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

Last update: 20 February 2014

## Chapter V: Plancherel measure

### The standard case

Let $\stackrel{ˆ}{T}$ be the character group of the discrete group $T\text{.}$ $\stackrel{ˆ}{T}$ 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 $u\left({a}^{\vee }\right)\in ℤ$ for all $a\in {\Sigma }_{0}\text{.}$ Let $ds$ be Haar measure on the compact group $\stackrel{ˆ}{T},$ normalized so that the total mass of $\stackrel{ˆ}{T}$ is $1\text{.}$

We shall assume in this section that $(5.1.1) qa/2≥1 for all a∈Σ0.$ Call this the standard case. The exceptional case, where ${q}_{a/2}<1$ for some $a\in {\Sigma }_{0},$ will be examined separately in (5.2). Here we shall prove

Assume that ${q}_{a/2}\ge 1$ for all $a\in {\Sigma }_{0}\text{.}$ Then the Plancherel measure $\mu$ (1.5.1) on the space ${\Omega }^{+}$ of positive definite spherical functions on $G$ relative to $K$ is concentrated on the set $\left\{{\omega }_{s}:s\in \stackrel{ˆ}{T}\right\}$ and is given there by $dμ(ωs)= Q(q-1) |W0| ·ds|c(s)|2$ where $|{W}_{0}|$ is the order of the Weyl group ${W}_{0}\text{.}$ Proof. For ${f}_{1},{f}_{2}\in ℒ\left(G,K\right)$ 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 $⟨{f}_{1},{f}_{2}⟩=⟨{\stackrel{ˆ}{f}}_{1},{\stackrel{ˆ}{f}}_{2}⟩\text{.}$ By linearity it is enough to take ${f}_{1},$ ${f}_{2}$ to be characteristic functions ${\chi }_{{t}_{1}},$ ${\chi }_{{t}_{2}}$ $\left({t}_{1},{t}_{2}\in {T}^{++}\right),$ and we may assume that ${t}_{1}\ge {t}_{2}$ with respect to the total ordering denned in (3.3.9). Clearly we have $(5.1.3) ⟨χt1,χt2⟩= { 0 if t1>t2, (Kt1K:K) if t1=t2.$ Now consider $⟨{\stackrel{ˆ}{\chi }}_{{t}_{1}},{\stackrel{ˆ}{\chi }}_{{t}_{2}}⟩\text{.}$ To compute this we shall express ${|c\left(s\right)|}^{-2}{\stackrel{ˆ}{\chi }}_{{t}_{1}}{\omega }_{s}$ as an infinite ascending series, and ${\stackrel{ˆ}{\chi }}_{{t}_{2}}\left({\omega }_{s}\right)$ as a descending polynomial, with at most one term in common. Term by term integration over the torus $\stackrel{ˆ}{T}$ will then give the required result. Since $\stackrel{‾}{s}={s}^{-1}$ we have $\stackrel{‾}{c\left(b,s\right)}=c\left(b,{s}^{-1}\right)=c\left(-b,s\right)$ for each $b\in {\Sigma }_{1},$ hence $|c(s)|-2 =(c(s)c(s-1))-1 =∏b∈Σ1c (b,s)-1,$ the product being over all the roots, positive and negative. This shows that ${|c\left(s\right)|}^{-2}$ is ${W}_{0}\text{-invariant,}$ hence $(5.1.4) |c(s)|-2= (c(ws)c(ws-1))-1$ for all $w\in {W}_{0}\text{.}$ Now $χˆt(ωs)= ∫Gχt(g)ωs (g-1)dg=vt ωs(z-1)$ where ${v}_{t}=\left(KtK:K\right),$ and $z$ is any element of ${\nu }^{-1}\left(t\right)\text{.}$ Hence if $s$ is nonsingular we have from (4.1.2) and (5.1.4) $1vt· χˆt(ωs) |c(s)|2 = δ(t)-1/2 Q(q-1) ∑w∈W0 (ws,t) c(ws-1)$ or equivalently $(5.1.5) 1vt· χˆt(ωs) |c(s)|2 = δ(t)-1/2 Q(q-1) ∑w∈W0 (ws,t) ∏b∈Σ1+ 1-qb/2-1/2(ws,tb) 1-qb/2-1/2qb-1(ws,tb) .$ Now if $b=a\in {\Sigma }_{0},$ then ${q}_{b/2}^{1/2}{q}_{b}={\left({q}_{a}{q}_{a+1}\right)}^{1/2}>1,$ and if $b=\frac{1}{2}a$ then ${q}_{b/2}^{1/2}{q}_{b}={q}_{a/2}\ne 1$ (because $b\in {\Sigma }_{1}\text{).}$ Hence the right-hand side of (5.1.5) is denned for all $s\in \stackrel{ˆ}{T},$ and therefore (5.1.5) is valid for all $s\in \stackrel{ˆ}{T}\text{.}$ Moreover, under the assumption (5.1.1) we have ${q}_{b/2}^{1/2}{q}_{b}>1$ for all $b\in {\Sigma }_{1},$ and therefore $∏b∈Σ1+ 1-qb/2-1/2(ws,tb) 1-qb/2-1/2qb-1(ws,tb) =∏b∈Σ1+ { 1-(qb-1) ∑n=1∞ qb/2-n/2 qb-n (ws,tb)n } ,$ these series being absolutely and uniformly convergent on the compact group $\stackrel{ˆ}{T}\text{.}$ Multiplying them all together we shall get from (5.1.5) $(5.1.6) 1vt· χˆt(ωs) |c(s)|2 = δ(t)-1/2 Q(q-1) ∑t′∈T++ att′ ∑w∈W0 (ws,t′)$ say, with coefficients ${a}_{tt\prime }$ independent of $s,$ and the summation being over all $t\prime \in {T}^{++}$ of the form $t′=t·∏b∈Σ1+ tbnb$ with integer exponents ${n}_{b}\ge 0\text{.}$ It follows that $t\prime \ge t$ with respect to the ordering (3.3.9); also it is easy to see that ${a}_{tt}=1,$ so that (replacing $t$ by ${t}_{1}\text{)}$ we have $(5.1.6') 1vt1· χˆt1(ωs) |c(s)|2 = δ(t1)-1/2 Q(q-1) ( ∑w∈W0 (ws,t1)+ ∑t1′>t1t1′∈T++ at1t1′ ∑w∈W0 (ws,t1′) )$ for all $t\in {T}^{++},$ the series on the right being uniformly convergent on $\stackrel{ˆ}{T}\text{.}$ On the other hand, from (3.3.4') and (3.3.8'), $(5.1.7) χˆt2(ωs)= s(χ∼t2)=δ (t2)1/2 (s,⟨t2⟩)+ ∑t2′ Furthermore, we have $(5.1.8) ∫Tˆ (ws,t1′) (s,⟨t2′⟩‾) ds= { 0 if t1′≠ t2′, 1 if t1′= t2′$ by orthogonality of characters on $\stackrel{ˆ}{T}\text{.}$ Hence, multiplying the infinite series (5.1.6') by the polynomial (5.1.7) and integrating term by term over $\stackrel{ˆ}{T},$ we shall have, using (5.1.8), $vt1-1∫Tˆ χˆt1(ωs) χˆt2(ωs)‾ |c(s)|-2ds= { 0 if t1>t2, |W0| Q(q-1) if t1=t2.$ Hence $⟨χˆt1,χˆt2⟩ =0=⟨χt1,χt2⟩ if t1>t2$ and $⟨χˆt1,χˆt1⟩ =(Kt1K:K)= ⟨χt1,χt1⟩.$ $\square$

### The exceptional case

In this section we shall consider the case excluded from the considerations of (5.1), namely where ${q}_{a/2}<1$ for some $a\in {\Sigma }_{0}\text{.}$ First of all, this implies that $\frac{a}{2}\in {\Sigma }_{1}$ for some $a\in {\Sigma }_{0},$ so that the root system ${\Sigma }_{1}$ is not reduced. Since it is irreducible, it must be of type $B{C}_{l}$ and can therefore be described as follows. There exists an orthonormal basis ${x}_{1},\dots ,{x}_{l}$ of $A$ such that, if ${e}_{1},\dots ,{e}_{l}\in {A}^{*}$ are the coordinate functions relative to this basis, the root system ${\Sigma }_{1}$ consists of $±ei(1≤i≤l), ±2ei (1≤i≤l),± ei±ej (1≤i and ${\Sigma }_{0}$ consists of $±2ei(1≤i≤l), ±ei±ej (1≤i and is of type ${C}_{l}\text{.}$ We choose the set ${\Pi }_{0}$ of simple roots as follows: $Π0= { e1-e2,e2- e3,…,el-1- el,2el } .$ The Weyl group ${W}_{0}$ is the group of all signed permutations of ${e}_{1},\dots ,{e}_{l},$ and has order ${2}^{l}l!\text{.}$

Let $q0=q±ei±ej, q1=q±ei, q2=q±2ei,$ so that ${q}_{1}<1\text{.}$

Next consider the translation group $T\text{.}$ Put $ti=t2ei (1≤i≤l).$ Then ${t}_{i}\left(0\right)={\left(2{e}_{i}\right)}^{\vee }={x}_{i},$ and hence $t±ei±ej= ti±1 tj±1.$ The ${t}_{i}$ are a basis of $T\text{.}$

If $s\in S,$ put $si=s(ti)∈ ℂ*.$ Then $c\left(s\right)$ is the product of the factors $(1+q1-1/2si-1) (1-q1-1/2q2-1si-1) 1-si-2$ for $i=1,\dots ,l,$ and the factors $1-q0-1si-1sj 1-si-1sj · 1-q0-1si-1sj-1 1-si-1sj-1$ for $1\le i

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 ${t}_{1},{t}_{2}\in {T}^{++}$ and ${t}_{1}\ge {t}_{2}\text{.}$ Proceeding as in (5.1), we have $χˆt1(ωs)= vt1· δ(t1)-1/2 Q(q-1) ∑w∈W0c(ws) (ws,t1).$ Since the Haar measure $ds$ on $\stackrel{ˆ}{T}$ is invariant under ${W}_{0},$ 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'), $χˆt2(ωs)= ∑t2′≤t2t2′∈T++ χˆt2(t2′) (s,⟨t2′⟩),$ so that (5.2.1) now takes the form $(5.2.2) vt2·δ (t1)-1/2 ∑t2′≤t2t2′∈T++ χ∼t2(t2′) ∫Tˆc(s) (s,t1) (s-1,⟨t2′⟩) ϕ(s)-1ds.$ In this expression, $\left({s}^{-1},⟨{t}_{2}^{\prime }⟩\right)$ is a sum of terms $\left({s}^{-1},w{t}_{2}^{\prime }{w}^{-1}\right)$ $\left(w\in {W}_{0}\right)\text{.}$ Since ${t}_{1}\ge {t}_{2}\ge {t}_{2}^{\prime }\ge w{t}_{2}^{\prime }{w}^{-1}$ for all $w\in {W}_{0},$ 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 $t\in T$ and $t\ge 0\text{.}$ Since $|{s}_{i}|=|s\left({t}_{i}\right)|=1,$ we may express (5.2.3) as a multiple contour integral, each contour being the unit circle $\gamma$ in the complex plane. If ${\theta }_{i}=\text{arg}\left({s}_{i}\right),$ we have $ds=∏j=1l (dθj/2π)$ and therefore $ds=∏j=1l 12πi dsjsj. (i=-1 here!)$ Consequently, if $t={t}_{1}^{{\lambda }_{1}}\cdots {t}_{l}^{{\lambda }_{l}},$ the integral (5.2.3) takes the form $(5.2.3') I=12πi∫γ s1λ1-1d s1⋯12πi ∫γslλl-1 dsl·1c(s-1).$ Since $t\ge 0,$ the first of the exponents ${\lambda }_{i}$ which is non-zero is positive.

We shall establish a reduction formula for $I\text{.}$ For this purpose we have to introduce more notation. Let $J=\left({j}_{1},\dots ,{j}_{r}\right)$ be a sequence of integers satisfying $1≤j1 and let ${\stackrel{ˆ}{T}}_{J}$ be the set of all $s\in S$ such that $sjα = -q0r-α q11/2, (1≤α≤r) |si| = 1if i∉J.$ Also define a function ${\varphi }_{J}\left(s\right)$ inductively as follows: $(5.2.4) ϕ∅(s) = ϕ(s)=c (s)c(s-1), ϕJ(s)= ϕj1,…,jr (s) = limsj1→-q0r-1q11/2 ϕj2,…,jr(s) 1+q01-rq1-1/2sj1 (r>0).$ ${\varphi }_{J}\left(s\right)$ is a function of the complex variables ${s}_{i},$ $i\notin J,$ but we regard it as a function on ${\stackrel{ˆ}{T}}_{J}\text{.}$ It is easily checked from the product which defines $\varphi \left(s\right)$ that ${\varphi }_{{j}_{2},\dots ,{j}_{r}}\left(s\right)$ has $\left(1+{q}_{0}^{1-r}{q}_{1}^{-1/2}{s}_{{j}_{1}}\right)$ 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 ${\stackrel{ˆ}{T}}_{J}$ is given by $ds=∏j∉J 12πi dsjsj.$ $\text{(}{\stackrel{ˆ}{T}}_{J}$ is a translate of a subtorus of $\stackrel{ˆ}{T},$ and this measure is the translation of the normalized Haar measure on that subtorus.)

Finally, let ${\varphi }^{1}$ (resp. ${c}^{1}\text{)}$ denote the product obtained from $\varphi$ (resp. $c\text{)}$ by deleting all factors involving ${s}_{1},$ and let ${T}^{1}$ be the subgroup of $T$ generated by ${t}_{2},\dots ,{t}_{l}\text{.}$ Define ${\stackrel{ˆ}{T}}_{J}^{1}$ and ${\varphi }_{J}^{1}$ as above for subsequences $J$ of $\left(2,3,\dots ,l\right),$ and put $IJ1=∫TˆJ1 c1(s)(s,t) ϕJ1(s)-1 ds$ for $t\in {T}^{1},$ the measure being here $ds=\prod _{j\ne 1}\frac{1}{2\pi i}\frac{d{s}_{j}}{{s}_{j}}\text{.}$

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 $1\text{.}$ So we define ${\epsilon }_{r}$ for $0\le r\le l$ as follows: ${\epsilon }_{0}=1$ and $(5.2.5) εr= { 1 if q0r-1 q11/2 <1, 0 otherwise$ for $1\le r\le l\text{.}$

Since ${q}_{0}>1,$ it follows that ${\epsilon }_{r}=1⇒{\epsilon }_{r-1}=1\text{.}$

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

Let $J=\left({j}_{1},\dots ,{j}_{r}\right)$ be a subsequence of $\left(2,3,\dots ,l\right)$ and put $J\prime =\left(1,{j}_{1},\dots ,{j}_{r}\right)\text{.}$ Then $εrIJ+ εr+1IJ′= { 0 if λ1>0, εrIJ1 if λ1=0.$ Proof. If ${\epsilon }_{r}=0$ then also ${\epsilon }_{r+1}=0$ (because ${q}_{0}>1\text{)}$ and there is nothing to prove. So assume that ${\epsilon }_{r}=1\text{.}$ Consider the integral ${I}_{J},$ and integrate it round the unit circle $\gamma$ with respect to the variable ${s}_{1}\text{.}$ To do this we must pick out the factors in the product $c\left(s\right){\varphi }_{J}{\left(s\right)}^{-1}$ which involve ${s}_{1}\text{.}$ An elementary calculation shows that they are, after some cancellations, $(5.2.7) 1-s12 (1+q0-rq1-1/2s1) (1-q1-1/2q2-1s1) · 1+q0r-1q11/2s1 1+q0-1q11/2s1 ·∏j∉J′ 1-s1sj±1 1-q0-1s1sj±1 .$ Hence, since $|{s}_{j}|=1$ for $j\notin J\prime ,$ the integrand $12πi c(s)(s,t) ϕJ(s) ds1s1$ considered as a function of ${s}_{1},$ has poles at the points $s1=-q0r q11/2, q11/2q2, -q0q1-1/2, q0si±1$ and also at the origin, if ${\lambda }_{1}=0\text{.}$ The points ${q}_{1}^{1/2}{q}_{2},$ $-{q}_{0}{q}_{1}^{-1/2},$ ${q}_{0}{s}_{i}^{±1}$ all lie outside the unit circle $\gamma$ (for ${q}_{1}^{1/2}{q}_{2}={\delta }^{1/2}\left({t}_{l}\right)>1,$ and ${q}_{0}$ and ${q}_{1}^{-1/2}$ are both $>1\text{).}$ The point $-{q}_{0}^{r}{q}_{1}^{1/2}$ lies inside $\gamma$ if and only if ${\epsilon }_{r+1}=1\text{.}$ (If ${q}_{0}^{r}{q}_{1}^{1/2}=1,$ then the factor $\left(1+{q}_{0}^{-\rho }{q}_{1}^{-1/2}{s}_{1}\right)$ in the denominator of (5.2.7) cancels into the factor $1-{s}_{1}^{2}$ 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. $\square$

$∑Jε|J| IJ= { 0 if t>1, 1 if t=1,$ where the sum is over all subsequences $J$ of $\left(1,2,\dots ,l\right),$ and $|J|=\text{Card}\left(J\right)\text{.}$ Proof. By induction on $l\text{.}$ The induction can start with $l=1,$ when the result is easily verified. If ${\lambda }_{1}>0,$ we have from (5.2.6) $∑Jε|J| IJ=0.$ If ${\lambda }_{1}=0,$ again from (5.2.6) we have $∑Jε|J| IJ=∑J1 ε|J1| IJ11,$ the sum on the right being over all subsequences ${J}^{1}$ of $\left(2,3,\dots ,l\right)\text{.}$ The result now follows from the inductive hypothesis. $\square$

Now let $\pi :S\to \Omega$ be the mapping $s↦{\omega }_{s}$ of $S$ onto the set $\Omega$ of all z.s.f. on $G$ relative to $K\text{.}$ The fibres of $\pi$ are the orbits of the action of ${W}_{0}$ on $S\text{.}$ Clearly $\pi \left({\stackrel{ˆ}{T}}_{J}\right)$ 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 ${\stackrel{ˆ}{T}}_{J},$ the z.s.f. belonging to ${\Omega }_{r}$ are the ${\omega }_{s}$ for which $r$ of the numbers $s\left({t}_{i}\right)$ $\left(1\le i\le l\right)$ take the values $-q11/2,- q0q11/2, …,-q0r-1 q11/2$ (or their inverses), and the remaining $s\left({t}_{i}\right)$ have absolute value $1\text{.}$ Since $\varphi \left(s\right)$ is invariant under the action of ${W}_{0},$ it follows that ${\varphi }_{J}\left(s\right)$ depends only on the number of elements in $J\text{:}$ so we may define $ϕr(ωs)= ϕJ(s)if |J|=r.$

The spherical functions belonging to ${\Omega }_{r}$ are positive definite if ${\epsilon }_{r}=1\text{.}$

We leave the proof until later.

We have already defined a measure $ds$ on each ${\stackrel{ˆ}{T}}_{J}$ giving it total mass $1\text{.}$ Now define a measure $d\omega$ on ${\Omega }_{r}$ as follows: $∫Ωrf(ω) dω=∫π-1(Ωr) f(ωs)ds$ for any continuous function $f$ on ${\Omega }_{r}$ with compact support.

The Plancherel measure $\mu$ on the space of positive definite spherical functions is concentrated on the sets ${\Omega }_{r}$ such that ${\epsilon }_{r}=1,$ where ${\epsilon }_{r}$ is defined in (5.2.5). On ${\Omega }_{r,\mu }$ is given by $dμ(ω)= Q(q-1) wr dωϕr(ω)$ where ${w}_{r}={2}^{l-r}\left(l-r\right)!$ is the order of the normalizer in ${W}_{0}$ of any of the sets ${\stackrel{ˆ}{T}}_{J}$ such that $|J|=r\text{.}$ Proof. For ${f}_{1},{f}_{2}\in ℒ\left(G,K\right),$ define $⟨fˆ1,fˆ2⟩r =∫Ωrfˆ1 (ω)fˆ2(ω)‾ dμ(ω) (0≤r≤l)$ where $d\mu \left(\omega \right)$ is as above. Then we have to show that $∑r=0lεr ⟨fˆ1,fˆ2⟩r =⟨f1,f2⟩.$ As before we take ${f}_{1}={\chi }_{{t}_{1}},$ ${f}_{2}={\chi }_{{t}_{2}}$ with ${t}_{1},{t}_{2}\in {T}^{++}$ and ${t}_{1}\ge {t}_{2}\text{.}$ Let ${J}_{r}$ be the sequence $\left(1,2,\dots ,r\right)\text{.}$ Then exactly as in (5.1) $⟨χˆt1,χˆt2⟩r = Q(q-1)wr ∫TˆJr χˆt1(ωs) χˆt2(ωs)‾ dsϕJr(s) = vt1·δ(t1)-1/2 wr ∑t2′≤t2t2′∈T++ χ∼t2(t2′) ∑w∈W0 ∫TˆJrc(ws) (ws,t1) (s-1,⟨t2′⟩) dsϕJr(s).$ Now it is easily verified that if $s\in {\stackrel{ˆ}{T}}_{{J}_{r}}$ and $w\in {W}_{0},$ then $c\left(ws\right)=0$ unless $w$ maps ${t}_{1},\dots ,{t}_{r}$ to ${t}_{{j}_{1}},\dots ,{t}_{{j}_{r}}$ respectively with ${j}_{1}<\cdots <{j}_{r}\text{.}$ Hence $1wr∑w∈W0 ∫TˆJrc (ws)(ws,t1) (s-1,⟨t2′⟩) ϕJr(s)-1 ds =∑|J|=r ∫TˆJc(s) (s,t1) (s-1,⟨t2′⟩) ϕJ(s)-1ds.$ Consequently $∑r=0lεr ⟨χˆt1,χˆt2⟩r =vt1·δ (t1)-1/2 ∑t2′≤t2t2′∈T++ χ∼t2(t2′) ∑Jε|J| ∫TˆJ c(s)(s,t1) (s-1,⟨t2′⟩) ds ϕJ(s)$ and by (5.2.8) the inner sum is $0$ or $1$ according as ${t}_{1}\ne {t}_{2}^{\prime }$ or ${t}_{1}={t}_{2}^{\prime }\text{.}$ Hence, finally, $∑r=0lεr ⟨χˆt1,χˆt2⟩r= { 0 if t1>t2, ⟨χt1,χt1⟩ if t1=t2$ since ${v}_{{t}_{1}}·\delta {\left({t}_{1}\right)}^{-1/2}{\stackrel{\sim }{\chi }}_{{t}_{1}}\left({t}_{1}\right)={v}_{{t}_{1}}=⟨{\chi }_{{t}_{1}},{\chi }_{{t}_{1}}⟩\text{.}$ $\square$

We have still to prove (5.2.9). Consider first the set ${\Omega }_{l}\text{:}$ this consists of a single spherical function, say ${\omega }_{{s}_{0}},$ where ${s}_{0}\in S$ is given by $s0(ti)=- q0i-1 q11/2 (1≤i≤l).$

Let $z\in {Z}^{++},$ $\nu \left(z\right)=t\text{.}$ Then $ωs0(z-1)= (δ-1/2s0,t).$ Proof. Consider $c\left(w{s}_{0}\right)$ where $w\in {W}_{0}$ and $w\ne 1\text{.}$ One sees immediately that some factor in the numerator vanishes, so that $c\left(w{s}_{0}\right)=0$ if $w\ne 1\text{.}$ Also a simple verification shows that $c\left({s}_{0}\right)=c\left({\delta }^{1/2}\right),$ and hence $c\left({s}_{0}\right)=Q\left({q}^{-1}\right)$ by (4.5.8). The lemma now follows from (4.1.2). $\square$

If ${\epsilon }_{l}=1$ then ${\omega }_{{s}_{0}}$ is square-integrable. Proof. We have $∫G|ωs0(g)|2 dg=∑t∈T++ vt(δ-1/2s0,t)2$ where ${v}_{t}=\left(KtK:K\right)=\delta \left(t\right){Q}^{t}\left({q}^{-1}\right)$ by (3.2.15). Hence we have to show that the series $(5.2.13) ∑t∈T++ Qt(q-1) (s0,t)2$ converges if $|{s}_{0}\left({t}_{i}\right)|<1$ for $1\le i\le l\text{.}$ Since $t\in {T}^{++}$ we have $t=\prod _{i=1}^{l}{t}_{i}^{{\lambda }_{i}}$ with ${\lambda }_{1}\ge {\lambda }_{2}\ge \dots \ge {\lambda }_{l}\ge 0\text{.}$ Let $\rho =\left({\rho }_{1},\dots ,{\rho }_{r}\right)$ be a sequence of integers such that $(5.2.14) 1\le {\rho }_{1}< ρ2<…<ρr ≤l$ and consider the $t=\prod {t}_{i}^{{\lambda }_{i}}$ such that $λ1=λ2=…= λρ1>λρ1+1 =…=λρ2>…> λρr+1=…= λl=0.$ ${Q}^{t}\left({q}^{-1}\right)$ will be the same for all these $t,$ because they all have the same stabilizer ${W}_{0t}$ in the Weyl group ${w}_{0}\text{.}$ Hence the series (5.2.13) splits up into a sum of series, one for each sequence $\rho$ satisfying (5.2.14), and each of these series is a product of geometric series whose common ratios are products of the ${s}_{0}{\left({t}_{i}\right)}^{2}\text{.}$ Hence all these series converge and therefore so does the series (5.2.13). $\square$

Remark. It is not difficult to show, using (5.2.11), that ${\omega }_{{s}_{0}}$ is a z.s.f. on $G$ relative to the Iwahori subgroup $B\text{.}$ By (1.2.6) the z.s.f. on $G$ relative to $B$ correspond bijectively to the $ℂ\text{-algebra}$ homomorphisms $ℒ\left(G,B\right)\to ℂ\text{.}$ Let ${\Pi }_{0}$ (resp. $\Pi \text{)}$ be the set of simple roots for ${\Sigma }_{0}$ (resp. $\Sigma \text{),}$ (so that ${\Pi }_{0}=\left\{{a}_{1},\dots ,{a}_{l}\right\},$ $\Pi =\left\{{\alpha }_{0},\dots ,{\alpha }_{l}\right\},$ where ${\alpha }_{0}=1-2{e}_{1},$ ${\alpha }_{i}={a}_{i}={e}_{i}-{e}_{i+1}$ $\left(1\le i\le l-1\right),$ ${\alpha }_{l}={a}_{l}=2{e}_{l}\text{).}$ Then $ℒ\left(G,B\right)$ is generated as $ℂ\text{-algebra}$ by the characteristic functions ${\chi }_{i}$ of $B{w}_{{\alpha }_{i}}B$ $\left(0\le i\le l\right),$ and the homomorphism ${\stackrel{ˆ}{\omega }}_{{s}_{0}}:ℒ\left(G,B\right)\to ℂ$ is given by ${\chi }_{i}↦q\left({w}_{{\alpha }_{i}}\right)$ $\left(1\le i\le l\right),$ ${\chi }_{0}↦-1\text{.}$

In general, the z.s.f. on $G$ relative to $B$ correspond naturally to the linear characters of the Weyl group $W\text{.}$ 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 ${\omega }_{s}\in {\Omega }_{r},$ where ${\epsilon }_{r}=1\text{.}$ We may assume that $s(ti) = -q0r-i q11/2 (1≤i≤r), |s(ti)| = 1if i>r.$ Let $A\prime$ (resp. $A″\text{)}$ be the subspace of $A$ spanned by the first $r$ basis vectors ${x}_{1},\dots ,{x}_{r}$ (resp. by ${x}_{r+1},\dots ,{x}_{l}\text{).}$ The restrictions to $A\prime$ of the roots $a\in {\Sigma }_{0}$ (resp. affine roots $\alpha \in \Sigma \text{)}$ which vanish identically on $A″$ form a subsystem ${\Sigma }_{0}^{\prime }$ of ${\Sigma }_{0}$ (resp. a subsystem $\Sigma \prime$ of $\Sigma \text{).}$ The Weyl group ${W}_{0}^{\prime }$ and affine Weyl group $W\prime$ of $\Sigma \prime$ can be regarded as subgroups of ${W}_{0}$ and $W$ respectively. The translation subgroup $T\prime$ of $W\prime$ is generated by ${t}_{1},\dots ,{t}_{r}\text{.}$ Likewise, starting with $A″$ we define ${\Sigma }_{0}^{″},$ $\Sigma ″,$ etc.

Let $G\prime$ be the subgroup of $G$ generated by the ${U}_{\alpha }$ with $\alpha \in \Sigma$ and by $N\prime ={\nu }^{-1}\left(W\prime \right),$ and define $G″$ similarly. Then $G\prime G″$ is a subgroup of $G\text{.}$

Let $s\prime =s|T\prime ,$ $s″=s|T″\text{.}$ Then ${\omega }_{s\prime }$ is a square-integrable spherical function on $G\prime$ 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 $G″\text{)}$ that ${\omega }_{s″}$ is positive definite.

Now let ${P}_{0}$ be the (parabolic) subgroup of $G$ generated by $G\prime ,$ $G″$ and ${U}^{-},$ and let ${\Delta }_{0}$ be the modulus function on ${P}_{0}\text{.}$ ${P}_{0}$ is the semi-direct product of $G\prime G″$ and the subgroup ${U}_{0}^{\prime }$ generated by the root subgroups ${U}_{\left(a\right)}$ corresponding to negative roots $a\in {\Sigma }_{0}$ which do not belong to either ${\Sigma }_{0}^{\prime }$ or ${\Sigma }_{0}^{″}$ (i.e. $a=-{e}_{i}±{e}_{j}$ with $i\le r$ and $j>r\text{).}$ The spherical function ${\Delta }_{0}^{-1/2}{\omega }_{s\prime }{\omega }_{s″}$ on $G\prime G″$ extends to a spherical function on ${P}_{0},$ say ${\omega }_{0}\text{.}$ By (1.4.7), the spherical function induced by ${\omega }_{0}$ on $G$ is positive definite, and by transitivity of induction (1.3.3) it is equal to ${\omega }_{s}\text{.}$ 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 $ℝ$ or $ℂ$ or a $p\text{-adic}$ field, and the additive group of $𝓀$ is self-dual. Associated canonically with $𝓀$ there is a meromorphic function ${\gamma }_{k}\left(s\right)$ of a complex variable $s,$ sometimes called the gamma-function of $𝓀\text{.}$ We recall its definition briefly (see Tate's thesis [Tat1967], where it is denoted by $\rho \left({|\phantom{\rule{1em}{0ex}}|}^{s}\right)\text{.)}$

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

We can compute ${\gamma }_{k}\left(s\right)$ from (5.3.1) by choosing the function $f$ intelligently.

$𝓀=ℝ\text{:}$ take $f\left(x\right)={e}^{-\pi {x}^{2}},$ then $\stackrel{ˆ}{f}=f$ and one finds that $γℝ(s)= π-s/2Γ(s/2) π(s-1)/2Γ((1-s)/2)$ so that $(5.3.2)ℝ γℝ(s) γℝ(-s)= B(12,12s) B(12,-12s)$ where $B$ is Euler's beta-function.

$𝓀=ℂ\text{:}$ take $f\left(x\right)={e}^{-2\pi {|x|}^{2}}$ $\text{(}|x|$ the ordinary absolute value on $ℂ\text{:}$ in fact $‖x‖={|x|}^{2}\text{),}$ then again $\stackrel{ˆ}{f}=f$ and we have $γℂ(s)= (2π)-sΓ(s) (2π)s-1Γ(1-s)$ so that $(5.3.2)ℂ γℂ(s) γℂ(-s)= -4π2/s2.$

$k=p\text{-adic field.}$ 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 $G\left({\Sigma }_{0},𝓀\right)$ as in (2.5), where $𝓀$ is any local field. If $𝓀=ℝ$ or $ℂ,$ let $K$ be any maximal compact subgroup of $G$ (they are all conjugate). If $𝓀$ is $p\text{-adic,}$ 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 $𝓀=ℝ$ or $ℂ,$ the z.s.f. on $G$ relative to $K$ are parametrized by the $ℝ\text{-linear}$ mappings $s:A\to ℂ,$ 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 $l=\text{dim} A\text{.}$ If $ds$ is a Euclidean measure on this space, then (Harish-Chandra [Har1966]) the Plancherel measure $\mu$ is of the form $dμ(ωs)= κ ds/|c(s)|2$ where $\kappa$ is a constant (depending on the choices of $ds$ and of Haar measure on $G\text{)}$ and $c(s)= ∏a∈Σ0+ B(12,12s(a∨)) if 𝓀=ℝ$ and $c(s)= ∏a∈Σ0+ s(a∨)-1 if 𝓀=ℂ.$ Hence, from (5.3.2), we have $(5.3.3)ℝ,ℂ c(s)c(-s) =∏a∈Σ0 γk(s(a∨))$ in both cases, apart from a constant factor in the case $𝓀=ℂ\text{.}$

On the other hand, if $𝓀$ is $p\text{-adic,}$ then as we have seen in (5.1) the support of the Plancherel measure is the character group $\stackrel{ˆ}{T}$ of $T\text{.}$ 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 ${s}_{0}\in S=\text{Hom}\left(T,{ℂ}^{*}\right),$ we define $s:A\to ℂ/\left(\frac{2\pi i}{\text{log} q}\right)ℤ$ by the rule $s0(ta)= q-s(a∨) (a∈Σ0)$ where as before $q$ is the number of elements in the residue field of $𝓀\text{.}$ Then from (5.1.2) the Plancherel measure $\mu$ is of the form $dμ(ωs)=κ ds/|c(s)|2$ where $ds$ is Euclidean measure on the space of pure imaginary $s,$ $\kappa$ is a constant, and $c(s)= ∏a∈Σ0+ 1-q-1-s(a∨) 1-q-s(a∨)$ (because in the present situation all the ${q}_{a}$ are equal to $q\text{).}$ Hence, from ${\text{(5.3.2)}}_{p\text{-adic}}$ we have $(5.3.3)p-adic c(s)c(-s) =∏a∈Σ0 γk(s(a∨)),$ apart from a constant factor depending only on $𝓀\text{.}$

So, to sum up:

If $𝓀$ is any local field and $G=G\left({\Sigma }_{0},𝓀\right)$ 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 $dμ(ωs)= κ·ds ∏a∈Σ0γk(s(a∨))$ where $\kappa$ is a constant.

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

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

$𝓀=ℝ\text{.}$ Let $G=KAN$ be an Iwasawa decomposition, $𝔤$ and $𝔞$ the Lie algebras of $G$ and $A$ respectively, ${\Sigma }_{1}$ the set of 'restricted roots' of $𝔤$ relative to $𝔞,$ and for each $b\in {\Sigma }_{1}$ let ${m}_{b}$ be the multiplicity of $b\text{.}$ Then the z.s.f. on $G$ relative to $K$ are parametrized by the elements $\lambda$ of ${\text{Hom}}_{ℝ}\left(𝔞,ℂ\right),$ and the Plancheral measure is supported on the subspace consisting of the pure imaginary $\lambda :𝔞\to iℝ\text{.}$ Write ${\omega }_{\lambda }$ for the z.s.f. corresponding to $\lambda \text{.}$ Then [Har1966] the Plancherel measure $\mu$ is given by $dμ(ωλ)= κ dλ/|c(λ)|2$ where $\kappa$ is a constant and (see [GKa1962]) $c\left(\lambda \right)$ is a product of beta-functions, namely $c(λ)= ∏b∈Σ1+ B ( 12mb,14 mb/2+12 λ(b∨) ) .$ Let $(5.3.5) ξℝ(s)= π-12s Γ(12s) (s∈ℂ)$ which is the local zeta-function $\zeta \left(f,s\right)$ for the function $f\left(x\right)={e}^{-\pi {x}^{2}}\text{.}$ In this notation we have $(5.3.6) c(λ)=κ ∏b∈Σ1+ ζℝ(12mb/2+λ(b∨)) ζℝ(mb+12mb/2+λ(b∨))$ where $\kappa$ is independent of $A\text{.}$

$𝓀$ $p\text{-adic.}$ If the number of elements in the residue field of $𝓀$ is $q,$ then each index ${q}_{b}$ $\left(b\in {\Sigma }_{1}\right)$ is a power of $q\text{.}$ We shall therefore write $qb=qmb (b∈Σ1)$ and call ${m}_{b}$ the formal multiplicity of the root $b\in {\Sigma }_{1}\text{.}$ Next, to bring our notation into line with that used above in the case $𝓀=ℝ,$ we shall replace the parameter $s\in \text{Hom}\left(T,{ℂ}^{*}\right)$ for the spherical functions by $\lambda \in {\text{Hom}}_{ℝ}\left(A,ℂ\right)$ defined by $s(ta)= qλ(a∨)$ for all $a\in {\Sigma }_{0}\text{.}$ $\lambda$ is not uniquely determined by $s,$ but it is unique modulo the lattice $\frac{2\pi i}{\text{log} q}L,$ where as in (2.5) $L$ is the lattice of all $u\in {A}^{*}$ such that $u\left({a}^{\vee }\right)\in ℤ$ for all $a\in {\Sigma }_{0}\text{.}$ In particular, if $s\in \stackrel{ˆ}{T}$ then $\lambda$ is pure imaginary. Writing $c\left(\lambda \right)$ in place of $c\left(s\right),$ we have from (4.1) $c(λ)= ∏b∈Σ1+ 1-q-(12mb/2+mb+λ(b∨)) 1-q(-12mb/2+λ(b∨))$ Now let $(5.3.7) ζk(s)= (1-q-s)-1 (s∈ℂ)$ which is the local zeta-function $\zeta \left(f,s\right)$ for the characteristic function of the ring of integers of $𝓀\text{.}$ Then in this notation we have

For $𝓀$ real or $p\text{-adic,}$ the Plancherel measure is $dμ(ωλ)= κ·dλ/ |c(λ)|2$ where $\kappa$ is a constant and $c(λ)= ∏b∈Σ1+ ζk(12mb/2+λ(b∨)) ζk(12mb/2+mb+λ(b∨))$ the functions ${\zeta }_{k}$ being defined by (5.3.5) and (5.3.7).

Remark. There is one important difference between the real and $p\text{-adic}$ cases: in the $p\text{-adic}$ case the formal multiplicity ${m}_{b}$ 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 $p\text{-adic}$ 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.