Spherical Functions on a Group of p-adic Type

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 13 February 2014

Chapter I: Basic properties of spherical functions

The algebra (G,K)

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

Let K be a compact subgroup of G and let 𝒞(G,K) be the subspace of 𝒞(G) consisting of the functions f which are bi-invariant with respect to K: f(gk)=f(gk)=f(g) for all gG and KK. Also let (G,K)=𝒞(G,K) (G) so that the functions in (G,K) are continuous, compactly-supported and bi-invariant with respect to K.

Fix a left Haar measure dg on G, and define a multiplication (convolution) on (G,K) as follows: (f1*f2)(g0) =Gf1(g0g) f2(g-1)dg for any f1,f2(G,K). In this way (G,K) becomes an associative -algebra.

For the time being we shall not assume that (G,K) is commutative, nor that G is unimodular.


(i) If f(G,K) and ϕ𝒞(G,K) then f*ϕ𝒞(G,K).
(ii) We shall be particularly concerned with the case where K is an open subgroup of G. In that case we shall always normalize the Haar measure on G so that Kdg=1.

Suppose that K is open in G. For each double coset ξ=KgK let χξ be the characteristic function of ξ. Then the χξ form a basis of (G,K) as a vector space over , and the algebra (G,K) has an identity element, namely χK.


Since K is open and compact, so is each double coset ξ=KgK, and therefore χξ(G,K). Conversely, any f(G,K) has compact support, which must therefore consist of a finite number of double cosets ξ, and hence f is a linear combination of the characteristic functions χξ. Clearly these are linearly independent, which proves the first assertion.

Next, consider χK*χξ. We have (χK*χξ)(g0) =GχK(g0g) χξ(g-1)dg and the integrand vanishes unless g0gK and g-1ξ, which implies that g0ξ. It follows that χK*χξ is a scalar multiple of χξ, and one sees immediately that the scalar multiple is 1, by virtue of our normalization of the Haar measure (Remark (ii) above). Likewise χξ*χK=χξ.

Now let dk be a Haar measure on the compact group K, normalized so that Kdk=1. For any ϕ(G) we define ϕ(G,K) by ϕ(g)=K Kϕ(k1gk2) dk1dk2, so that ϕϕ is a projection of (G) onto (G,K).

Let ω𝒞(G,K) be such that Gf(g)ω (g-1)dg=0 for all f(G,K). Then ω=0.


Let ϕ(G). Then Gϕ(g)ω (g-1)dg = Gϕ(k1gk2) ω(g-1)dg for anyk1,k2K, = G ( K Kϕ (k1gk2)d k1dk2 ) ω(g-1)dg = Gϕ(g) ω(g-1)dg = 0by hypothesis. Since ω is continuous, it follows that ω=0.

Spherical measures and spherical functions

For each g(G) the norm f= supgG |f(g)| is defined, since f is continuous and compactly supported. If C is any compact subset of G, let C(G) be the subspace of (G) consisting of the functions which vanish outside C. A (complex-valued) measure μ on G is by definition a linear functional μ:(G) whose restriction to each C(G) is continuous with respect to the topology defined by the norm . Equivalently, for each compact subset C of G there exists a positive real number mC such that |μ(f)|mC f for all fC(G). As usual, we write Gf(g)dμ(g) in place of μ(f), whenever convenient.

A non-zero measure μ on G is said to be spherical (with respect to the compact subgroup K) if

(1) μ is bi-invariant with respect to K, in other words dμ(kg)=dμ(gk)=dμ(g) for all kK;
(2) μ:(G,K) is a ring homomorphism, i.e. μ(f1*f2)= μ(f1)·μ(f2) for all f1,f2(G,K).

For example, fGf(g-1)dg is a spherical measure.

If μ is spherical and f(G), then μ(f)=μ(f).

Indeed, this is true if μ is merely bi-invariant. with respect to K. It shows that a spherical measure is uniquely determined by its restriction to (G,K).

Suppose that K is open in G. Then every non-zero -algebra homomorphism ν:(G,K) determines a spherical measure on G.


For all f(G) define μ(f)=ν(f). We have to check that μ satisfies the continuity condition above.

Let C be a compact subset of G. Since the double cosets KgK are open and mutually disjoint, C intersects only a finite number of them, say KgiK (1in). Let χi be the characteristic function of KgiK. If f(G) has support contained in C, then f vanishes outside KCK=i=1nKgiK, and hence f is a linear combination of the χi, say f=i=1n λiχi(λi) Hence f= maxi=1n |λi|. (1)

Next, we have f f (2) because for any gG, |f(g)| KK |f(k1gk2)| dk1dk2 f. Hence |μ(f)|= |ν(f)| = i=1nλi ν(χi) i=1n |λi|· |ν(χi)| mCf mCf by (1) and (2), where mC=|ν(χi)| depends only on C. Hence μ is a measure on G, hence is a spherical measure.

Let μ be a spherical measure on G. Then there exists a unique ω𝒞(G,K) such that dμ(g)=ω(g-1)dg, i.e. such that μ(f)=(f*ω)(1) for all f(G).


Since μ0 there exists f0(G,K) such that μ(f0)0. Let f(G,K). Then μ(f*f0)=G f(g1) ( Gf0 (g1-1g) dμ(g) ) dg1. Since also μ(f*f0)=μ(f)μ(f0) 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 ω so defined is continuous and bi-invariant with respect to K, i.e. ω𝒞(G,K). If now ϕ(G) 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 ω. Hence dμ(g)=ω(g-1)dg, and since ω is continuous it is the only function with this property.

A function ω𝒞(G,K) is a zonal spherical function (z.s.f. for short) on G relative to K if ω(g-1)dg is a spherical measure. The following proposition gives other equivalent definitions:

The following are equivalent, for a complex-valued function ω on G:

(1) ω is a z.s.f. on G relative to K;
(2) ω𝒞(G,K); ω(1)=1; and f*ω=λfω for all f(G,K), where λf is a scalar (depending on f);
(3) ω is continuous, not identically zero, and ω(g1)ω(g2) =Kω(g1kg2) dk for all g1,g2G.


(1) (2). From the formula (1.2.4) it is clear that ω𝒞(G,K) and ω(1)=1. Also, for all f,f0(G,K) we have Gf0(g) (f*ω)(g-1) dg = (f0*f*ω)(1) = μ(f0*f)= μ(f)μ(f0) where μ is the associated spherical measure. Hence, putting h=f*ω-μ(f)ω, we have Gf0(g)h (g-1)dg=0 for all f0(G,K), and therefore h=0 by (1.1.2).

(2) (3). We remark first that if f(G,K) 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 ωg𝒞(G,K) and Gf(g) ωg(g-1) dg = λfω(g) = Gf(g)ω (g-1)ω (g)dg. Since this holds for all f(G,K), it follows from (1.1.2) that ω(g-1)ω(g) =ωg(g-1)= Kω(g-1kg) dk for all g,gG.

(3) (1). Let μ(f)=Gf(g)ω(g-1)dg. We have to show that μ is a spherical measure. It is immediate from (3) that ω, and therefore μ, is bi-invariant
with respect to K. If f1,f2(G,K) then μ(f1*f2) = G ( Gf1(g1) f2(g1-1g) dg1 ) ω(g-1)dg = GGf1(g1) f2(g2)ω (g2-1g1-1) dg1dg2 = GGf1(g1) f2(g2) ( K ω(g2-1kg1-1) dk ) dg1dg2 = GGf1(g1) f2(g2) ω(g1-1) ω(g2-1) dg1dg2 = μ(f1)· μ(f2).

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 -algebra homomorphisms (G,K): namely, the zonal spherical function ω corresponds to the homomorphism ωˆ:(G,K) defined by ωˆ(f)=G f(g)ω(g-1) dg.

If ω is a z.s.f. then so is the function gω(g-1). 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 ω:G* whose kernel contains K is a z.s.f. relative to K, again by (1.2.5)(3).

Induction of spherical functions

Let G0 be a closed subgroup of G such that G=KG0, and let ω0 be a zonal spherical function on G0 relative to K0=KG0. Define a function ϕ0 on G by the rule ϕ0(kg0)= ω0(g0) (kK,g0G0). Then the function ω on G defined by ω(g)=K ϕ0(gk)dk is a z.s.f. on G relative to K.


We remark first that ϕ0 is well defined, for if kg0=kg0 (kK,g0G0), then k-1k=g0g-1K0, so that g0K0g0 and therefore ω0(g0)=ω0(g0).

Clearly ω𝒞(G,K) and ω(1)=1. Hence it is enough to show that f*ω to is a scalar multiple of ω, for each f(G,K).

From the definition of ϕ0 we have ϕ0(kg)=ϕ0(g) for all kK and gG. Hence, if gG and g0G0, K0ϕ0 (gk0g0)d k0=K0 ϕ0(g0k0g0) dk0 if g=kg0 (kK,g0G0), = K0ω0 (g0k0g0) dk0 = ω0(g0) ω0(g0) by (1.2.5) so that K0ϕ0 (gk0g0)dk= ϕ0(g)ω0 (g0). (1) Hence, if f(G,K) and g=kg0, (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 λf=(f*ϕ)(1). 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 = λfKϕ0 (gk)dkby (2) = λfω(g).

The spherical function ω is said to be induced by ω0.

With the hypotheses of (1.3.1) the function gKϕ0 (g-1k)dk is a zonal spherical function on G.


(1.3.1), (1.2.7).

(Transitivity of induction). Let G0G1 be closed subgroups of G such that G=KG0. Let ω0 be a z.s.f. on G0 relative to K0=KG0 and let ω1 be the z.s.f. on G1 relative to K1=KG1 induced by ω0. Then the z.s.f. on G relative to K induced by ω0 and ω1 are equal.


As in (1.3.1) define ϕ0 by the rule ϕ0(kg0)= ω0(g0) (kK,g0G0) and ϕ1 by the rule ϕ1(kg1) = ω1(g1) (kK,g1G1) = K1ϕ0 (g1k1) dk1. Let ω be the z.s.f. on G relative to K induced by ω1, so that ω(g)=K ϕ1(gk)dk. Writing gk=kg1(k) where kK and g1(k)G1, 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 ω is also the spherical function on G induced by ω0.

Positive definite spherical functions and representations of class 1

A function ϕ:G is said to be positive definite if ϕ is continuous, not identically zero, and if i,j=1nλi λjϕ (gi-1gj) is real and 0 for all choices of g1,,gnG and λ1,,λn.

If ϕ is positive definite, then ϕ(1) is real and >0; |ϕ(g)|ϕ(1) for all gG; and ϕ(g-1)=ϕ(g).


Taking n=1 in the definition we see that ϕ(1) is real and 0. Taking n=2, the matrix ( ϕ(1) ϕ(g) ϕ(g-1) ϕ(1) ) must be Hermitian and positive, whence ϕ(g-1)=ϕ(g) and |ϕ(g)|ϕ(1). Since ϕ is not identically zero, it follows that ϕ(1)>0.

Let be a separable complex Hilbert space, B() the set of all continuous linear mappings . A representation T of G on is a mapping gTg of G into B() such that

(1) Tg1g2=Tg1Tg2 for all g1,g2G; T1=1.
(2) For each h the mapping gTg(h) is a continuous map of G into .
The representation T is irreducible if no closed subspace of , other than {0} and , is invariant under Tg for all gG. T is unitary if each Tg is unitary.

(i) Let T be a unitary representation of G on and let h, h0. Then the function ϕ defined by ϕ(g)= h,Tg(h) (where , is the scalar product on ) is positive definite.
(ii) Conversely, let ϕ be a positive definite function on G. Then there exists a unitary representation Tϕ of G on a Hilbert space and a non-zero vector h such that ϕ(g)= h,Tgϕ(h) for all gG.


(i) is a straightforward verification: clearly ϕ is continuous and ϕ(1)0; 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 0.

(ii) We shall only sketch the proof. For more details see Helgason [Hel1962], p. 413. Let Vϕ be the subspace of 𝒞(G) spanned by all the left-translates of ϕ, that is to say by all functions gϕ (gG) defined by gϕ(g)=ϕ(g-1g). Define a scalar product on Vϕ by g1ϕ,g2ϕ =ϕ(g1-1g2) extended by linearity. The set N={fVϕ:f,f=0} is a subspace of Vϕ; the quotient Vϕ/N inherits a scalar product from Vϕ and hence can be completed to a Hilbert space . Each gG defines an endomorphism fgf of Vϕ which on passing to the quotient induces a unitary operator Tgϕ:. One checks that Tϕ is a representation of G. Finally, if h is the image of ϕVϕ, then h,Tgϕ(h) =ϕ,gϕ= ϕ(g).

Suppose ϕ is a positive definite spherical function on G relative to K. Then there exists h such that ϕ(g)= h,Tgϕ(h) for all gG and such that Tkϕ(h)=h for all kK.

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

Assume that G is unimodular and (G,K) commutative.

(i) Let ω be a positive definite z.s.f. on G. Then the representation Tω constructed from ω as in (1.4.2) is of class 1.
(ii) Conversely, if T is a representation of G of class 1 and if h is a unit vector left fixed by Tk for each kK, 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 Tω.

We return to the situation of (1.3): G0 is a closed subgroup of G and G=KG0. Assume also that K is open in G, so that K0=KG0 is an open compact subgroup of G0. Let dξ be left Haar measure on G0, normalized so that K0dξ=1. The right Haar measure on G0, normalized in the same way, will be of the form Δ(ξ)dξ, where Δ is a continuous homomorphism of G0 into the multiplicative group +* of positive real numbers. Δ is trivial on K0, because K0 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.

Let K=K/K0 and let g1 be a fixed element of G. For kK, write g1-1k=kη (kK,ηG0). Then the cosets k=kK0 and K0ηK0 are uniquely determined by g1 and k=kK0. If dk is a relatively invariant measure on K, we have dk=Δ(η) dk.


Let g=kξ (kK,ξG0). Then g1-1g=g1-1 kξ=kηξ so that d(g1-1g)= dk·Δ(ηξ)d(ηξ)= dk·Δ(η)Δ(ξ)dξ by (1.4.4); but also d(g1-1g)=dg=dk·Δ(ξ)dξ. Hence the result.

Let ω0 be a z.s.f. on G0 relative to K0, and ω the induced z.s.f. on G relative to K (1.3.1). Suppose that Δ12ω0 (i.e. ξΔ(ξ)12ω0(ξ)) is positive definite. Then ω is positive definite.


By (1.4.2) and (1.4.3) there exists a unitary representation T of G on a Hilbert space 0 and a vector h00 such that Δ(ξ)12 ω0(ξ)= h0,Tξh0 for all ξG0, and such that Tk0(h0)=h0 for all k0K0.

Let be the Hilbert space of all classes of measurable functions f:G0 which satisfy

(i) f(gξ)=Δ(ξ)-12Tξ(f(g)) for all gG and ξG0;
(ii) Kf(k)2dk<.
The inner product on is f1,f2= Kf1(k),f2(k) dk. Let g1G. For each f define a function Ug1f on G by the rule (Ug1f)(g)= f(g1-1g). Clearly Ug1f satisfies (i) above. Also we have Ug1f2= K(Ug1f)(k)2 dk=K f(g1-1k)2 dk. Now from (i) and the fact that Δ is trivial on K0 it follows that (for fixed g1) f(g1-1k)2 depends only on the image of k in K=K/K0. Hence Ug1f2 = K f(g1-1k)2 dk = K f(kη)2 Δ(η)dk by (1.4.6) = K Tηf(k)2 dk = f2 because T is unitary. Hence Ug1f and the operator Ug1 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 be the function G0 defined by h(kξ)=Δ (ξ)-12 Tξ(h0) (kK,ξG) (h is well defined because T/K0 fixes h0.) Now calculate h,Ug-1h: h,Ug-1h =K h(k),h(gk)dk: putting gk=kξ (kK,ξG0) this becomes K h0,Δ(ξ)-12 Tξ(h0) dk = Kω0(ξ)dk= ω(g). Hence ω(g)=h,Ug-1h; since Ug is unitary, the calculation in (1.4.2) shows that ω is positive definite.

Let ω be a z.s.f. on G relative to K, and suppose that ω is square-integrable. Then ω is positive definite.


G acts on L2(G) by left-translations: (Tg0f)(g)=f(g0-1g). Calculate the scalar product of ω and Tg0ω: 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)ω2. Since Tg0 is unitary (by the left-invariance of the measure), it follows as in (1.4.2) that ω is positive definite.

Fourier analysis

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

Let Ω+ denote the set of all positive definite z.s.f. on G relative to K. For each f1(G,K), the Fourier transform of f is the function on Ω+ defined by fˆ(ω)=Gf (g)ω(g-1) dg. The set Ω+ is given the weakest topology for which all the Fourier transforms of functions in 1(G,K) are continuous. Equivalently, the topology on Ω+ is the compact-open topology: the sets N(C,U)={ωΩ+:ω(C)U}, where CG is compact and U is open, form a basis for the topology of Ω+.

With this topology, Ω+ is Hausdorff and locally compact, and the Fourier transforms of functions in 1(G,K) tend to 0 at infinity. We have Ω+is compact Kis open inG; Ω+is discrete Gis compact. The fundamental result is

(Plancherel-Godement theorem). There exists a unique positive measure μ on Ω+ such that

(1) f(G,K)fˆ2(Ω+,μ),
(2) G|f(g)|2dg= Ω+|fˆ(ω)|2dμ(ω) for all f(G,K).
Moreover, the mapping ffˆ extends to an isomorphism of Hilbert spaces L2(G,K)L2(Ω,μ).

This measure μ is called the Plancherel measure on Ω+. 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 (G,K) becomes function multiplication on Ω+: (f1*f2)=fˆ1·fˆ2.

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={1}; 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*, and since positive definite z.s.f. are bounded (by (1.4.1)), they are just the characters of G. So in this case Ω+ is the character group Gˆ, and the Plancherel measure μ is Haar measure on Gˆ. For case (ii), let Γ be a compact group, take G=Γ×Γ, and take K to be the diagonal subgroup. Then the bi-invariant functions on G may be identified with the class functions on Γ; all the z.s.f. on G relative to K are positive definite, and with the above identification they are precisely the functions γχ(γ)/χ(1), where χ is an irreducible character of Γ.

For this and other examples (including G=SO3(), K=SO2(), 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-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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