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

Last update: 13 February 2014

## Chapter I: Basic properties of spherical functions

### The algebra $ℒ\left(G,K\right)$

Let $G$ be a locally compact Hausdorff topological group with a countable basis of open sets. Let $𝒞\left(G\right)$ denote the vector space of all continuous complex-valued functions on $G,$ and let $ℒ\left(G\right)$ be the subspace of $𝒞\left(G\right)$ consisting of the functions with compact support.

Let $K$ be a compact subgroup of $G$ and let $𝒞\left(G,K\right)$ be the subspace of $𝒞\left(G\right)$ consisting of the functions $f$ which are bi-invariant with respect to $K\text{:}$ $f\left(gk\right)=f\left(gk\right)=f\left(g\right)$ for all $g\in G$ and $K\in K\text{.}$ Also let $ℒ(G,K)=𝒞(G,K) ∩ℒ(G)$ so that the functions in $ℒ\left(G,K\right)$ are continuous, compactly-supported and bi-invariant with respect to $K\text{.}$

Fix a left Haar measure $dg$ on $G,$ and define a multiplication (convolution) on $ℒ\left(G,K\right)$ as follows: $(f1*f2)(g0) =∫Gf1(g0g) f2(g-1)dg$ for any ${f}_{1},{f}_{2}\in ℒ\left(G,K\right)\text{.}$ In this way $ℒ\left(G,K\right)$ becomes an associative $ℂ\text{-algebra.}$

For the time being we shall not assume that $ℒ\left(G,K\right)$ is commutative, nor that $G$ is unimodular.

Remarks.

 (i) If $f\in ℒ\left(G,K\right)$ and $\varphi \in 𝒞\left(G,K\right)$ then $f*\varphi \in 𝒞\left(G,K\right)\text{.}$ (ii) We shall be particularly concerned with the case where $K$ is an open subgroup of $G\text{.}$ In that case we shall always normalize the Haar measure on $G$ so that ${\int }_{K}dg=1\text{.}$

Suppose that $K$ is open in $G\text{.}$ For each double coset $\xi =KgK$ let ${\chi }_{\xi }$ be the characteristic function of $\xi \text{.}$ Then the ${\chi }_{\xi }$ form a basis of $ℒ\left(G,K\right)$ as a vector space over $ℂ,$ and the algebra $ℒ\left(G,K\right)$ has an identity element, namely ${\chi }_{K}\text{.}$ Proof. Since $K$ is open and compact, so is each double coset $\xi =KgK,$ and therefore ${\chi }_{\xi }\in ℒ\left(G,K\right)\text{.}$ Conversely, any $f\in ℒ\left(G,K\right)$ has compact support, which must therefore consist of a finite number of double cosets $\xi ,$ and hence $f$ is a linear combination of the characteristic functions ${\chi }_{\xi }\text{.}$ Clearly these are linearly independent, which proves the first assertion. Next, consider ${\chi }_{K}*{\chi }_{\xi }\text{.}$ We have $(χK*χξ)(g0) =∫GχK(g0g) χξ(g-1)dg$ and the integrand vanishes unless ${g}_{0}g\in K$ and ${g}^{-1}\in \xi ,$ which implies that ${g}_{0}\in \xi \text{.}$ It follows that ${\chi }_{K}*{\chi }_{\xi }$ is a scalar multiple of ${\chi }_{\xi },$ and one sees immediately that the scalar multiple is $1,$ by virtue of our normalization of the Haar measure (Remark (ii) above). Likewise ${\chi }_{\xi }*{\chi }_{K}={\chi }_{\xi }\text{.}$ $\square$

Now let $dk$ be a Haar measure on the compact group $K,$ normalized so that ${\int }_{K}dk=1\text{.}$ For any $\varphi \in ℒ\left(G\right)$ we define ${\varphi }^{♮}\in ℒ\left(G,K\right)$ by $ϕ♮(g)=∫K ∫Kϕ(k1gk2) dk1dk2,$ so that $\varphi \to {\varphi }^{♮}$ is a projection of $ℒ\left(G\right)$ onto $ℒ\left(G,K\right)\text{.}$

Let $\omega \in 𝒞\left(G,K\right)$ be such that $∫Gf(g)ω (g-1)dg=0$ for all $f\in ℒ\left(G,K\right)\text{.}$ Then $\omega =0\text{.}$ Proof. Let $\varphi \in ℒ\left(G\right)\text{.}$ Then $∫Gϕ(g)ω (g-1)dg = ∫Gϕ(k1gk2) ω(g-1)dg for any k1,k2∈K, = ∫G ( ∫K ∫Kϕ (k1gk2)d k1dk2 ) ω(g-1)dg = ∫Gϕ♮(g) ω(g-1)dg = 0by hypothesis.$ Since $\omega$ is continuous, it follows that $\omega =0\text{.}$ $\square$

### Spherical measures and spherical functions

For each $g\in ℒ\left(G\right)$ the norm $‖f‖∞= supg∈G |f(g)|$ is defined, since $f$ is continuous and compactly supported. If $C$ is any compact subset of $G,$ let ${ℒ}_{C}\left(G\right)$ be the subspace of $ℒ\left(G\right)$ consisting of the functions which vanish outside $C\text{.}$ A (complex-valued) measure $\mu$ on $G$ is by definition a linear functional $\mu :ℒ\left(G\right)\to ℂ$ whose restriction to each ${ℒ}_{C}\left(G\right)$ is continuous with respect to the topology defined by the norm ${‖ ‖}_{\infty }\text{.}$ Equivalently, for each compact subset $C$ of $G$ there exists a positive real number ${m}_{C}$ such that $|μ(f)|≤mC ‖f‖∞$ for all $f\in {ℒ}_{C}\left(G\right)\text{.}$ As usual, we write ${\int }_{G}f\left(g\right)d\mu \left(g\right)$ in place of $\mu \left(f\right),$ whenever convenient.

A non-zero measure $\mu$ on $G$ is said to be spherical (with respect to the compact subgroup $K\text{)}$ if

 (1) $\mu$ is bi-invariant with respect to $K,$ in other words $dμ(kg)=dμ(gk)=dμ(g)$ for all $k\in K\text{;}$ (2) $\mu :ℒ\left(G,K\right)\to ℂ$ is a ring homomorphism, i.e. $μ(f1*f2)= μ(f1)·μ(f2)$ for all ${f}_{1},{f}_{2}\in ℒ\left(G,K\right)\text{.}$

For example, $f↦{\int }_{G}f\left({g}^{-1}\right)dg$ is a spherical measure.

If $\mu$ is spherical and $f\in ℒ\left(G\right),$ then $\mu \left(f\right)=\mu \left({f}^{♮}\right)\text{.}$

Indeed, this is true if $\mu$ is merely bi-invariant. with respect to $K\text{.}$ It shows that a spherical measure is uniquely determined by its restriction to $ℒ\left(G,K\right)\text{.}$

Suppose that $K$ is open in $G\text{.}$ Then every non-zero $ℂ\text{-algebra}$ homomorphism $\nu :ℒ\left(G,K\right)\to ℂ$ determines a spherical measure on $G\text{.}$ Proof. For all $f\in ℒ\left(G\right)$ define $μ(f)=ν(f♮).$ We have to check that $\mu$ satisfies the continuity condition above. Let $C$ be a compact subset of $G\text{.}$ Since the double cosets $KgK$ are open and mutually disjoint, $C$ intersects only a finite number of them, say $K{g}_{i}K$ $\left(1\le i\le n\right)\text{.}$ Let ${\chi }_{i}$ be the characteristic function of $K{g}_{i}K\text{.}$ If $f\in ℒ\left(G\right)$ has support contained in $C,$ then ${f}^{♮}$ vanishes outside $KCK=\bigcup _{i=1}^{n}K{g}_{i}K,$ and hence ${f}^{♮}$ is a linear combination of the ${\chi }_{i},$ say $f♮=∑i=1n λiχi(λi∈ℂ)$ Hence $‖f♮‖∞= maxi=1n |λi|. (1)$ Next, we have $‖f♮‖∞≤ ‖f‖∞ (2)$ because for any $g\in G,$ $|f♮(g)|≤ ∫K∫K |f(k1gk2)| dk1dk2≤ ‖f‖∞.$ Hence $|μ(f)|= |ν(f♮)| = ∑i=1nλi ν(χi) ≤ ∑i=1n |λi|· |ν(χi)| ≤ mC‖f♮‖∞ ≤mC‖f‖∞$ by (1) and (2), where ${m}_{C}=\sum |\nu \left({\chi }_{i}\right)|$ depends only on $C\text{.}$ Hence $\mu$ is a measure on $G,$ hence is a spherical measure. $\square$

Let $\mu$ be a spherical measure on $G\text{.}$ Then there exists a unique $\omega \in 𝒞\left(G,K\right)$ such that ${d}_{\mu }\left(g\right)=\omega \left({g}^{-1}\right)dg,$ i.e. such that $μ(f)=(f*ω)(1)$ for all $f\in ℒ\left(G\right)\text{.}$ Proof. Since $\mu \ne 0$ there exists ${f}_{0}\in ℒ\left(G,K\right)$ such that $\mu \left({f}_{0}\right)\ne 0\text{.}$ Let $f\in ℒ\left(G,K\right)\text{.}$ Then $μ(f*f0)=∫G f(g1) ( ∫Gf0 (g1-1g) dμ(g) ) dg1.$ Since also $\mu \left(f*{f}_{0}\right)=\mu \left(f\right)\mu \left({f}_{0}\right)$ it follows that $μ(f)=∫Gf(g) ω(g1-1)dg1,$ where $ω(g1)=μ (f0)-1 ∫Gf0(g1g) dμ(g). (1.2.4)$ Clearly the function $\omega$ so defined is continuous and bi-invariant with respect to $K,$ i.e. $\omega \in 𝒞\left(G,K\right)\text{.}$ If now $\varphi \in ℒ\left(G\right)$ then by (1.2.1) $μ(ϕ)=μ (ϕ♮) = ∫Gϕ♮(g) ω(g-1)dg = ∫Gϕ(g)ω (g-1)dg$ by the bi-invariance of $\omega \text{.}$ Hence $d\mu \left(g\right)=\omega \left({g}^{-1}\right)dg,$ and since $\omega$ is continuous it is the only function with this property. $\square$

A function $\omega \in 𝒞\left(G,K\right)$ is a zonal spherical function (z.s.f. for short) on $G$ relative to $K$ if $\omega \left({g}^{-1}\right)dg$ is a spherical measure. The following proposition gives other equivalent definitions:

The following are equivalent, for a complex-valued function $\omega$ on $G\text{:}$

 (1) $\omega$ is a z.s.f. on $G$ relative to $K\text{;}$ (2) $\omega \in 𝒞\left(G,K\right)\text{;}$ $\omega \left(1\right)=1\text{;}$ and $f*\omega ={\lambda }_{f}\omega$ for all $f\in ℒ\left(G,K\right),$ where ${\lambda }_{f}$ is a scalar (depending on $f\text{);}$ (3) $\omega$ is continuous, not identically zero, and $ω(g1)ω(g2) =∫Kω(g1kg2) dk$ for all ${g}_{1},{g}_{2}\in G\text{.}$ Proof. (1) $⇒$ (2). From the formula (1.2.4) it is clear that $\omega \in 𝒞\left(G,K\right)$ and $\omega \left(1\right)=1\text{.}$ Also, for all $f,{f}_{0}\in ℒ\left(G,K\right)$ we have $∫Gf0(g) (f*ω)(g-1) dg = (f0*f*ω)(1) = μ(f0*f)= μ(f)μ(f0)$ where $\mu$ is the associated spherical measure. Hence, putting $h=f*\omega -\mu \left(f\right)\omega ,$ we have $∫Gf0(g)h (g-1)dg=0$ for all ${f}_{0}\in ℒ\left(G,K\right),$ and therefore $h=0$ by (1.1.2). (2) $⇒$ (3). We remark first that if $f\in ℒ\left(G,K\right)$ then $∫Gf(g)ω(g-1) dg=(f*ω)(1)= λfω(1)=λf.$ Next, we have $λfω(g′)= (f*ω)(g′) = ∫Gf(g)ω (g-1g′)dg = ∫Gf(g) ( ∫Kω(g-1kg′) dk ) dg.$ Hence, if we put $ωg′(g)= ∫Kω(gkg′)dk$ then ${\omega }_{g\prime }\in 𝒞\left(G,K\right)$ and $∫Gf(g) ωg′(g-1) dg = λfω(g′) = ∫Gf(g)ω (g-1)ω (g′)dg.$ Since this holds for all $f\in ℒ\left(G,K\right),$ it follows from (1.1.2) that $ω(g-1)ω(g′) =ωg′(g-1)= ∫Kω(g-1kg′) dk$ for all $g,g\prime \in G\text{.}$ (3) $⇒$ (1). Let $\mu \left(f\right)={\int }_{G}f\left(g\right)\omega \left({g}^{-1}\right)dg\text{.}$ We have to show that $\mu$ is a spherical measure. It is immediate from (3) that $\omega ,$ and therefore $\mu ,$ is bi-invariant with respect to $K\text{.}$ If ${f}_{1},{f}_{2}\in ℒ\left(G,K\right)$ then $μ(f1*f2) = ∫G ( ∫Gf1(g1) f2(g1-1g) dg1 ) ω(g-1)dg = ∫G∫Gf1(g1) f2(g2)ω (g2-1g1-1) dg1dg2 = ∫G∫Gf1(g1) f2(g2) ( ∫K ω(g2-1kg1-1) dk ) dg1dg2 = ∫G∫Gf1(g1) f2(g2) ω(g1-1) ω(g2-1) dg1dg2 = μ(f1)· μ(f2).$ $\square$

We shall now forget about spherical measures and work only with the spherical functions, for which we may take (2) or (3) of (1.2.5) as equivalent definitions. In terms of spherical functions, (1.2.2) takes the form

If $K$ is open in $G,$ there is a one-one correspondence between the set of z.s.f. on $G$ relative to $K$ and the set of non-zero $ℂ\text{-algebra}$ homomorphisms $ℒ\left(G,K\right)\to ℂ\text{:}$ namely, the zonal spherical function $\omega$ corresponds to the homomorphism $\stackrel{ˆ}{\omega }:ℒ\left(G,K\right)\to ℂ$ defined by $ωˆ(f)=∫G f(g)ω(g-1) dg.$

If $\omega$ is a z.s.f. then so is the function $g↦\omega \left({g}^{-1}\right)\text{.}$ This is clear from (3) of (1.2.5) [since $K,$ being compact, is unimodular]. Also the constant function $1$ on $G$ is a z.s.f. More generally, any continuous homomorphism $\omega :G\to {ℂ}^{*}$ whose kernel contains $K$ is a z.s.f. relative to $K,$ again by (1.2.5)(3).

### Induction of spherical functions

Let ${G}_{0}$ be a closed subgroup of $G$ such that $G=K{G}_{0},$ and let ${\omega }_{0}$ be a zonal spherical function on ${G}_{0}$ relative to ${K}_{0}=K\cap {G}_{0}\text{.}$ Define a function ${\varphi }_{0}$ on $G$ by the rule $ϕ0(kg0)= ω0(g0) (k∈K,g0∈G0).$ Then the function $\omega$ on $G$ defined by $ω(g)=∫K ϕ0(gk)dk$ is a z.s.f. on $G$ relative to $K\text{.}$ Proof. We remark first that ${\varphi }_{0}$ is well defined, for if $k{g}_{0}=k\prime {g}_{0}^{\prime }$ $\left(k\prime \in K,{g}_{0}^{\prime }\in {G}_{0}\right),$ then ${k\prime }^{-1}k={g}_{0}^{\prime }{g}^{-1}\in {K}_{0},$ so that ${g}_{0}^{\prime }\in {K}_{0}{g}_{0}$ and therefore ${\omega }_{0}\left({g}_{0}^{\prime }\right)={\omega }_{0}\left({g}_{0}\right)\text{.}$ Clearly $\omega \in 𝒞\left(G,K\right)$ and $\omega \left(1\right)=1\text{.}$ Hence it is enough to show that $f*\omega$ to is a scalar multiple of $\omega ,$ for each $f\in ℒ\left(G,K\right)\text{.}$ From the definition of ${\varphi }_{0}$ we have ${\varphi }_{0}\left(kg\right)={\varphi }_{0}\left(g\right)$ for all $k\in K$ and $g\in G\text{.}$ Hence, if $g\in G$ and ${g}_{0}\in {G}_{0},$ $∫K0ϕ0 (gk0g0)d k0=∫K0 ϕ0(g0′k0g0) dk0$ if $g=k{g}_{0}^{\prime }$ $\left(k\in K,{g}_{0}^{\prime }\in {G}_{0}\right),$ $= ∫K0ω0 (g0′k0g0) dk0 = ω0(g0′) ω0(g0) by (1.2.5)$ so that $∫K0ϕ0 (gk0g0)dk= ϕ0(g)ω0 (g0). (1)$ Hence, if $f\in ℒ\left(G,K\right)$ and $g=k{g}_{0},$ $(f*ϕ0)(g) = ∫Gf(kg0g1) ϕ0(g1-1) dg1 = ∫Gf(g2) ϕ0(g2-1g0) dg2 (g2=g0g1) = ∫Gf(g2) ( ∫K0ϕ0 (g2-1k0g0) dk0 ) dg2 = ∫Gf(g2)ϕ0 (g2-1)ω0 (g0)dg2 by (1) above = λfϕ0(g) (2)$ where ${\lambda }_{f}=\left(f*\varphi \right)\left(1\right)\text{.}$ So, finally, $(f*ω)(g) = ∫Gf(gg1) ( ∫Kϕ0 (g1-1k) dk ) dg1 = ∫Gf(g2) ( ∫Kϕ0 (g2-1gk) dk ) dg2 (g2=gg1) = ∫K(f*ϕ0) (gk)dk = λf∫Kϕ0 (gk)dkby (2) = λfω(g).$ $\square$

The spherical function $\omega$ is said to be induced by ${\omega }_{0}\text{.}$

With the hypotheses of (1.3.1) the function $g⟼∫Kϕ0 (g-1k)dk$ is a zonal spherical function on $G\text{.}$ Proof. (1.3.1), (1.2.7). $\square$

(Transitivity of induction). Let ${G}_{0}\subset {G}_{1}$ be closed subgroups of $G$ such that $G=K{G}_{0}\text{.}$ Let ${\omega }_{0}$ be a z.s.f. on ${G}_{0}$ relative to ${K}_{0}=K\cap {G}_{0}$ and let ${\omega }_{1}$ be the z.s.f. on ${G}_{1}$ relative to ${K}_{1}=K\cap {G}_{1}$ induced by ${\omega }_{0}\text{.}$ Then the z.s.f. on $G$ relative to $K$ induced by ${\omega }_{0}$ and ${\omega }_{1}$ are equal. Proof. As in (1.3.1) define ${\varphi }_{0}$ by the rule $ϕ0(kg0)= ω0(g0) (k∈K,g0∈G0)$ and ${\varphi }_{1}$ by the rule $ϕ1(kg1) = ω1(g1) (k∈K,g1∈G1) = ∫K1ϕ0 (g1k1) dk1.$ Let $\omega$ be the z.s.f. on $G$ relative to $K$ induced by ${\omega }_{1},$ so that $ω(g)=∫K ϕ1(gk)dk.$ Writing $gk=k\prime {g}_{1}\left(k\right)$ where $k\prime \in K$ and ${g}_{1}\left(k\right)\in {G}_{1},$ we have $ω(g) = ∫Kω1(g1(k)) dk = ∫K ( ∫K1ϕ0 (g1(k)k1) dk1 ) dk = ∫K1 ( ∫K ϕ0(g1(kk1)) dk ) dk1(since g1(kk1)= g1(k)k1) = ∫Kϕ0(g1(k)) dk = ∫Kϕ0(gk)dk.$ Hence $\omega$ is also the spherical function on $G$ induced by ${\omega }_{0}\text{.}$ $\square$

### Positive definite spherical functions and representations of class 1

A function $\varphi :G\to ℂ$ is said to be positive definite if $\varphi$ is continuous, not identically zero, and if $∑i,j=1nλi λj‾ϕ (gi-1gj)$ is real and $\ge 0$ for all choices of ${g}_{1},\dots ,{g}_{n}\in G$ and ${\lambda }_{1},\dots ,{\lambda }_{n}\in ℂ\text{.}$

If $\varphi$ is positive definite, then $\varphi \left(1\right)$ is real and $>0\text{;}$ $|\varphi \left(g\right)|\le \varphi \left(1\right)$ for all $g\in G\text{;}$ and $\varphi \left({g}^{-1}\right)=\stackrel{‾}{\varphi \left(g\right)}\text{.}$ Proof. Taking $n=1$ in the definition we see that $\varphi \left(1\right)$ is real and $\ge 0\text{.}$ Taking $n=2,$ the matrix $\left(\begin{array}{cc}\varphi \left(1\right)& \varphi \left(g\right)\\ \varphi \left({g}^{-1}\right)& \varphi \left(1\right)\end{array}\right)$ must be Hermitian and positive, whence $\varphi \left({g}^{-1}\right)=\stackrel{‾}{\varphi \left(g\right)}$ and $|\varphi \left(g\right)|\le \varphi \left(1\right)\text{.}$ Since $\varphi$ is not identically zero, it follows that $\varphi \left(1\right)>0\text{.}$ $\square$

Let $ℋ$ be a separable complex Hilbert space, $B\left(ℋ\right)$ the set of all continuous linear mappings $ℋ\to ℋ\text{.}$ A representation $T$ of $G$ on $ℋ$ is a mapping $g↦{T}_{g}$ of $G$ into $B\left(ℋ\right)$ such that

 (1) ${T}_{{g}_{1}{g}_{2}}={T}_{{g}_{1}}\circ {T}_{{g}_{2}}$ for all ${g}_{1},{g}_{2}\in G\text{;}$ ${T}_{1}=1\text{.}$ (2) For each $h\in ℋ$ the mapping $g↦{T}_{g}\left(h\right)$ is a continuous map of $G$ into $ℋ\text{.}$
The representation $T$ is irreducible if no closed subspace of $ℋ,$ other than $\left\{0\right\}$ and $ℋ,$ is invariant under ${T}_{g}$ for all $g\in G\text{.}$ $T$ is unitary if each ${T}_{g}$ is unitary.

 (i) Let $T$ be a unitary representation of $G$ on $ℋ$ and let $h\in ℋ,$ $h\ne 0\text{.}$ Then the function $\varphi$ defined by $ϕ(g)= ⟨h,Tg(h)⟩$ (where $⟨,⟩$ is the scalar product on $ℋ\text{)}$ is positive definite. (ii) Conversely, let $\varphi$ be a positive definite function on $G\text{.}$ Then there exists a unitary representation ${T}^{\varphi }$ of $G$ on a Hilbert space $ℋ$ and a non-zero vector $h\in ℋ$ such that $ϕ(g)= ⟨h,Tgϕ(h)⟩$ for all $g\in G\text{.}$ Proof. (i) is a straightforward verification: clearly $\varphi$ is continuous and $\varphi \left(1\right)\ne 0\text{;}$ also $∑i,j=1n λiλj‾ ϕ(gi-1gj) = ∑i,jλi λj‾ ⟨ h,Tgi-1 Tgj(h) ⟩ = ⟨ ∑iλi Tgi(h), ∑iλi Tgi(h) ⟩ = ‖∑λiTgi(h)‖2$ which is real and $\ge 0\text{.}$ (ii) We shall only sketch the proof. For more details see Helgason [Hel1962], p. 413. Let ${V}_{\varphi }$ be the subspace of $𝒞\left(G\right)$ spanned by all the left-translates of $\varphi ,$ that is to say by all functions ${}^{g}\varphi$ $\left(g\in G\right)$ defined by ${}^{g}\varphi \left(g\prime \right)=\varphi \left({g}^{-1}g\prime \right)\text{.}$ Define a scalar product on ${V}_{\varphi }$ by $⟨g1ϕ,g2ϕ⟩ =ϕ(g1-1g2)$ extended by linearity. The set $N=\left\{f\in {V}_{\varphi }:⟨f,f⟩=0\right\}$ is a subspace of ${V}_{\varphi }\text{;}$ the quotient ${V}_{\varphi }/N$ inherits a scalar product from ${V}_{\varphi }$ and hence can be completed to a Hilbert space $ℋ\text{.}$ Each $g\in G$ defines an endomorphism $f↦{}^{g}f$ of ${V}_{\varphi }$ which on passing to the quotient induces a unitary operator ${T}_{g}^{\varphi }:ℋ\to ℋ\text{.}$ One checks that ${T}_{\varphi }$ is a representation of $G\text{.}$ Finally, if $h\in ℋ$ is the image of $\varphi \in {V}_{\varphi },$ then $⟨h,Tgϕ(h)⟩ =⟨ϕ,gϕ⟩= ϕ(g).$ $\square$

Suppose $\varphi$ is a positive definite spherical function on $G$ relative to $K\text{.}$ Then there exists $h\in ℋ$ such that $ϕ(g)= ⟨h,Tgϕ(h)⟩$ for all $g\in G$ and such that ${T}_{k}^{\varphi }\left(h\right)=h$ for all $k\in K\text{.}$

In fact, much more is true if we assume that $G$ is unimodular and $ℒ\left(G,K\right)$ a commutative ring. A representation $T$ of $G$ on a Hilbert space $ℋ$ is said to be of class $1$ (relative to the compact subgroup $K\text{)}$ if $T$ is irreducible and unitary and if there exists a non-zero vector $h\in ℋ$ such that ${T}_{k}\left(h\right)=h$ for all $k\in K\text{:}$ that is to say, if the restriction of $T$ to $K$ contains the identity representation of $K\text{.}$ Then there is the following result, which we shall not prove here:

Assume that $G$ is unimodular and $ℒ\left(G,K\right)$ commutative.

 (i) Let $\omega$ be a positive definite z.s.f. on $G\text{.}$ Then the representation ${T}^{\omega }$ constructed from $\omega$ as in (1.4.2) is of class $1\text{.}$ (ii) Conversely, if $T$ is a representation of $G$ of class $1$ and if $h\in ℋ$ is a unit vector left fixed by ${T}_{k}$ for each $k\in K,$ then $h$ is unique up to a scalar multiple. Choosing $h$ so that $‖h‖=1,$ the function $ω(g)= ⟨h,Tg(h)⟩$ is a positive definite spherical function, and finally $T$ is equivalent to ${T}^{\omega }\text{.}$

We return to the situation of (1.3): ${G}_{0}$ is a closed subgroup of $G$ and $G=K{G}_{0}\text{.}$ Assume also that $K$ is open in $G,$ so that ${K}_{0}=K\cap {G}_{0}$ is an open compact subgroup of ${G}_{0}\text{.}$ Let $d\xi$ be left Haar measure on ${G}_{0},$ normalized so that ${\int }_{{K}_{0}}d\xi =1\text{.}$ The right Haar measure on ${G}_{0},$ normalized in the same way, will be of the form $\Delta \left(\xi \right)d\xi ,$ where $\Delta$ is a continuous homomorphism of ${G}_{0}$ into the multiplicative group ${ℝ}_{+}^{*}$ of positive real numbers. $\Delta$ is trivial on ${K}_{0},$ because ${K}_{0}$ is compact.

If $dk$ is normalized Haar measure on $K$ then we have $∫Gf(g)dg=∫K ( ∫G0f(kξ) Δ(ξ)dξ ) dk (1.4.5)$ for any integrable function $f$ on $G\text{.}$

Let $\stackrel{‾}{K}=K/{K}_{0}$ and let ${g}_{1}$ be a fixed element of $G\text{.}$ For $k\in K,$ write $g1-1k=k′η (k′∈K,η∈G0).$ Then the cosets $\stackrel{‾}{k}\prime =k\prime {K}_{0}$ and ${K}_{0}\eta {K}_{0}$ are uniquely determined by ${g}_{1}$ and $\stackrel{‾}{k}=k{K}_{0}\text{.}$ If $d\stackrel{‾}{k}$ is a relatively invariant measure on $\stackrel{‾}{K},$ we have $dk‾=Δ(η) dk‾′.$ Proof. Let $g=k\xi$ $\left(k\in K,\xi \in {G}_{0}\right)\text{.}$ Then $g1-1g=g1-1 kξ=k′ηξ$ so that $d\left({g}_{1}^{-1}g\right)=dk\prime ·\Delta \left(\eta \xi \right)d\left(\eta \xi \right)=dk\prime ·\Delta \left(\eta \right)\Delta \left(\xi \right)d\xi$ by (1.4.4); but also $d\left({g}_{1}^{-1}g\right)=dg=dk·\Delta \left(\xi \right)d\xi \text{.}$ Hence the result. $\square$

Let ${\omega }_{0}$ be a z.s.f. on ${G}_{0}$ relative to ${K}_{0},$ and $\omega$ the induced z.s.f. on $G$ relative to $K$ (1.3.1). Suppose that ${\Delta }^{\frac{1}{2}}{\omega }_{0}$ (i.e. $\xi ↦\Delta {\left(\xi \right)}^{\frac{1}{2}}{\omega }_{0}\left(\xi \right)\text{)}$ is positive definite. Then $\omega$ is positive definite. Proof.

By (1.4.2) and (1.4.3) there exists a unitary representation $T$ of $G$ on a Hilbert space ${ℋ}_{0}$ and a vector ${h}_{0}\in {ℋ}_{0}$ such that $Δ(ξ)12 ω0(ξ)= ⟨h0,Tξh0⟩$ for all $\xi \in {G}_{0},$ and such that ${T}_{{k}_{0}}\left({h}_{0}\right)={h}_{0}$ for all ${k}_{0}\in {K}_{0}\text{.}$

Let $ℋ$ be the Hilbert space of all classes of measurable functions $f:G\to {ℋ}_{0}$ which satisfy

 (i) $f\left(g\xi \right)=\Delta {\left(\xi \right)}^{-\frac{1}{2}}{T}_{\xi }\left(f\left(g\right)\right)$ for all $g\in G$ and $\xi \in {G}_{0}\text{;}$ (ii) ${\int }_{K}{‖f\left(k\right)‖}^{2}dk<\infty \text{.}$
The inner product on $ℋ$ is $⟨f1,f2⟩= ∫K⟨f1(k),f2(k)⟩ dk.$ Let ${g}_{1}\in G\text{.}$ For each $f\in ℋ$ define a function ${U}_{{g}_{1}}f$ on $G$ by the rule $(Ug1f)(g)= f(g1-1g).$ Clearly ${U}_{{g}_{1}}f$ satisfies (i) above. Also we have $‖Ug1f‖2= ∫K‖(Ug1f)(k)‖2 dk=∫K ‖f(g1-1k)‖2 dk.$ Now from (i) and the fact that $\Delta$ is trivial on ${K}_{0}$ it follows that (for fixed ${g}_{1}\text{)}$ ${‖f\left({g}_{1}^{-1}k\right)‖}^{2}$ depends only on the image of $k$ in $\stackrel{‾}{K}=K/{K}_{0}\text{.}$ Hence $‖Ug1f‖2 = ∫K‾ ‖f(g1-1k)‖2 dk‾ = ∫K‾ ‖f(k′η)‖2 Δ(η)dk‾′ by (1.4.6) = ∫K‾ ‖Tηf(k′)‖2 dk‾′ = ‖f‖2$ because $T$ is unitary. Hence ${U}_{{g}_{1}}f\in ℋ$ and the operator ${U}_{{g}_{1}}$ is unitary. (One can go on to show without any difficulty that $U$ is a unitary representation of $G,$ but we shall not need to use this fact.)

Let $h\in ℋ$ be the function $G\to {ℋ}_{0}$ defined by $h(kξ)=Δ (ξ)-12 Tξ(h0) (k∈K,ξ∈G)$ $\text{(}h$ is well defined because $T/{K}_{0}$ fixes ${h}_{0}\text{.)}$ Now calculate $⟨h,{U}_{g}^{-1}h⟩\text{:}$ $⟨h,Ug-1h⟩ =∫K ⟨h(k),h(gk)⟩dk:$ putting $gk=k\prime \xi$ $\left(k\prime \in K,\xi \in {G}_{0}\right)$ this becomes $∫K ⟨ h0,Δ(ξ)-12 Tξ(h0) ⟩ dk = ∫Kω0(ξ)dk= ω(g).$ Hence $\omega \left(g\right)=⟨h,{U}_{g}^{-1}h⟩\text{;}$ since ${U}_{g}$ is unitary, the calculation in (1.4.2) shows that $\omega$ is positive definite.

$\square$

Let $\omega$ be a z.s.f. on $G$ relative to $K,$ and suppose that $\omega$ is square-integrable. Then $\omega$ is positive definite. Proof. $G$ acts on ${L}^{2}\left(G\right)$ by left-translations: $\left({T}_{{g}_{0}}f\right)\left(g\right)=f\left({g}_{0}^{-1}g\right)\text{.}$ Calculate the scalar product of $\omega$ and ${T}_{{g}_{0}}\omega \text{:}$ $⟨ω,Tg0ω⟩ = ∫Gω(g) ω(g0-1g)‾ dg = ∫Gω(g0g) ω(g)‾dg = ∫G ( ∫Kω(g0kg) dk ) ω(g)‾dg = ∫Gω(g0)ω (g)ω(g)‾ dgby (1.2.5) = ω(g0)‖ω‖2.$ Since ${T}_{{g}_{0}}$ is unitary (by the left-invariance of the measure), it follows as in (1.4.2) that $\omega$ is positive definite. $\square$

### Fourier analysis

In this section we shall assume that $G$ is unimodular and that the algebra $ℒ\left(G,K\right)$ is commutative. Let ${ℒ}^{1}\left(G,K\right)$ (resp. ${ℒ}^{2}\left(G,K\right)\text{)}$ denote the space of integrable (resp. square-integrable) functions on $G$ which are bi-invariant with respect to $K\text{.}$

Let ${\Omega }^{+}$ denote the set of all positive definite z.s.f. on $G$ relative to $K\text{.}$ For each $f\in {ℒ}^{1}\left(G,K\right),$ the Fourier transform of $f$ is the function on ${\Omega }^{+}$ defined by $fˆ(ω)=∫Gf (g)ω(g-1) dg.$ The set ${\Omega }^{+}$ is given the weakest topology for which all the Fourier transforms of functions in ${ℒ}^{1}\left(G,K\right)$ are continuous. Equivalently, the topology on ${\Omega }^{+}$ is the compact-open topology: the sets $N\left(C,U\right)=\left\{\omega \in {\Omega }^{+}:\omega \left(C\right)\subset U\right\},$ where $C\subset G$ is compact and $U\subset ℂ$ is open, form a basis for the topology of ${\Omega }^{+}\text{.}$

With this topology, ${\Omega }^{+}$ is Hausdorff and locally compact, and the Fourier transforms of functions in ${ℒ}^{1}\left(G,K\right)$ tend to $0$ at infinity. We have $Ω+ is compact ⇔ K is open in G; Ω+ is discrete ⇔ G is compact.$ The fundamental result is

(Plancherel-Godement theorem). There exists a unique positive measure $\mu$ on ${\Omega }^{+}$ such that

 (1) $f\in ℒ\left(G,K\right)⇒\stackrel{ˆ}{f}\in {ℒ}^{2}\left({\Omega }^{+},\mu \right),$ (2) ${\int }_{G}{|f\left(g\right)|}^{2}dg={\int }_{{\Omega }^{+}}{|\stackrel{ˆ}{f}\left(\omega \right)|}^{2}d\mu \left(\omega \right)$ for all $f\in ℒ\left(G,K\right)\text{.}$
Moreover, the mapping $f↦\stackrel{ˆ}{f}$ extends to an isomorphism of Hilbert spaces ${L}^{2}\left(G,K\right)\to {L}^{2}\left(\Omega ,\mu \right)\text{.}$

This measure $\mu$ is called the Plancherel measure on ${\Omega }^{+}\text{.}$ The proof of (1.5.1) (or rather of a more general result which includes it as a special case) is sketched in [God1957] and given in detail in [Die1967], Chapter XV.

From (1.5.1) there follows the usual duality of Fourier analysis, just as in the case of locally compact abelian groups. Convolution in $ℒ\left(G,K\right)$ becomes function multiplication on ${\Omega }^{+}\text{:}$ ${\left({f}_{1}*{f}_{2}\right)}^{\wedge }={\stackrel{ˆ}{f}}_{1}·{\stackrel{ˆ}{f}}_{2}\text{.}$

This theory includes as particular cases: (i) Fourier analysis on locally compact abelian groups; (ii) characters of compact groups. For (i), let $G$ be abelian and $K=\left\{1\right\}\text{;}$ then the functional equation (1.2.5)(3) shows that the z.s.f. on $G$ in this case are just the continuous homomorphisms $G\to {ℂ}^{*},$ and since positive definite z.s.f. are bounded (by (1.4.1)), they are just the characters of $G\text{.}$ So in this case ${\Omega }^{+}$ is the character group $\stackrel{ˆ}{G},$ and the Plancherel measure $\mu$ is Haar measure on $\stackrel{ˆ}{G}\text{.}$ For case (ii), let $\Gamma$ be a compact group, take $G=\Gamma ×\Gamma ,$ and take $K$ to be the diagonal subgroup. Then the bi-invariant functions on $G$ may be identified with the class functions on $\Gamma \text{;}$ all the z.s.f. on $G$ relative to $K$ are positive definite, and with the above identification they are precisely the functions $\gamma ↦\chi \left(\gamma \right)/\chi \left(1\right),$ where $\chi$ is an irreducible character of $\Gamma \text{.}$

For this and other examples (including $G={SO}_{3}\left(ℝ\right),$ $K={SO}_{2}\left(ℝ\right),$ which is the origin of the name 'zonal spherical functions'), we refer to Helgason [Hel1962], Chapter X.

## 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.