Spherical Functions on a Group of $p\text{-adic}$ 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 $\hat{T}$ be the character group of the discrete group $T\text{.}$ $\hat{T}$ 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 $u\left(a^{\vee }\right)\in \mathbb{Z}$ for all $a\in {\Sigma }_0\text{.}$ Let $ds$ be Haar measure on the compact group $\hat{T},$ normalized so that the total mass of $\hat{T}$ is $1\text{.}$

We shall assume in this section that $$\begin{array}{cc}\text{(5.1.1)}& q_{a/2}\ge 1\text{for all}\hspace{0.17em}a\in {\Sigma }_0\text{.}\end{array}$$ 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 $\{{\omega }_s:s\in \hat{T}\}$ and is given there by $$d\mu \left({\omega }_s\right)=\frac{Q\left(q^{-1}\right)}{\left|W_0\right|}·\frac{ds}{{\left|c\left(s\right)\right|}^2}$$ where $\left|W_0\right|$ is the order of the Weyl group $W_0\text{.}$

		Proof.
	For $f_1,f_2\in ℒ(G,K)$ we define $$\begin{array}{ccc}⟨f_1,f_2⟩& =& {\int }_G{f}_1\left(g\right)\overline{f_2\left(g\right)}dg,\\ ⟨{\hat{f}}_1,{\hat{f}}_2⟩& =& \frac{Q\left(q^{-1}\right)}{\left|W_0\right|}{\int }_{\hat{T}}{\hat{f}}_1\left({\omega }_s\right)\overline{{\stackrel{ˆ}{f}}_2\left({\omega }_s\right)}{\left|c\left(s\right)\right|}^{-2}ds\text{.}\end{array}$$ We have to show that $⟨f_1,f_2⟩=⟨{\hat{f}}_1,{\hat{f}}_2⟩\text{.}$ By linearity it is enough to take $f_1,$ $f_2$ to be characteristic functions ${\chi }_{t_1},$ ${\chi }_{t_2}$ $(t_1,t_2\in T^{++}),$ and we may assume that $t_1\ge t_2$ with respect to the total ordering denned in (3.3.9). Clearly we have $$\begin{array}{cc}\text{(5.1.3)}& ⟨{\chi }_{t_1},{\chi }_{t_2}⟩=\{\begin{array}{cc}0& \text{if}\hspace{0.17em}{t}_1>t_2,\\ (K{t}_{1}K:K)& \text{if}\hspace{0.17em}{t}_1=t_2\text{.}\end{array}\end{array}$$ Now consider $⟨{\hat{\chi }}_{t_1},{\hat{\chi }}_{t_2}⟩\text{.}$ To compute this we shall express ${\left|c\left(s\right)\right|}^{-2}{\hat{\chi }}_{t_1}{\omega }_s$ as an infinite ascending series, and ${\hat{\chi }}_{t_2}\left({\omega }_s\right)$ as a descending polynomial, with at most one term in common. Term by term integration over the torus $\hat{T}$ will then give the required result. Since $\bar{s}=s^{-1}$ we have $\overline{c(b,s)}=c(b,s^{-1})=c(-b,s)$ for each $b\in {\Sigma }_1,$ hence $${\left|c\left(s\right)\right|}^{-2}={\left(c\left(s\right)c\left(s^{-1}\right)\right)}^{-1}=\prod _{b\in {\Sigma }_1}c{(b,s)}^{-1},$$ the product being over all the roots, positive and negative. This shows that ${\left|c\left(s\right)\right|}^{-2}$ is $W_0\text{-invariant,}$ hence $$\begin{array}{cc}\text{(5.1.4)}& {\left|c\left(s\right)\right|}^{-2}={\left(c\left(ws\right)c\left(w{s}^{-1}\right)\right)}^{-1}\end{array}$$ for all $w\in W_0\text{.}$ Now $${\hat{\chi }}_t\left({\omega }_s\right)={\int }_G{\chi }_t\left(g\right){\omega }_s\left(g^{-1}\right)dg=v_t{\omega }_s\left(z^{-1}\right)$$ where $v_t=(KtK:K),$ 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) $$\frac{1}{v_t}·\frac{{\hat{\chi }}_t\left({\omega }_s\right)}{{\left|c\left(s\right)\right|}^2}=\frac{\delta {\left(t\right)}^{-1/2}}{Q\left(q^{-1}\right)}\sum _{w\in W_0}\frac{(ws,t)}{c\left(w{s}^{-1}\right)}$$ or equivalently $$\begin{array}{cc}\text{(5.1.5)}& \frac{1}{v_t}·\frac{{\hat{\chi }}_t\left({\omega }_s\right)}{{\left|c\left(s\right)\right|}^2}=\frac{\delta {\left(t\right)}^{-1/2}}{Q\left(q^{-1}\right)}\sum _{w\in W_0}(ws,t)\prod _{b\in {\Sigma }_1^{+}}\frac{1-q_{b/2}^{-1/2}(ws,t_b)}{1-q_{b/2}^{-1/2}{q}_b^{-1}(ws,t_b)}\text{.}\end{array}$$ 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 $s\in \hat{T},$ and therefore (5.1.5) is valid for all $s\in \hat{T}\text{.}$ 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 $$\prod _{b\in {\Sigma }_1^{+}}\frac{1-q_{b/2}^{-1/2}(ws,t_b)}{1-q_{b/2}^{-1/2}{q}_b^{-1}(ws,t_b)}=\prod _{b\in {\Sigma }_1^{+}}\{1-(q_b-1)\sum _{n=1}^{\infty }{q}_{b/2}^{-n/2}{q}_b^{-n}{(ws,t_b)}^n\},$$ these series being absolutely and uniformly convergent on the compact group $\hat{T}\text{.}$ Multiplying them all together we shall get from (5.1.5) $$\begin{array}{cc}\text{(5.1.6)}& \frac{1}{v_t}·\frac{{\hat{\chi }}_t\left({\omega }_s\right)}{{\left|c\left(s\right)\right|}^2}=\frac{\delta {\left(t\right)}^{-1/2}}{Q\left(q^{-1}\right)}\sum _{t\prime \in T^{++}}{a}_{tt\prime }\sum _{w\in W_0}(ws,t\prime )\end{array}$$ say, with coefficients $a_{tt\prime }$ independent of $s,$ and the summation being over all $t\prime \in T^{++}$ of the form $$t\prime =t·\prod _{b\in {\Sigma }_1^{+}}{t}_b^{n_b}$$ 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 $$\begin{array}{cc}\text{(5.1.6')}& \frac{1}{v_{t_1}}·\frac{{\hat{\chi }}_{t_1}\left({\omega }_s\right)}{{\left|c\left(s\right)\right|}^2}=\frac{\delta {\left(t_1\right)}^{-1/2}}{Q\left(q^{-1}\right)}(\sum _{w\in W_0}(ws,t_1)+\sum _{\genfrac{}{}{0ex}{}{t_1^{\prime }>t_1}{t_1^{\prime }\in T^{++}}}{a}_{t_1{t}_1^{\prime }}\sum _{w\in W_0}(ws,t_1^{\prime }))\end{array}$$ for all $t\in T^{++},$ the series on the right being uniformly convergent on $\hat{T}\text{.}$ On the other hand, from (3.3.4') and (3.3.8'), $$\begin{array}{cc}\text{(5.1.7)}& {\hat{\chi }}_{t_2}\left({\omega }_s\right)=s\left({\stackrel{\sim }{\chi }}_{t_2}\right)=\delta {\left(t_2\right)}^{1/2}(s,⟨t_2⟩)+\sum _{\genfrac{}{}{0ex}{}{t_2^{\prime }<t_2}{t_2^{\prime }\in T^{++}}}{\hat{\chi }}_{t_2}\left(t_2^{\prime }\right)(s,⟨t_2^{\prime }⟩)\text{.}\end{array}$$ Furthermore, we have $$\begin{array}{cc}\text{(5.1.8)}& {\int }_{\hat{T}}(ws,t_1^{\prime })\left(\overline{s,⟨t_2^{\prime }⟩}\right)ds=\{\begin{array}{cc}0& \text{if}\hspace{0.17em}{t}_1^{\prime }\ne t_2^{\prime },\\ 1& \text{if}\hspace{0.17em}{t}_1^{\prime }=t_2^{\prime }\end{array}\end{array}$$ by orthogonality of characters on $\hat{T}\text{.}$ Hence, multiplying the infinite series (5.1.6') by the polynomial (5.1.7) and integrating term by term over $\hat{T},$ we shall have, using (5.1.8), $$v_{t_1}^{-1}{\int }_{\hat{T}}{\hat{\chi }}_{t_1}\left({\omega }_s\right)\overline{{\stackrel{ˆ}{\chi }}_{t_2}\left({\omega }_s\right)}{\left|c\left(s\right)\right|}^{-2}ds=\{\begin{array}{cc}0& \text{if}\hspace{0.17em}{t}_1>t_2,\\ \frac{\left|W_0\right|}{Q\left(q^{-1}\right)}& \text{if}\hspace{0.17em}{t}_1=t_2\text{.}\end{array}$$ Hence $$⟨{\hat{\chi }}_{t_1},{\hat{\chi }}_{t_2}⟩=0=⟨{\chi }_{t_1},{\chi }_{t_2}⟩\text{if}\hspace{0.17em}{t}_1>t_2$$ and $$⟨{\hat{\chi }}_{t_1},{\hat{\chi }}_{t_1}⟩=(K{t}_{1}K:K)=⟨{\chi }_{t_1},{\chi }_{t_1}⟩\text{.}$$ $\square$

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 $$\pm e_i(1\le i\le l),\pm 2e_i(1\le i\le l),\pm e_i\pm e_j(1\le i<j\le l)$$ and ${\Sigma }_0$ consists of $$\pm 2e_i(1\le i\le l),\pm e_i\pm e_j(1\le i<j\le l)$$ and is of type $C_l\text{.}$ We choose the set ${\Pi }_0$ of simple roots as follows: $${\Pi }_0=\{e_1-e_2,e_2-e_3,\dots ,e_{l-1}-e_l,2e_l\}\text{.}$$ 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 $$q_0=q_{\pm e_i\pm e_j},q_1=q_{\pm e_i},q_2=q_{\pm 2e_i},$$ so that $q_1<1\text{.}$

Next consider the translation group $T\text{.}$ Put $$t_i=t_{2e_i}(1\le i\le l)\text{.}$$ Then $t_i\left(0\right)={\left(2e_i\right)}^{\vee }=x_i,$ and hence $$t_{\pm e_i\pm e_j}=t_i^{\pm 1}{t}_j^{\pm 1}\text{.}$$ The $t_i$ are a basis of $T\text{.}$

If $s\in S,$ put $$s_i=s\left(t_i\right)\in {ℂ}^{*}\text{.}$$ Then $c\left(s\right)$ is the product of the factors $$\frac{(1+q_1^{-1/2}{s}_i^{-1})(1-q_1^{-1/2}{q}_2^{-1}{s}_i^{-1})}{1-s_i^{-2}}$$ for $i=1,\dots ,l,$ and the factors $$\frac{1-q_0^{-1}{s}_i^{-1}{s}_j}{1-s_i^{-1}{s}_j}·\frac{1-q_0^{-1}{s}_i^{-1}{s}_j^{-1}}{1-s_i^{-1}{s}_j^{-1}}$$ for $1\le i<j\le l\text{.}$

Finally, let $$\varphi \left(s\right)=c\left(s\right)c\left(s^{-1}\right)\text{.}$$ Then we have to compute the integral $$\begin{array}{cc}\text{(5.2.1)}& \frac{Q\left(q^{-1}\right)}{\left|W_0\right|}{\int }_{\hat{T}}{\hat{\chi }}_{t_1}\left({\omega }_s\right)\overline{{\stackrel{ˆ}{\chi }}_{t_2}\left({\omega }_s\right)}·\varphi {\left(s\right)}^{-1}ds\end{array}$$ where $t_1,t_2\in T^{++}$ and $t_1\ge t_2\text{.}$ Proceeding as in (5.1), we have $${\hat{\chi }}_{t_1}\left({\omega }_s\right)=v_{t_1}·\frac{\delta {\left(t_1\right)}^{-1/2}}{Q\left(q^{-1}\right)}\sum _{w\in W_0}c\left(ws\right)(ws,t_1)\text{.}$$ Since the Haar measure $ds$ on $\hat{T}$ is invariant under $W_0,$ it follows that the integral (5.2.1) is equal to $$v_{t_1}·\delta {\left(t_1\right)}^{-1/2}{\int }_{\hat{T}}c\left(s\right)(s,t_1)·\overline{{\stackrel{ˆ}{\chi }}_{t_2}\left({\omega }_s\right)}\varphi {\left(s\right)}^{-1}ds\text{.}$$ On the other hand, from (3.3.8'), $${\hat{\chi }}_{t_2}\left({\omega }_s\right)=\sum _{\genfrac{}{}{0ex}{}{t_2^{\prime }\le t_2}{t_2^{\prime }\in T^{++}}}{\hat{\chi }}_{t_2}\left(t_2^{\prime }\right)(s,⟨t_2^{\prime }⟩),$$ so that (5.2.1) now takes the form $$\begin{array}{cc}\text{(5.2.2)}& v_{t_2}·\delta {\left(t_1\right)}^{-1/2}\sum _{\genfrac{}{}{0ex}{}{t_2^{\prime }\le t_2}{t_2^{\prime }\in T^{++}}}{\stackrel{\sim }{\chi }}_{t_2}\left(t_2^{\prime }\right){\int }_{\hat{T}}c\left(s\right)(s,t_1)(s^{-1},⟨t_2^{\prime }⟩)\varphi {\left(s\right)}^{-1}ds\text{.}\end{array}$$ In this expression, $(s^{-1},⟨t_2^{\prime }⟩)$ is a sum of terms $(s^{-1},w{t}_2^{\prime }{w}^{-1})$ $(w\in W_0)\text{.}$ 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 $$\begin{array}{cc}\text{(5.2.3)}& I={\int }_{\hat{T}}c\left(s\right)(s,t)\varphi {\left(s\right)}^{-1}ds\end{array}$$ where $t\in T$ and $t\ge 0\text{.}$ Since $\left|s_i\right|=\left|s\left(t_i\right)\right|=1,$ 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 $$ds=\prod _{j=1}^l(d{\theta }_j/2\pi )$$ and therefore $$ds=\prod _{j=1}^l\frac{1}{2\pi i}\frac{d{s}_j}{s_j}\text{.}(i=\sqrt{-1}\hspace{0.17em}\text{here!})$$ Consequently, if $t=t_1^{{\lambda }_1}\cdots t_l^{{\lambda }_l},$ the integral (5.2.3) takes the form $$\begin{array}{cc}\text{(5.2.3')}& I=\frac{1}{2\pi i}{\int }_{\gamma }{s}_1^{{\lambda }_1-1}d{s}_1\cdots \frac{1}{2\pi i}{\int }_{\gamma }{s}_l^{{\lambda }_l-1}d{s}_l·\frac{1}{c\left(s^{-1}\right)}\text{.}\end{array}$$ 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 $J=(j_1,\dots ,j_r)$ be a sequence of integers satisfying $$1\le j_1<j_2<\cdots <j_r\le l$$ and let ${\hat{T}}_J$ be the set of all $s\in S$ such that $$\begin{array}{ccc}{s}_{j_{\alpha }}& =& -q_0^{r-\alpha }{q}_1^{1/2},(1\le \alpha \le r)\\ \left|s_i\right|& =& 1\text{if}\hspace{0.17em}i\notin J\text{.}\end{array}$$ Also define a function ${\varphi }_J\left(s\right)$ inductively as follows: $$\begin{array}{cc}\text{(5.2.4)}& \begin{array}{ccc}{\varphi }_{\varnothing }\left(s\right)& =& \varphi \left(s\right)=c\left(s\right)c\left(s^{-1}\right),\\ {\varphi }_J\left(s\right)={\varphi }_{j_1,\dots ,j_r}\left(s\right)& =& \underset{s_{j_1}\to -q_0^{r-1}{q}_1^{1/2}}{\text{lim}}\frac{{\varphi }_{j_2,\dots ,j_r}\left(s\right)}{1+q_0^{1-r}{q}_1^{-1/2}{s}_{j_1}}(r>0)\text{.}\end{array}\end{array}$$ ${\varphi }_J\left(s\right)$ is a function of the complex variables $s_i,$ $i\notin J,$ but we regard it as a function on ${\hat{T}}_J\text{.}$ It is easily checked from the product which defines $\varphi \left(s\right)$ that ${\varphi }_{j_2,\dots ,j_r}\left(s\right)$ has $(1+q_0^{1-r}{q}_1^{-1/2}{s}_{j_1})$ as a factor in the numerator, so that the right-hand side of (5.2.4) is well defined.

Let $$I_J=I_{j_1,\dots ,j_r}={\int }_{{\hat{T}}_J}c\left(s\right)(s,t){\varphi }_J{\left(s\right)}^{-1}ds$$ where the measure $ds$ on ${\hat{T}}_J$ is given by $$ds=\prod _{j\notin J}\frac{1}{2\pi i}\frac{d{s}_j}{s_j}\text{.}$$ $\text{(}{\hat{T}}_J$ is a translate of a subtorus of $\hat{T},$ 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 ${\hat{T}}_J^1$ and ${\varphi }_J^1$ as above for subsequences $J$ of $(2,3,\dots ,l),$ and put $$I_J^1={\int }_{{\hat{T}}_J^1}{c}^1\left(s\right)(s,t){\varphi }_J^1{\left(s\right)}^{-1}ds$$ 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 $$q_1^{1/2},q_0{q}_1^{1/2},\dots ,q_0^{l-1}{q}_1^{1/2}$$ are less than $1\text{.}$ So we define ${\epsilon }_r$ for $0\le r\le l$ as follows: ${\epsilon }_0=1$ and $$\begin{array}{cc}\text{(5.2.5)}& {\epsilon }_r=\{\begin{array}{cc}1& \text{if}\hspace{0.17em}{q}_0^{r-1}{q}_1^{1/2}<1,\\ 0& \text{otherwise}\end{array}\end{array}$$ for $1\le r\le l\text{.}$

Since $q_0>1,$ it follows that ${\epsilon }_r=1\Rightarrow {\epsilon }_{r-1}=1\text{.}$

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

Let $J=(j_1,\dots ,j_r)$ be a subsequence of $(2,3,\dots ,l)$ and put $J\prime =(1,j_1,\dots ,j_r)\text{.}$ Then $${\epsilon }_r{I}_J+{\epsilon }_{r+1}{I}_{J\prime }=\{\begin{array}{cc}0& \text{if}\hspace{0.17em}{\lambda }_1>0,\\ {\epsilon }_r{I}_J^1& \text{if}\hspace{0.17em}{\lambda }_1=0\text{.}\end{array}$$

		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, $$\begin{array}{cc}\text{(5.2.7)}& \frac{1-s_1^2}{(1+q_0^{-r}{q}_1^{-1/2}{s}_1)(1-q_1^{-1/2}{q}_2^{-1}{s}_1)}·\frac{1+q_0^{r-1}{q}_1^{1/2}{s}_1}{1+q_0^{-1}{q}_1^{1/2}{s}_1}·\prod _{j\notin J\prime }\frac{1-s_1{s}_j^{\pm 1}}{1-q_0^{-1}{s}_1{s}_j^{\pm 1}}\text{.}\end{array}$$ Hence, since $\left|s_j\right|=1$ for $j\notin J\prime ,$ the integrand $$\frac{1}{2\pi i}\frac{c\left(s\right)(s,t)}{{\varphi }_J\left(s\right)}\frac{d{s}_1}{s_1}$$ considered as a function of $s_1,$ has poles at the points $$s_1=-q_0^r{q}_1^{1/2},q_1^{1/2}{q}_2,-q_0{q}_1^{-1/2},q_0{s}_i^{\pm 1}$$ and also at the origin, if ${\lambda }_1=0\text{.}$ The points $q_1^{1/2}{q}_2,$ $-q_0{q}_1^{-1/2},$ $q_0{s}_i^{\pm 1}$ 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 $(1+q_0^{-\rho }{q}_1^{-1/2}{s}_1)$ 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. $\square$

$$\sum _J{\epsilon }_{\left|J\right|}{I}_J=\{\begin{array}{cc}0& \text{if}\hspace{0.17em}t>1,\\ 1& \text{if}\hspace{0.17em}t=1,\end{array}$$ where the sum is over all subsequences $J$ of $(1,2,\dots ,l),$ and $\left|J\right|=\text{Card}\left(J\right)\text{.}$

		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) $$\sum _J{\epsilon }_{\left|J\right|}{I}_J=0\text{.}$$ If ${\lambda }_1=0,$ again from (5.2.6) we have $$\sum _J{\epsilon }_{\left|J\right|}{I}_J=\sum _{J^1}{\epsilon }_{\left|J^1\right|}{I}_{J^1}^1,$$ the sum on the right being over all subsequences $J^1$ of $(2,3,\dots ,l)\text{.}$ The result now follows from the inductive hypothesis. $\square$

Now let $\pi :S\to \Omega$ be the mapping $s\mapsto {\omega }_s$ 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 $\pi \left({\hat{T}}_J\right)$ depends only on the number of elements in $J,$ and not on the particular sequence. Let $${\Omega }_r=\pi \left({\hat{T}}_J\right)\text{if}\hspace{0.17em}\left|J\right|=r\text{.}$$

From the definition of ${\hat{T}}_J,$ the z.s.f. belonging to ${\Omega }_r$ are the ${\omega }_s$ for which $r$ of the numbers $s\left(t_i\right)$ $(1\le i\le l)$ take the values $$-q_1^{1/2},-q_0{q}_1^{1/2},\dots ,-q_0^{r-1}{q}_1^{1/2}$$ (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 $${\varphi }_r\left({\omega }_s\right)={\varphi }_J\left(s\right)\text{if}\hspace{0.17em}\left|J\right|=r\text{.}$$

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 ${\hat{T}}_J$ giving it total mass $1\text{.}$ Now define a measure $d\omega$ on ${\Omega }_r$ as follows: $${\int }_{{\Omega }_r}f\left(\omega \right)d\omega ={\int }_{{\pi }^{-1}\left({\Omega }_r\right)}f\left({\omega }_s\right)ds$$ 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 $$d\mu \left(\omega \right)=\frac{Q\left(q^{-1}\right)}{w_r}\frac{d\omega }{{\varphi }_r\left(\omega \right)}$$ where $w_r=2^{l-r}(l-r)!$ is the order of the normalizer in $W_0$ of any of the sets ${\hat{T}}_J$ such that $\left|J\right|=r\text{.}$

		Proof.
	For $f_1,f_2\in ℒ(G,K),$ define $${⟨{\hat{f}}_1,{\hat{f}}_2⟩}_r={\int }_{{\Omega }_r}{\hat{f}}_1\left(\omega \right)\overline{{\stackrel{ˆ}{f}}_2\left(\omega \right)}d\mu \left(\omega \right)(0\le r\le l)$$ where $d\mu \left(\omega \right)$ is as above. Then we have to show that $$\sum _{r=0}^l{\epsilon }_r{⟨{\hat{f}}_1,{\hat{f}}_2⟩}_r=⟨f_1,f_2⟩\text{.}$$ 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 $(1,2,\dots ,r)\text{.}$ Then exactly as in (5.1) $$\begin{array}{ccc}{⟨{\hat{\chi }}_{t_1},{\hat{\chi }}_{t_2}⟩}_r& =& \frac{Q\left(q^{-1}\right)}{w_r}{\int }_{{\hat{T}}_{J_r}}{\hat{\chi }}_{t_1}\left({\omega }_s\right)\overline{{\stackrel{ˆ}{\chi }}_{t_2}\left({\omega }_s\right)}\frac{ds}{{\varphi }_{J_r}\left(s\right)}\\ & =& \frac{v_{t_1}·\delta {\left(t_1\right)}^{-1/2}}{w_r}\sum _{\genfrac{}{}{0ex}{}{t_2^{\prime }\le t_2}{t_2^{\prime }\in T^{++}}}{\stackrel{\sim }{\chi }}_{t_2}\left(t_2^{\prime }\right)\sum _{w\in W_0}{\int }_{{\hat{T}}_{J_r}}c\left(ws\right)(ws,t_1)(s^{-1},⟨t_2^{\prime }⟩)\frac{ds}{{\varphi }_{J_r}\left(s\right)}\text{.}\end{array}$$ Now it is easily verified that if $s\in {\hat{T}}_{J_r}$ 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 $$\begin{array}{c}\frac{1}{w_r}\sum _{w\in W_0}{\int }_{{\hat{T}}_{J_r}}c\left(ws\right)(ws,t_1)(s^{-1},⟨t_2^{\prime }⟩){\varphi }_{J_r}{\left(s\right)}^{-1}ds\\ =\sum _{\left|J\right|=r}{\int }_{{\hat{T}}_J}c\left(s\right)(s,t_1)(s^{-1},⟨t_2^{\prime }⟩){\varphi }_J{\left(s\right)}^{-1}ds\text{.}\end{array}$$ Consequently $$\sum _{r=0}^l{\epsilon }_r{⟨{\hat{\chi }}_{t_1},{\hat{\chi }}_{t_2}⟩}_r=v_{t_1}·\delta {\left(t_1\right)}^{-1/2}\sum _{\genfrac{}{}{0ex}{}{t_2^{\prime }\le t_2}{t_2^{\prime }\in T^{++}}}{\stackrel{\sim }{\chi }}_{t_2}\left(t_2^{\prime }\right)\sum _J{\epsilon }_{\left|J\right|}{\int }_{{\hat{T}}_J}\frac{c\left(s\right)(s,t_1)(s^{-1},⟨t_2^{\prime }⟩)ds}{{\varphi }_J\left(s\right)}$$ 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, $$\sum _{r=0}^l{\epsilon }_r{⟨{\hat{\chi }}_{t_1},{\hat{\chi }}_{t_2}⟩}_r=\{\begin{array}{cc}0& \text{if}\hspace{0.17em}{t}_1>t_2,\\ ⟨{\chi }_{t_1},{\chi }_{t_1}⟩& \text{if}\hspace{0.17em}{t}_1=t_2\end{array}$$ 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 $$s_0\left(t_i\right)=-q_0^{i-1}{q}_1^{1/2}(1\le i\le l)\text{.}$$

Let $z\in Z^{++},$ $\nu \left(z\right)=t\text{.}$ Then $${\omega }_{s_0}\left(z^{-1}\right)=({\delta }^{-1/2}{s}_0,t)\text{.}$$

		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 $${\int }_G{\left|{\omega }_{s_0}\left(g\right)\right|}^{2}dg=\sum _{t\in T^{++}}{v}_t{({\delta }^{-1/2}{s}_0,t)}^2$$ where $v_t=(KtK:K)=\delta \left(t\right)Q^t\left(q^{-1}\right)$ by (3.2.15). Hence we have to show that the series $$\begin{array}{cc}\text{(5.2.13)}& \sum _{t\in T^{++}}{Q}^t\left(q^{-1}\right){(s_0,t)}^2\end{array}$$ converges if $\left|s_0\left(t_i\right)\right|<1$ 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 $\rho =({\rho }_1,\dots ,{\rho }_r)$ be a sequence of integers such that $$\begin{array}{cc}\text{(5.2.14)}& $ 1\le {\rho }_1$<{\rho }_2<\dots <{\rho }_r\le l\end{array}$$ and consider the $t=\prod t_i^{{\lambda }_i}$ such that $${\lambda }_1={\lambda }_2=\dots ={\lambda }_{{\rho }_1}>{\lambda }_{{\rho }_1+1}=\dots ={\lambda }_{{\rho }_2}>\dots >{\lambda }_{{\rho }_r+1}=\dots ={\lambda }_l=0\text{.}$$ $Q^t\left(q^{-1}\right)$ 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 $ℒ(G,B)\to ℂ\text{.}$ Let ${\Pi }_0$ (resp. $\Pi \text{)}$ be the set of simple roots for ${\Sigma }_0$ (resp. $\Sigma \text{),}$ (so that ${\Pi }_0=\{a_1,\dots ,a_l\},$ $\Pi =\{{\alpha }_0,\dots ,{\alpha }_l\},$ where ${\alpha }_0=1-2e_1,$ ${\alpha }_i=a_i=e_i-e_{i+1}$ $(1\le i\le l-1),$ ${\alpha }_l=a_l=2e_l\text{).}$ Then $ℒ(G,B)$ is generated as $ℂ\text{-algebra}$ by the characteristic functions ${\chi }_i$ of $B{w}_{{\alpha }_i}B$ $(0\le i\le l),$ and the homomorphism ${\hat{\omega }}_{s_0}:ℒ(G,B)\to ℂ$ is given by ${\chi }_i\mapsto q\left(w_{{\alpha }_i}\right)$ $(1\le i\le l),$ ${\chi }_0\mapsto -1\text{.}$

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 $$\begin{array}{ccc}s\left(t_i\right)& =& -q_0^{r-i}{q}_1^{1/2}(1\le i\le r),\\ \left|s\left(t_i\right)\right|& =& 1\text{if}\hspace{0.17em}i>r\text{.}\end{array}$$ 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. $a=-e_i\pm e_j$ 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 $\rho \left({\left|\right|}^s\right)\text{.)}$

If $f$ is any well-behaved function on $𝓀,$ let $\hat{f}$ be its Fourier transform with respect to the additive group structure. Since ${𝓀}^{+}$ is self-dual, $\hat{f}$ is a function on ${𝓀}^{+}\text{.}$ (The Haar measure is normalized so that $\widehat{\stackrel{ˆ}{f}}\left(x\right)=f(-x)\text{.)}$ If $\text{Re}\left(s\right)>0$ define $$\zeta (f,s)={\int }_{k^{\times }}f\left(x\right){\Vert x\Vert }^s{d}^{\times }x,$$ where $\Vert x\Vert$ is the normalized absolute value on $𝓀$ (i.e. $d\left(ax\right)=\Vert a\Vert dx$ where $dx$ is additive Haar measure) and $d^{\times }x$ is a Haar measure on the multiplicative group ${𝓀}^{\times }\text{.}$ Then $\zeta (f,s)$ has a functional equation $$\begin{array}{cc}\text{(5.3.1)}& \zeta (f,s)={\gamma }_k\left(s\right)\zeta (\hat{f},1-s)\end{array}$$ with ${\gamma }_k\left(s\right)$ independent of $f\text{.}$ This equation defines ${\gamma }_k\left(s\right)$ for $\text{Re}\left(S\right)\in (0,1),$ 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 $\hat{f}=f$ and one finds that $${\gamma }_{ℝ}\left(s\right)=\frac{{\pi }^{-s/2}\Gamma (s/2)}{{\pi }^{(s-1)/2}\Gamma ((1-s)/2)}$$ so that $$\begin{array}{cc}{\text{(5.3.2)}}_{ℝ}& {\gamma }_{ℝ}\left(s\right){\gamma }_{ℝ}(-s)=B(\frac{1}{2},\frac{1}{2}s)B(\frac{1}{2},-\frac{1}{2}s)\end{array}$$ where $B$ is Euler's beta-function.

$𝓀=ℂ\text{:}$ take $f\left(x\right)=e^{-2\pi {\left|x\right|}^2}$ $\text{(}\left|x\right|$ the ordinary absolute value on $ℂ\text{:}$ in fact $\Vert x\Vert ={\left|x\right|}^2\text{),}$ then again $\hat{f}=f$ and we have $${\gamma }_{ℂ}\left(s\right)=\frac{{\left(2\pi \right)}^{-s}\Gamma \left(s\right)}{{\left(2\pi \right)}^{s-1}\Gamma (1-s)}$$ so that $$\begin{array}{cc}{\text{(5.3.2)}}_{ℂ}& {\gamma }_{ℂ}\left(s\right){\gamma }_{ℂ}(-s)=-4{\pi }^2/s^2\text{.}\end{array}$$

$k=p\text{-adic field.}$ Taking $f$ to be the characteristic function of the ring of integers of $𝓀,$ one finds that $${\gamma }_k\left(s\right)=d^{s-\frac{1}{2}}\frac{1-q^{s-1}}{1-q^{-s}}$$ where $d$ is the discriminant of $𝓀$ and $q$ is the number of elements in the residue field. Hence in this case we have $$\begin{array}{cc}{\text{(5.3.2)}}_{p\text{-adic}}& {\gamma }_k\left(s\right){\gamma }_k(-s)=d^{-1}\frac{1-q^{-1-s}}{1-q^{-s}}·\frac{1-q^{-1+s}}{1-q^s}\text{.}\end{array}$$

Now let $G$ be the universal Chevalley group $G({\Sigma }_0,𝓀)$ 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 $l=\text{dim}\hspace{0.17em}A\text{.}$ If $ds$ is a Euclidean measure on this space, then (Harish-Chandra [Har1966]) the Plancherel measure $\mu$ is of the form $$d\mu \left({\omega }_s\right)=\kappa \hspace{0.17em}ds/{\left|c\left(s\right)\right|}^2$$ where $\kappa$ is a constant (depending on the choices of $ds$ and of Haar measure on $G\text{)}$ and $$c\left(s\right)=\prod _{a\in {\Sigma }_0^{+}}B(\frac{1}{2},\frac{1}{2}s\left(a^{\vee }\right))\text{if}\hspace{0.17em}𝓀=ℝ$$ and $$c\left(s\right)=\prod _{a\in {\Sigma }_0^{+}}s{\left(a^{\vee }\right)}^{-1}\text{if}\hspace{0.17em}𝓀=ℂ\text{.}$$ Hence, from (5.3.2), we have $$\begin{array}{cc}{\text{(5.3.3)}}_{ℝ,ℂ}& c\left(s\right)c(-s)=\prod _{a\in {\Sigma }_0}{\gamma }_k\left(s\left(a^{\vee }\right)\right)\end{array}$$ 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 $\hat{T}$ 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 $s_0\in S=\text{Hom}(T,{ℂ}^{*}),$ we define $s:A\to ℂ/\left(\frac{2\pi i}{\text{log}\hspace{0.17em}q}\right)\mathbb{Z}$ by the rule $$s_0\left(t_a\right)=q^{-s\left(a^{\vee }\right)}(a\in {\Sigma }_0)$$ 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 $$d\mu \left({\omega }_s\right)=\kappa \hspace{0.17em}ds/{\left|c\left(s\right)\right|}^2$$ where $ds$ is Euclidean measure on the space of pure imaginary $s,$ $\kappa$ is a constant, and $$c\left(s\right)=\prod _{a\in {\Sigma }_0^{+}}\frac{1-q^{-1-s\left(a^{\vee }\right)}}{1-q^{-s\left(a^{\vee }\right)}}$$ (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 $$\begin{array}{cc}{\text{(5.3.3)}}_{p\text{-adic}}& c\left(s\right)c(-s)=\prod _{a\in {\Sigma }_0}{\gamma }_k\left(s\left(a^{\vee }\right)\right),\end{array}$$ apart from a constant factor depending only on $𝓀\text{.}$

So, to sum up:

If $𝓀$ is any local field and $G=G({\Sigma }_0,𝓀)$ 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 $$d\mu \left({\omega }_s\right)=\frac{\kappa ·ds}{\prod _{a\in {\Sigma }_0}{\gamma }_k\left(s\left(a^{\vee }\right)\right)}$$ 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 ${\text{Hom}}_{ℝ}(𝔞,ℂ),$ 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 $$d\mu \left({\omega }_{\lambda }\right)=\kappa \hspace{0.17em}d\lambda /{\left|c\left(\lambda \right)\right|}^2$$ where $\kappa$ is a constant and (see [GKa1962]) $c\left(\lambda \right)$ is a product of beta-functions, namely $$c\left(\lambda \right)=\prod _{b\in {\Sigma }_1^{+}}B(\frac{1}{2}{m}_b,\frac{1}{4}{m}_{b/2}+\frac{1}{2}\lambda \left(b^{\vee }\right))\text{.}$$ Let $$\begin{array}{cc}\text{(5.3.5)}& {\xi }_{ℝ}\left(s\right)={\pi }^{-\frac{1}{2}s}\Gamma \left(\frac{1}{2}s\right)(s\in ℂ)\end{array}$$ which is the local zeta-function $\zeta (f,s)$ for the function $f\left(x\right)=e^{-\pi x^2}\text{.}$ In this notation we have $$\begin{array}{cc}\text{(5.3.6)}& c\left(\lambda \right)=\kappa \prod _{b\in {\Sigma }_1^{+}}\frac{{\zeta }_{ℝ}(\frac{1}{2}{m}_{b/2}+\lambda \left(b^{\vee }\right))}{{\zeta }_{ℝ}(m_b+\frac{1}{2}{m}_{b/2}+\lambda \left(b^{\vee }\right))}\end{array}$$ 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 $q_b$ $(b\in {\Sigma }_1)$ is a power of $q\text{.}$ We shall therefore write $$q_b=q^{m_b}(b\in {\Sigma }_1)$$ 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 $s\in \text{Hom}(T,{ℂ}^{*})$ for the spherical functions by $\lambda \in {\text{Hom}}_{ℝ}(A,ℂ)$ defined by $$s\left(t_a\right)=q^{\lambda \left(a^{\vee }\right)}$$ for all $a\in {\Sigma }_0\text{.}$ $\lambda$ is not uniquely determined by $s,$ but it is unique modulo the lattice $\frac{2\pi i}{\text{log}\hspace{0.17em}q}L,$ where as in (2.5) $L$ is the lattice of all $u\in A^{*}$ such that $u\left(a^{\vee }\right)\in \mathbb{Z}$ for all $a\in {\Sigma }_0\text{.}$ In particular, if $s\in \hat{T}$ then $\lambda$ is pure imaginary. Writing $c\left(\lambda \right)$ in place of $c\left(s\right),$ we have from (4.1) $$c\left(\lambda \right)=\prod _{b\in {\Sigma }_1^{+}}\frac{1-q^{-(\frac{1}{2}{m}_{b/2}+m_b+\lambda \left(b^{\vee }\right))}}{1-q^{(-\frac{1}{2}{m}_{b/2}+\lambda \left(b^{\vee }\right))}}$$ Now let $$\begin{array}{cc}\text{(5.3.7)}& {\zeta }_k\left(s\right)={(1-q^{-s})}^{-1}(s\in ℂ)\end{array}$$ which is the local zeta-function $\zeta (f,s)$ 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 $$d\mu \left({\omega }_{\lambda }\right)=\kappa ·d\lambda /{\left|c\left(\lambda \right)\right|}^2$$ where $\kappa$ is a constant and $$c\left(\lambda \right)=\prod _{b\in {\Sigma }_1^{+}}\frac{{\zeta }_k(\frac{1}{2}{m}_{b/2}+\lambda \left(b^{\vee }\right))}{{\zeta }_k(\frac{1}{2}{m}_{b/2}+m_b+\lambda \left(b^{\vee }\right))}$$ 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.

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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