Spherical Functions on a Group of p-adic Type

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 20 February 2014

Chapter V: Plancherel measure

The standard case

Let Tˆ be the character group of the discrete group T. Tˆ is the product of l circles, and may be identified with the torus A*/L, where as in (3.3.13) L is the lattice of linear forms u on A such that u(a) for all aΣ0. Let ds be Haar measure on the compact group Tˆ, normalized so that the total mass of Tˆ is 1.

We shall assume in this section that (5.1.1) qa/21 for allaΣ0. Call this the standard case. The exceptional case, where qa/2<1 for some aΣ0, will be examined separately in (5.2). Here we shall prove

Assume that qa/21 for all aΣ0. Then the Plancherel measure μ (1.5.1) on the space Ω+ of positive definite spherical functions on G relative to K is concentrated on the set {ωs:sTˆ} and is given there by dμ(ωs)= Q(q-1) |W0| ·ds|c(s)|2 where |W0| is the order of the Weyl group W0.

Proof.

For f1,f2(G,K) we define f1,f2 = Gf1(g) f2(g) dg, fˆ1,fˆ2 = Q(q-1)|W0| Tˆfˆ1 (ωs) fˆ2(ωs) |c(s)|-2 ds. We have to show that f1,f2=fˆ1,fˆ2. By linearity it is enough to take f1, f2 to be characteristic functions χt1, χt2 (t1,t2T++), and we may assume that t1t2 with respect to the total ordering denned in (3.3.9). Clearly we have (5.1.3) χt1,χt2= { 0 ift1>t2, (Kt1K:K) ift1=t2.

Now consider χˆt1,χˆt2. To compute this we shall express |c(s)|-2χˆt1ωs as an infinite ascending series, and χˆt2(ωs) as a descending polynomial, with at most one term in common. Term by term integration over the torus Tˆ will then give the required result.

Since s=s-1 we have c(b,s)=c(b,s-1)=c(-b,s) for each bΣ1, hence |c(s)|-2 =(c(s)c(s-1))-1 =bΣ1c (b,s)-1, the product being over all the roots, positive and negative. This shows that |c(s)|-2 is W0-invariant, hence (5.1.4) |c(s)|-2= (c(ws)c(ws-1))-1 for all wW0.

Now χˆt(ωs)= Gχt(g)ωs (g-1)dg=vt ωs(z-1) where vt=(KtK:K), and z is any element of ν-1(t). Hence if s is nonsingular we have from (4.1.2) and (5.1.4) 1vt· χˆt(ωs) |c(s)|2 = δ(t)-1/2 Q(q-1) wW0 (ws,t) c(ws-1) or equivalently (5.1.5) 1vt· χˆt(ωs) |c(s)|2 = δ(t)-1/2 Q(q-1) wW0 (ws,t) bΣ1+ 1-qb/2-1/2(ws,tb) 1-qb/2-1/2qb-1(ws,tb) . Now if b=aΣ0, then qb/21/2qb=(qaqa+1)1/2>1, and if b=12a then qb/21/2qb=qa/21 (because bΣ1). Hence the right-hand side of (5.1.5) is denned for all sTˆ, and therefore (5.1.5) is valid for all sTˆ.

Moreover, under the assumption (5.1.1) we have qb/21/2qb>1 for all bΣ1, and therefore bΣ1+ 1-qb/2-1/2(ws,tb) 1-qb/2-1/2qb-1(ws,tb) =bΣ1+ { 1-(qb-1) n=1 qb/2-n/2 qb-n (ws,tb)n } , these series being absolutely and uniformly convergent on the compact group Tˆ. Multiplying them all together we shall get from (5.1.5) (5.1.6) 1vt· χˆt(ωs) |c(s)|2 = δ(t)-1/2 Q(q-1) tT++ att wW0 (ws,t) say, with coefficients att independent of s, and the summation being over all tT++ of the form t=t·bΣ1+ tbnb with integer exponents nb0. It follows that tt with respect to the ordering (3.3.9); also it is easy to see that att=1, so that (replacing t by t1) we have (5.1.6') 1vt1· χˆt1(ωs) |c(s)|2 = δ(t1)-1/2 Q(q-1) ( wW0 (ws,t1)+ t1>t1t1T++ at1t1 wW0 (ws,t1) ) for all tT++, the series on the right being uniformly convergent on Tˆ.

On the other hand, from (3.3.4') and (3.3.8'), (5.1.7) χˆt2(ωs)= s(χt2)=δ (t2)1/2 (s,t2)+ t2<t2t2T++ χˆt2(t2) (s,t2).

Furthermore, we have (5.1.8) Tˆ (ws,t1) (s,t2) ds= { 0 ift1 t2, 1 ift1= t2 by orthogonality of characters on Tˆ. Hence, multiplying the infinite series (5.1.6') by the polynomial (5.1.7) and integrating term by term over Tˆ, we shall have, using (5.1.8), vt1-1Tˆ χˆt1(ωs) χˆt2(ωs) |c(s)|-2ds= { 0 ift1>t2, |W0| Q(q-1) ift1=t2. Hence χˆt1,χˆt2 =0=χt1,χt2 ift1>t2 and χˆt1,χˆt1 =(Kt1K:K)= χt1,χt1.

The exceptional case

In this section we shall consider the case excluded from the considerations of (5.1), namely where qa/2<1 for some aΣ0. First of all, this implies that a2Σ1 for some aΣ0, so that the root system Σ1 is not reduced. Since it is irreducible, it must be of type BCl and can therefore be described as follows. There exists an orthonormal basis x1,,xl of A such that, if e1,,elA* are the coordinate functions relative to this basis, the root system Σ1 consists of ±ei(1il), ±2ei (1il),± ei±ej (1i<jl) and Σ0 consists of ±2ei(1il), ±ei±ej (1i<jl) and is of type Cl. We choose the set Π0 of simple roots as follows: Π0= { e1-e2,e2- e3,,el-1- el,2el } . The Weyl group W0 is the group of all signed permutations of e1,,el, and has order 2ll!.

Let q0=q±ei±ej, q1=q±ei, q2=q±2ei, so that q1<1.

Next consider the translation group T. Put ti=t2ei (1il). Then ti(0)=(2ei)=xi, and hence t±ei±ej= ti±1 tj±1. The ti are a basis of T.

If sS, put si=s(ti) *. Then c(s) is the product of the factors (1+q1-1/2si-1) (1-q1-1/2q2-1si-1) 1-si-2 for i=1,,l, and the factors 1-q0-1si-1sj 1-si-1sj · 1-q0-1si-1sj-1 1-si-1sj-1 for 1i<jl.

Finally, let ϕ(s)=c(s) c(s-1). Then we have to compute the integral (5.2.1) Q(q-1)|W0| Tˆ χˆt1(ωs) χˆt2(ωs) ·ϕ(s)-1ds where t1,t2T++ and t1t2. Proceeding as in (5.1), we have χˆt1(ωs)= vt1· δ(t1)-1/2 Q(q-1) wW0c(ws) (ws,t1). Since the Haar measure ds on Tˆ is invariant under W0, it follows that the integral (5.2.1) is equal to vt1·δ (t1)-1/2 Tˆc(s) (s,t1)· χˆt2(ωs) ϕ(s)-1ds. On the other hand, from (3.3.8'), χˆt2(ωs)= t2t2t2T++ χˆt2(t2) (s,t2), so that (5.2.1) now takes the form (5.2.2) vt2·δ (t1)-1/2 t2t2t2T++ χt2(t2) Tˆc(s) (s,t1) (s-1,t2) ϕ(s)-1ds. In this expression, (s-1,t2) is a sum of terms (s-1,wt2w-1) (wW0). Since t1t2t2wt2w-1 for all wW0, in order to compute (5.2.2) we have to compute integrals of the form (5.2.3) I=Tˆc(s) (s,t)ϕ(s)-1 ds where tT and t0. Since |si|=|s(ti)|=1, we may express (5.2.3) as a multiple contour integral, each contour being the unit circle γ in the complex plane. If θi=arg(si), we have ds=j=1l (dθj/2π) and therefore ds=j=1l 12πi dsjsj. (i=-1here!) Consequently, if t=t1λ1tlλl, the integral (5.2.3) takes the form (5.2.3') I=12πiγ s1λ1-1d s112πi γslλl-1 dsl·1c(s-1). Since t0, the first of the exponents λi which is non-zero is positive.

We shall establish a reduction formula for I. For this purpose we have to introduce more notation. Let J=(j1,,jr) be a sequence of integers satisfying 1j1<j2<< jrl and let TˆJ be the set of all sS such that sjα = -q0r-α q11/2, (1αr) |si| = 1ifiJ. Also define a function ϕJ(s) inductively as follows: (5.2.4) ϕ(s) = ϕ(s)=c (s)c(s-1), ϕJ(s)= ϕj1,,jr (s) = limsj1-q0r-1q11/2 ϕj2,,jr(s) 1+q01-rq1-1/2sj1 (r>0). ϕJ(s) is a function of the complex variables si, iJ, but we regard it as a function on TˆJ. It is easily checked from the product which defines ϕ(s) that ϕj2,,jr(s) has (1+q01-rq1-1/2sj1) as a factor in the numerator, so that the right-hand side of (5.2.4) is well defined.

Let IJ=Ij1,,jr =TˆJc(s) (s,t)ϕJ (s)-1ds where the measure ds on TˆJ is given by ds=jJ 12πi dsjsj. (TˆJ is a translate of a subtorus of Tˆ, and this measure is the translation of the normalized Haar measure on that subtorus.)

Finally, let ϕ1 (resp. c1) denote the product obtained from ϕ (resp. c) by deleting all factors involving s1, and let T1 be the subgroup of T generated by t2,,tl. Define TˆJ1 and ϕJ1 as above for subsequences J of (2,3,,l), and put IJ1=TˆJ1 c1(s)(s,t) ϕJ1(s)-1 ds for tT1, the measure being here ds=j112πidsjsj.

The final form of the Plancherel measure will depend on how many of the numbers q11/2,q0 q11/2,, q0l-1 q11/2 are less than 1. So we define εr for 0rl as follows: ε0=1 and (5.2.5) εr= { 1 ifq0r-1 q11/2 <1, 0 otherwise for 1rl.

Since q0>1, it follows that εr=1εr-1=1.

Having set all this up, the key to the calculation of the integral I in (5.2.3) is the following lemma:

Let J=(j1,,jr) be a subsequence of (2,3,,l) and put J=(1,j1,,jr). Then εrIJ+ εr+1IJ= { 0 ifλ1>0, εrIJ1 ifλ1=0.

Proof.

If εr=0 then also εr+1=0 (because q0>1) and there is nothing to prove. So assume that εr=1. Consider the integral IJ, and integrate it round the unit circle γ with respect to the variable s1. To do this we must pick out the factors in the product c(s)ϕJ(s)-1 which involve s1. An elementary calculation shows that they are, after some cancellations, (5.2.7) 1-s12 (1+q0-rq1-1/2s1) (1-q1-1/2q2-1s1) · 1+q0r-1q11/2s1 1+q0-1q11/2s1 ·jJ 1-s1sj±1 1-q0-1s1sj±1 . Hence, since |sj|=1 for jJ, the integrand 12πi c(s)(s,t) ϕJ(s) ds1s1 considered as a function of s1, has poles at the points s1=-q0r q11/2, q11/2q2, -q0q1-1/2, q0si±1 and also at the origin, if λ1=0.

The points q11/2q2, -q0q1-1/2, q0si±1 all lie outside the unit circle γ (for q11/2q2=δ1/2(tl)>1, and q0 and q1-1/2 are both >1). The point -q0rq11/2 lies inside γ if and only if εr+1=1. (If q0rq11/2=1, then the factor (1+q0-ρq1-1/2s1) in the denominator of (5.2.7) cancels into the factor 1-s12 in the numerator, so there is never a pole on the unit circle.) The lemma now follows by computing the residues at the poles in the usual way.

Jε|J| IJ= { 0 ift>1, 1 ift=1, where the sum is over all subsequences J of (1,2,,l), and |J|=Card(J).

Proof.

By induction on l. The induction can start with l=1, when the result is easily verified.

If λ1>0, we have from (5.2.6) Jε|J| IJ=0.

If λ1=0, again from (5.2.6) we have Jε|J| IJ=J1 ε|J1| IJ11, the sum on the right being over all subsequences J1 of (2,3,,l). The result now follows from the inductive hypothesis.

Now let π:SΩ be the mapping sωs of S onto the set Ω of all z.s.f. on G relative to K. The fibres of π are the orbits of the action of W0 on S. Clearly π(TˆJ) depends only on the number of elements in J, and not on the particular sequence. Let Ωr=π (TˆJ)if |J|=r.

From the definition of TˆJ, the z.s.f. belonging to Ωr are the ωs for which r of the numbers s(ti) (1il) take the values -q11/2,- q0q11/2, ,-q0r-1 q11/2 (or their inverses), and the remaining s(ti) have absolute value 1. Since ϕ(s) is invariant under the action of W0, it follows that ϕJ(s) depends only on the number of elements in J: so we may define ϕr(ωs)= ϕJ(s)if |J|=r.

The spherical functions belonging to Ωr are positive definite if εr=1.

We leave the proof until later.

We have already defined a measure ds on each TˆJ giving it total mass 1. Now define a measure dω on Ωr as follows: Ωrf(ω) dω=π-1(Ωr) f(ωs)ds for any continuous function f on Ωr with compact support.

The Plancherel measure μ on the space of positive definite spherical functions is concentrated on the sets Ωr such that εr=1, where εr is defined in (5.2.5). On Ωr,μ is given by dμ(ω)= Q(q-1) wr dωϕr(ω) where wr=2l-r(l-r)! is the order of the normalizer in W0 of any of the sets TˆJ such that |J|=r.

Proof.

For f1,f2(G,K), define fˆ1,fˆ2r =Ωrfˆ1 (ω)fˆ2(ω) dμ(ω) (0rl) where dμ(ω) is as above. Then we have to show that r=0lεr fˆ1,fˆ2r =f1,f2. As before we take f1=χt1, f2=χt2 with t1,t2T++ and t1t2. Let Jr be the sequence (1,2,,r). Then exactly as in (5.1) χˆt1,χˆt2r = Q(q-1)wr TˆJr χˆt1(ωs) χˆt2(ωs) dsϕJr(s) = vt1·δ(t1)-1/2 wr t2t2t2T++ χt2(t2) wW0 TˆJrc(ws) (ws,t1) (s-1,t2) dsϕJr(s). Now it is easily verified that if sTˆJr and wW0, then c(ws)=0 unless w maps t1,,tr to tj1,,tjr respectively with j1<<jr. Hence 1wrwW0 TˆJrc (ws)(ws,t1) (s-1,t2) ϕJr(s)-1 ds =|J|=r TˆJc(s) (s,t1) (s-1,t2) ϕJ(s)-1ds. Consequently r=0lεr χˆt1,χˆt2r =vt1·δ (t1)-1/2 t2t2t2T++ χt2(t2) Jε|J| TˆJ c(s)(s,t1) (s-1,t2) ds ϕJ(s) and by (5.2.8) the inner sum is 0 or 1 according as t1t2 or t1=t2. Hence,‚Ä®finally, r=0lεr χˆt1,χˆt2r= { 0 ift1>t2, χt1,χt1 ift1=t2 since vt1·δ(t1)-1/2χt1(t1)=vt1=χt1,χt1.

We have still to prove (5.2.9). Consider first the set Ωl: this consists of a single spherical function, say ωs0, where s0S is given by s0(ti)=- q0i-1 q11/2 (1il).

Let zZ++, ν(z)=t. Then ωs0(z-1)= (δ-1/2s0,t).

Proof.

Consider c(ws0) where wW0 and w1. One sees immediately that some factor in the numerator vanishes, so that c(ws0)=0 if w1. Also a simple verification shows that c(s0)=c(δ1/2), and hence c(s0)=Q(q-1) by (4.5.8). The lemma now follows from (4.1.2).

If εl=1 then ωs0 is square-integrable.

Proof.

We have G|ωs0(g)|2 dg=tT++ vt(δ-1/2s0,t)2 where vt=(KtK:K)=δ(t)Qt(q-1) by (3.2.15). Hence we have to show that the series (5.2.13) tT++ Qt(q-1) (s0,t)2 converges if |s0(ti)|<1 for 1il.

Since tT++ we have t=i=1ltiλi with λ1λ2λl0. Let ρ=(ρ1,,ρr) be a sequence of integers such that (5.2.14) 1ρ1< ρ2<<ρr l and consider the t=tiλi such that λ1=λ2== λρ1>λρ1+1 ==λρ2>> λρr+1== λl=0. Qt(q-1) will be the same for all these t, because they all have the same stabilizer W0t in the Weyl group w0. Hence the series (5.2.13) splits up into a sum of series, one for each sequence ρ satisfying (5.2.14), and each of these series is a product of geometric series whose common ratios are products of the s0(ti)2. Hence all these series converge and therefore so does the series (5.2.13).

Remark. It is not difficult to show, using (5.2.11), that ωs0 is a z.s.f. on G relative to the Iwahori subgroup B. By (1.2.6) the z.s.f. on G relative to B correspond bijectively to the -algebra homomorphisms (G,B). Let Π0 (resp. Π) be the set of simple roots for Σ0 (resp. Σ), (so that Π0={a1,,al}, Π={α0,,αl}, where α0=1-2e1, αi=ai=ei-ei+1 (1il-1), αl=al=2el). Then (G,B) is generated as -algebra by the characteristic functions χi of BwαiB (0il), and the homomorphism ωˆs0:(G,B) is given by χiq(wαi) (1il), χ0-1.

In general, the z.s.f. on G relative to B correspond naturally to the linear characters of the Weyl group W. So they are always finite in number, equal to the order of W made abelian.

We shall now sketch a proof of (5.2.9). Consider a zonal spherical function ωsΩr, where εr=1. We may assume that s(ti) = -q0r-i q11/2 (1ir), |s(ti)| = 1ifi>r. Let A (resp. A) be the subspace of A spanned by the first r basis vectors x1,,xr (resp. by xr+1,,xl). The restrictions to A of the roots aΣ0 (resp. affine roots αΣ) which vanish identically on A form a subsystem Σ0 of Σ0 (resp. a subsystem Σ of Σ). The Weyl group W0 and affine Weyl group W of Σ can be regarded as subgroups of W0 and W respectively. The translation subgroup T of W is generated by t1,,tr. Likewise, starting with A we define Σ0, Σ, etc.

Let G be the subgroup of G generated by the Uα with αΣ and by N=ν-1(W), and define G similarly. Then GG is a subgroup of G.

Let s=s|T, s=s|T. Then ωs is a square-integrable spherical function on G by (5.2.12), and hence by (1.4.8) is positive definite. Again, since s is a character of T, it follows from (3.3.1) (applied to G) that ωs is positive definite.

Now let P0 be the (parabolic) subgroup of G generated by G, G and U-, and let Δ0 be the modulus function on P0. P0 is the semi-direct product of GG and the subgroup U0 generated by the root subgroups U(a) corresponding to negative roots aΣ0 which do not belong to either Σ0 or Σ0 (i.e. a=-ei±ej with ir and j>r). The spherical function Δ0-1/2ωsωs on GG extends to a spherical function on P0, say ω0. By (1.4.7), the spherical function induced by ω0 on G is positive definite, and by transitivity of induction (1.3.3) it is equal to ωs. This completes the proof of (5.2.9) and hence of (5.2.10).

Comparison with the real and complex cases

Let 𝓀 be a local field, that is to say a non-discrete locally compact field. Then 𝓀 is either or or a p-adic field, and the additive group of 𝓀 is self-dual. Associated canonically with 𝓀 there is a meromorphic function γk(s) of a complex variable s, sometimes called the gamma-function of 𝓀. We recall its definition briefly (see Tate's thesis [Tat1967], where it is denoted by ρ(||s).)

If f is any well-behaved function on 𝓀, let fˆ be its Fourier transform with respect to the additive group structure. Since 𝓀+ is self-dual, fˆ is a function on 𝓀+. (The Haar measure is normalized so that fˆˆ(x)=f(-x).) If Re(s)>0 define ζ(f,s)=k× f(x)xs d×x, where x is the normalized absolute value on 𝓀 (i.e. d(ax)=adx where dx is additive Haar measure) and d×x is a Haar measure on the multiplicative group 𝓀×. Then ζ(f,s) has a functional equation (5.3.1) ζ(f,s)= γk(s)ζ (fˆ,1-s) with γk(s) independent of f. This equation defines γk(s) for Re(S)(0,1), and γk is then extended by analytic continuation to a meromorphic function in the whole complex plane.

We can compute γk(s) from (5.3.1) by choosing the function f intelligently.

𝓀=: take f(x)=e-πx2, then fˆ=f and one finds that γ(s)= π-s/2Γ(s/2) π(s-1)/2Γ((1-s)/2) so that (5.3.2) γ(s) γ(-s)= B(12,12s) B(12,-12s) where B is Euler's beta-function.

𝓀=: take f(x)=e-2π|x|2 (|x| the ordinary absolute value on : in fact x=|x|2), then again fˆ=f and we have γ(s)= (2π)-sΓ(s) (2π)s-1Γ(1-s) so that (5.3.2) γ(s) γ(-s)= -4π2/s2.

k=p-adic field. Taking f to be the characteristic function of the ring of integers of 𝓀, one finds that γk(s)= ds-12 1-qs-1 1-q-s where d is the discriminant of 𝓀 and q is the number of elements in the residue field. Hence in this case we have (5.3.2)p-adic γk(s) γk(-s)= d-1 1-q-1-s 1-q-s · 1-q-1+s 1-qs .

Now let G be the universal Chevalley group G(Σ0,𝓀) as in (2.5), where 𝓀 is any local field. If 𝓀= or , let K be any maximal compact subgroup of G (they are all conjugate). If 𝓀 is p-adic, let K be the maximal compact subgroup of G defined as hitherto in terms of the affine root structure on G given in (2.5).

If 𝓀= or , the z.s.f. on G relative to K are parametrized by the -linear mappings s:A, and the Plancherel measure for the positive definite spherical functions is supported on the space of pure imaginary s, which is naturally a real vector space of dimension l=dimA. If ds is a Euclidean measure on this space, then (Harish-Chandra [Har1966]) the Plancherel measure μ is of the form dμ(ωs)= κds/|c(s)|2 where κ is a constant (depending on the choices of ds and of Haar measure on G) and c(s)= aΣ0+ B(12,12s(a)) if𝓀= and c(s)= aΣ0+ s(a)-1 if𝓀=. Hence, from (5.3.2), we have (5.3.3), c(s)c(-s) =aΣ0 γk(s(a)) in both cases, apart from a constant factor in the case 𝓀=.

On the other hand, if 𝓀 is p-adic, then as we have seen in (5.1) the support of the Plancherel measure is the character group Tˆ of T. To bring out the analogy with the real and complex cases we shall replace the multiplicative parametrization of the spherical functions, which we have used hitherto, by an additive parametrization. If s0S=Hom(T,*), we define s:A/(2πilogq) by the rule s0(ta)= q-s(a) (aΣ0) where as before q is the number of elements in the residue field of 𝓀. Then from (5.1.2) the Plancherel measure μ is of the form dμ(ωs)=κ ds/|c(s)|2 where ds is Euclidean measure on the space of pure imaginary s, κ is a constant, and c(s)= aΣ0+ 1-q-1-s(a) 1-q-s(a) (because in the present situation all the qa are equal to q). Hence, from (5.3.2)p-adic we have (5.3.3)p-adic c(s)c(-s) =aΣ0 γk(s(a)), apart from a constant factor depending only on 𝓀.

So, to sum up:

If 𝓀 is any local field and G=G(Σ0,𝓀) is a universal Chevalley group, then the Plancherel measure on the space of positive definite spherical functions on G relative to the maximal compact subgroup K is of the form dμ(ωs)= κ·ds aΣ0γk(s(a)) where κ is a constant.

Also for non-split groups there is a strong resemblance between the Plancherel measure in the real case and in the p-adic case.

Let 𝒢 be a simply-connected simple algebraic group denned over a local field 𝓀, and let G=𝒢(k). (We can exclude the case 𝓀= from here on, because in that case 𝒢 is necessarily split.)

𝓀=. Let G=KAN be an Iwasawa decomposition, 𝔤 and 𝔞 the Lie algebras of G and A respectively, Σ1 the set of 'restricted roots' of 𝔤 relative to 𝔞, and for each bΣ1 let mb be the multiplicity of b. Then the z.s.f. on G relative to K are parametrized by the elements λ of Hom(𝔞,), and the Plancheral measure is supported on the subspace consisting of the pure imaginary λ:𝔞i. Write ωλ for the z.s.f. corresponding to λ. Then [Har1966] the Plancherel measure μ is given by dμ(ωλ)= κdλ/|c(λ)|2 where κ is a constant and (see [GKa1962]) c(λ) is a product of beta-functions, namely c(λ)= bΣ1+ B ( 12mb,14 mb/2+12 λ(b) ) . Let (5.3.5) ξ(s)= π-12s Γ(12s) (s) which is the local zeta-function ζ(f,s) for the function f(x)=e-πx2. In this notation we have (5.3.6) c(λ)=κ bΣ1+ ζ(12mb/2+λ(b)) ζ(mb+12mb/2+λ(b)) where κ is independent of A.

𝓀 p-adic. If the number of elements in the residue field of 𝓀 is q, then each index qb (bΣ1) is a power of q. We shall therefore write qb=qmb (bΣ1) and call mb the formal multiplicity of the root bΣ1. Next, to bring our notation into line with that used above in the case 𝓀=, we shall replace the parameter sHom(T,*) for the spherical functions by λHom(A,) defined by s(ta)= qλ(a) for all aΣ0. λ is not uniquely determined by s, but it is unique modulo the lattice 2πilogqL, where as in (2.5) L is the lattice of all uA* such that u(a) for all aΣ0. In particular, if sTˆ then λ is pure imaginary. Writing c(λ) in place of c(s), we have from (4.1) c(λ)= bΣ1+ 1-q-(12mb/2+mb+λ(b)) 1-q(-12mb/2+λ(b)) Now let (5.3.7) ζk(s)= (1-q-s)-1 (s) which is the local zeta-function ζ(f,s) for the characteristic function of the ring of integers of 𝓀. Then in this notation we have

For 𝓀 real or p-adic, the Plancherel measure is dμ(ωλ)= κ·dλ/ |c(λ)|2 where κ is a constant and c(λ)= bΣ1+ ζk(12mb/2+λ(b)) ζk(12mb/2+mb+λ(b)) the functions ζk being defined by (5.3.5) and (5.3.7).

Remark. There is one important difference between the real and p-adic cases: in the p-adic case the formal multiplicity mb can be negative if 2b is also a root. This is precisely the 'exceptional case' dealt with in (5.2).

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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