa2 by (3.2.4), since ta(0)=a.

Let tT++, wW0. Then δ(t)12/ δ(wtw-1)12 is a positive integer.


First consider the case where w=wa (aΠ0), and let t be any element of T. Put x=t(0). Then (watwa)(0)=wa(x)=x-a(x)a so that watwa=t·ta-a(x) and therefore (3.2.9) δ(t)/δ (watwa) = δ(ta)a(x) = qa/2a(x) qa2a(x) by (3.2.7) = δa(t)2 Hence δ(t)12/δ(watwa)12 is a positive integer, by (3.2.4), provided that a(x)0.

Now let tT++ and let w be any element of W0. Let w=warwa1 be a reduced word for w, where aiΠ0 (1ir). Put ti=waiwa1twa1wai and put xi=ti(0)=waiwa1(x) where x=t(0). Then ai+1(xi)= (wa1waiai+1) (x)0 because wa1waiai+1 is a positive root (it is one of the positive roots which is negative on w-1C0, see (2.1)) and xC0. Hence from the first part of the proof we have δ(ti)12/δ(ti+1)12 and therefore δ(t)12/δ(wtw-1)12=δ(t0)12/δ(tr)12.

For the group U- we have analogously (3.2.1') du-=i=1n du-ai and the right Haar measure on B0-=ZU- is dr(zu-)=Δ(z)dzdu-, where Δ(z)=d(zu-z-1)/du-=Δa(z) over the negative roots. By (3.2.3) and (3.2.4) this is equal to Δ(z)-1. Hence (3.2.10) dr(zu-)=Δ (z)-1dzdu- and (3.2.2') dg=Δ(z)-1 dkdzdu-.

In the case of a Chevalley group (2.5) all the qa are equal to q, the number of elements in the residue field, and all the qa/2 are equal to 1, so that Σ1=Σ0. So in this case we have δ(t)12= qr(t(0)) where r=12aΣ0+a.

We shall need later the value of the index vt=(KtK:K), where tT++. The computation fits in conveniently here.

Let C be any chamber of Σ and let B=P(C) be the corresponding Iwahori subgroup. Then (BtB:B)= δ(t) for all tT++.


By (2.6.4) we have B=U(C)-HH(C)+. If α=-a+r is positive on C, where aΣ0+, then tα=-a+r+a(t(0)) is also positive on C, because a(t(0))0. Hence tU(C)-t-1B and therefore BtB=BtU(C)+. Consequently (BtB:B) = (BtU(C)+t-1:B) = (tU(C)+t-1:BtU(C)+t-1) = (tU(C)+t-1:U(C)+) by (2.6.5) = δ(t).

Let tT++. Then q(wtw-1)=δ(t) for all wW0.


q(wtw-1)= (Bwtw-1B:B) =(BtB:B) where B=w-1Bw.

Let tT++ and let W0t be the centralizer of t in W0, or equivalently the subgroup of W0 which fixes the point t(0)A. Put (3.2.14) Qt(ξ)= wW0tξ(w) in the notation of (3.1), and let Qt(ξ)=Q(ξ) /Qt(ξ) which is also a polynomial.

vt=(KtK:K)=δ(t)Qt(q-1) for all tT++.


By (2.3.5) we have KtK=BW0tW0B, which is the disjoint union of the sets Bt1W0B as t1 runs through the set {wtw-1:wW0}. For each such t1 there exists a unique w1W0 such that t1-1Cw1C0, i.e., Ct1w1C0. Thus we have (Bt1W0B:B) = wW0 (Bt1w1wB:B) = wW0 q(t1w1w). Now by (2.2.2) and (3.1.7) we have q(t1w1w)=q(t1w1)·q(w). In particular, taking w=w1-1, it follows that q(t1w1)=q (t1)·q(w1)-1 and therefore (Bt1W0B:B)= wW0q(t1) ·q(w1)-1q (w) or, making use of (3.2.12), (3.2.16) (Bt1W0B:B)= δ(t)q(w1)-1 Q(q).

Now the subgroup W0t of W0 has a unique set W0t of left coset representatives with the property that wtW0tand wtW0tl (wtwt)=l (wt)+l(wt).

Let w0 be the unique element of W0 of maximum length, and let w0t be the unique element of maximum length in W0t. Then, with w1 as above, we have



We have t1=wtw-1 and we may assume that wW0t. Since wt-1w-1Cw1C0, we have wt-1(0)w1C0, hence w1-1wt-1(0)C0; but also w0t-1(0)C0, because tT++. Hence w1-1w and w0 both map t-1(0) to the same point, because C0 is a fundamental domain for W0. It follows that w0w1-1w belongs to W0t. We want to show that it is equal to w0t, i.e. that it changes the sign of every aΣ0+ such that a(t(0))=0.

So let a>0, a(t(0))=0 and let yC. Then z=w1-1t1-1yC0. We have w1-1wa(z)= a(t-1w-1y) = a(w-1y-t(0)) = wa(y)because a(t(0))=0 and since wW0t we have wa>0, hence wa(y)>0. Hence w1-1wa>0 and therefore w0w1-1wa<0, as required.

So we have w1w0=wt w0t for some wtW0t. Since l(wtw0t) = l(wt)+ l(w0t), l(w1w0) = l(w0)- l(w1) it follows that q(wt) q(w0t)= q(w0) q(w1)-1 and therefore (3.2.18) q(w1)-1 = q(w0t)q(w0) q(wt) = Qt(q-1).

Hence finally (KtK:K) = (K:B)-1 wW0t (Bwtw-1W0B:B) = Q(q)-1δ(t) q(w1)-1Q(q) by (3.2.16) = δ(t)Qt(q-1) by (3.2.18).

Zonal spherical functions on G relative to K

Let (G,K) be the algebra of K-bi-invariant functions on G defined in (1.1), and let (G,K) be the subring consisting of the -valued functions in (G,K). (G,K) is called the Hecke ring of G relative to K. Clearly (G,K)(G,K).

For each tT++ let χt be the characteristic function of the double coset KtK. By (1.1.1) and (2.6.11) the χt form a -basis of (G,K) and a -basis of (G,K). The identity element is χ1, the characteristic function of K.

Let s:T* be a homomorphism. Then (sδ12)ν is a continuous homomorphism of Z into *, which we extend trivially across U- to a continuous homomorphism ϕs:B0-*: ϕs(zu-)= (sδ12,ν(z)). [To avoid large collections of brackets, we shall sometimes write (s,t) for s(t), where sHom(T,*) and tT.] Now extend ϕs to a function on the whole of G by the rule ϕs(kzu-)= (sδ12,ν(z)) (kK, zZ, u-U-).

For each sHom(T,*) the function ωs(g)=K ϕs(g-1k) dk is a zonal spherical function on G relative to K. Moreover, if s is a character of T, i.e. if |s(t)|=1 for all tT, then ωs is positive definite.


This is a direct consequence of the results of Chapter I. Since KB0-HU- (see the proof of (2.6.11)(2)) it follows that ϕs restricted to B0- is trivial on KB0- and hence (1.2.7) is a zonal spherical function on B0- relative to KB0-. Hence ωs is a z.s.f. on G relative to K, by (1.3.2). The second assertion follows from (1.4.7) and (3.2.10).

For all g0G and sHom(T,*), ωs-1(g0) =ωs(g0-1).


Let K=K/(KB0-) and let dk be a relatively invariant measure on K with total mass 1. If g0-1k=kzu- (kK, zZ, u-U-) then as in (1.4.6) we have dk=Δ(z)-1dk, and hence ωs-1(g0) = K ϕs-1 (g0-1k) dk = K (s-1δ12,ν(z)) Δ(z)-1d k = K(sδ12,ν(z-1)) dk = Kϕs (g0k)dk =ωs(g0-1).

The group W0 acts on T by inner automorphisms, hence acts on Hom(T,*): namely (ws,t)= (s,w-1tw) (sHom(T,*), tT, wW0).

For all wW0 and sHom(T,*), ωs=ωws.

This result is a consequence of the calculations we shall undertake in Chapter IV, and we shall postpone its proof to (4.5). The reader can check that there is no vicious circle involved here.

Now let f(G,K). Then if ωˆs is the spherical measure associated with ωs, ωˆs(f) = (f*ωs)(1) = Gf(g) ωs(g-1) dg = Gf(g) (Kϕs(gk)dk) dg = Gf(g)ϕs (g)dg = KZU-f (kzu-) (sδ12,ν(z)) Δ(z)-1 dkdzdu- = Z(sδ-12,ν(z)) (U-f(zu-)du-) dz.

The integral Δ(z)-12U-f(zu-)du- depends only on f and ν(z)=t say. We shall therefore write f(t)=δ (t)-12 U-f(zu-) du- for tT and any zν-1(t). Then the calculation above shows that (3.3.4) ωˆs(f)= tTf (t)s(t).

The f(t) should be regarded as the Fourier coefficients of f(G,K). There are no convergence difficulties here: since f has compact support and the double cosets KtU- are open (because K is open) and mutually disjoint, it follows that f takes non-zero values on only finitely many of these double cosets, i.e. f has finite support in T.

Let [T] be the group algebra of T over . Since T is a free abelian group generated by the ta with aΠ0, it follows that [T]=[ta±1:aΠ0], hence is a commutative integral domain. Consequently [T] is the coordinate ring (or affine algebra) of an affine algebraic variety, say S, whose points are the -algebra homomorphisms s:[T]. Restricting these homomorphisms to T gives a bisection of S onto Hom(T,*), and we shall identify Hom(T,*) with S in this way. The elements of [T] can therefore be regarded as functions on S=Hom(T,*). Hence, by the Nullstellensatz, if ϕ[T], (3.3.5) ϕ=0s(ϕ)=0 for allsS.

We regard the affine Weyl group W as a discrete group. Then, in the notation of (1.1), (T) is the algebra of functions on T with finite support. The mapping ϕtT ϕ(t)t is a -algebra isomorphism of (T) onto [T]. Hence sS defines a homomorphism (T), namely s(ϕ)=tT ϕ(t)s(t). The formula (3.3.4) now takes the form (3.3.4') ωˆs(f)= s(f).

The group W0 acts on (T) by transposition: (wϕ)(t)=ϕ(w-1tw) (ϕ(T), tT, wW0). Let (T)w0 denote the subring of W0-invariant functions in (T). Then we have

The mapping ff is a -algebra isomorphism of (G,K) onto (T)w0.

(G,K) is commutative.


First of all, let us see that f(T)W0 whenever f(G,K). Let sS, wW0. Then s(f)=ωˆs (f)=ωˆws (f)=(ws) (f)=s(w-1f) by (3.3.3) and (3.3.4'). Hence f=w-1f by (3.3.5).

Next, let f1,f2(G,K). Then s(f1*f2) =ωˆs (f1*f2)= ωˆs(f1)· ωˆs(d2) = s(f1)· s(f2) = s(f1*f2) by (3.3.4'). Again by (3.3.5) we deduce that (f1*f2)=f1*f2, and hence ff is a -algebra homomorphism.

It remains to show that the mapping ff of (G,K) into (T)W0 is bijective. Since the characteristic functions χt (tT++) form a -basis of (G,K), it is enough to show that the χt form a -basis of (T)W0. The last two parts of (2.6.11) were designed expressly for this purpose.

For each tT, the orbit of t under W0 (acting by inner automorphisms) has a unique representative in T++. Let t be the characteristic function of the W0-orbit of t. Then these functions t, as t runs through T++, form a -basis of (T)W0. Hence we can write (3.3.8) χt= t χt (t) t summed over tT++.

We define a total ordering on T as follows. Let Π0={a1,,al} be the set of simple roots of Σ0, arranged in some order. For each tT we have t(0)=i=1lmiai with mi. We define (3.3.9) { t0 the first non-zeromi is positive, t1t2 t1t2-1 0. In particular, the elements of T+ are 0 for this ordering.

Now if tT++ and tT we have KtU- χt(g)dg= U-χt (zu-) Δ(z)-1 du- for any zν-1(t), =δ(t)-12 χt(t). Hence we have (3.3.10) χt(t)= δ(t)12 vol.(KtKKtU-). So if χt(t)0 it follows from (2.6.11)(3) that tt-1T+ and hence that tt. Again, from (2.6.11)(4) we have χt(t)=δ(t)12. Consequently (3.3.8) now takes the form (3.3.8') χt=δ (t)12 t+ t<ttT++ χt(t) t from which it follows immediately (since δ(t) is never 0) that the χt form a basis of (T)W0.

(a) Consider the algebra (W,W0). Restriction of functions to the subgroup T defines an isomorphism ρ:(W,W0)(T)W0. Hence (3.3.6) provides a canonical isomorphism of (G,K) onto (W,W0), namely fρ-1(f). (The emphasis here is on the word 'canonical'.)

(b) If G is a universal Chevalley group (2.5) over a p-adic field 𝓀, there is
a much simpler proof that (G,K) is commutative. For there exists an
 nvolutory antiautomorphism i:GG such that i(ua(ξ))=u-a(ξ) for all aΣ0 and ξ𝓀, and i(z)=z for all zZ. It follows that i(K)=K and that i preserves Haar measure. Now let f(G,K) and define fi=fi. Since i
 is an antiautomorphism, we have (f1*f2)i=f2i*f1i. On the other hand, i preserves all double cosets KzK and therefore f=fi for all f(G,K). Hence f1*f2=f2*f1.

The same argument will work whenever G has an antiautomorphism i such that i(Ua+r)=U-a+r for all aΣ0 and r, and i(z)zH for all zZ. This will be the case, for example, whenever the central inversion w0:x-x is in the Weyl group W0. For then we may take i(g)=n0g-1n0-1, where n0ν-1(w0).

(c)From (3.3.8') it follows that the image of (G,K) in (T)W0 under
 ff is the -module spanned by the elements δ(t)12t with tT++
(because δ(t)-12χt(t) by (3.3.10)). If Σ0=Σ1, then δ(t)12 by (3.2.7), hence in this case the isomorphism of (G,K) onto (W,W0) in (a) above
embeds (G,K) in (W,W0), the Hecke ring of W relative to W0.

(1) Every zonal spherical function ω on G relative to K is equal to ωs for some sS=Hom(T,*).
(2) ωs=ωss=ws for some wW0.


(1) The mapping fωˆ(f) is a -algebra homomorphism (T)W0, which since (T) is integral over (T)W0 can be extended to a homomorphism (T). This in turn defines by restriction a homomorphism s:T*, and we have ωˆ(f)=s(f)=ωˆs(f) for f(G,K). Hence ωˆ=ωˆs and therefore ω=ωs by (1.7).

(2) Suppose that ωs=ωs. Then s and s agree on (T)W0, by (3.3.4'), and therefore their kernels m, m, say, are maximal ideals of (T) lying over the same maximal ideal of (T)W0. By a well-known theorem of commutative algebra (see for example [AMa1969], Chapter V, Ex. 13) there exists wW0 transforming m into m. Hence s=ws.

Since *×(/) (the isomorphism being re2πiθ(logr,θ)) we have S=Hom(T,*)Hom(T,)×Hom(T,/). Now Hom(T,) can be identified with A*: if λHom(T,), the corresponding element of A* is the linear form which maps t(0) to λ(t) for all tT. This identification induces an identification of Hom(T,/) with A*/L, where (as in (2.5)) L is the lattice of linear forms u on A such that u(a) for all aΣ0. Hence the space Ω of z.s.f. may be identified with the orbit space (A*×(A*/L))/W0. Alternatively, Ω may be identified with S/W0, the affine algebraic variety whose coordinate ring is (T)W0[T]W0.

Let t1,,tm be a set of generators of the semigroup T++. Then (G,K)=[χt1,,χtm]. (Of course, in general the χti will not be algebraically independent.)


For each tT++ let θt=δ (t)12 t so that the image of (G,K) under ff is the -module freely generated by the θt, by (3.3.11)(c). Let Mt=t<tθt. From (3.3.8') we have χtθt(modMt).

Also θt1* θt2 θt1t2 (modMt1t2). For θt1*θt2-θt1t2 is a sum of terms of the form δ(t1t2)12 δ(t3)-12 θts with t3=t1t2T++ and t1=w1t1w1-1, t2=w2t2w2-1 (w1,w2W0) and either t1t1 or t2t2. By (3.2.8) the coefficient δ(t1t2)12δ(t3)-12 is a rational integer, and we have t3<t1t2.

Hence if t=i=1mtiμiT++ it follows that χtθt i=1m θtiμi i=1m χtiμi (modMt) and so, by induction on t, that χt[χt1,,χtm]. Hence χt[χt1,,χtm].

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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