International Journal of Analysis and Applications

Volume 17, Number 6 (2019), 928-939

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-928

HARMONIC ANALYSIS ON INTERNALLY GELFAND PAIRS ASSOCIATED TO

GROUPOIDS

IBRAHIMA TOURE∗, KINVI KANGNI

UFR de Mathématiques et Informatique

Université Félix Houphouet-Boigny; 22 BP 582 Abidjan 22, COTE-D’IVOIRE

∗Corresponding author: ibrahima.toure@univ-ufhb.edu.ci; toure@aims.ac.za

Abstract. Let G be a topological locally compact, Hausdorff and second countable groupoid with a Haar

system and K a proper subgroupoid of G with a Haar system too. (G, K) is an internally Gelfand pair

if for any u in the unit space, the algebra of bi-K(u)-invariant functions on G(u) is commutative under

convolution. In this work, we give some characterizations of these pairs and extend to this context some

classical results of harmonic analysis.

1. Introduction

The notion of Gelfand pair, introduced by I.M.Gelfand, has been extensively studied on groups in papers

such as [1, 4, 5, 7–10]. It has permitted to extend many results of commutative harmonic analysis to non-

commutative case. The notion of groupoid is an extension of the notion of group. In [21, 22], we have

extended the notion of Gelfand pair from groups to groupoids. In these papers, our analysis is done on a

transitive locally compact groupoid, G, and a compact subgroupoid, K. For instance, in [21] we have proved

that (G,K) is a Gelfand pair if and only if for any u ∈ G(0) the pair of isotropy groups (G(u),K(u)) is

a Gelfand pair in group sense. Thanks to this result, we have extended some results of harmonic analysis

from groups to groupoids. But this result is not true in general. For instance, the groupoid algebra is

Received 2018-11-29; accepted 2019-01-30; published 2019-11-01.

2010 Mathematics Subject Classification. 22A22, 46J10.

Key words and phrases. groupoids; groupoid representation; internally Gelfand pairs.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

928

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-928


Int. J. Anal. Appl. 17 (6) (2019) 929

not necessarily commutative even for abelian groupoids that means groupoids with abelian isotropy groups.

Nonetheless, there is a ”nice” harmonic analysis on abelian groupoids [3,11,16]. So our purpose is, in order to

do harmonic analysis on pairs (G,K) for general locally compact groupoids, to define an alternative notion

of Gelfand pairs on groupoids taking into account only the isotropy groups. Since any compact groupoid

is proper and if K is proper then the isotropy groups K(u) are compact, we have giving our definition for

proper subgroupoid, K. In fact, for a locally compact groupoid G and a proper subgroupoid K, (G,K)

is an internally Gelfand pair if for any u in the unit space, the algebra of bi-K(u)-invariant functions on

G(u) is commutative under convolution. After notations and setup in the next section, we give in section 3

some characterizations of internally Gelfand pairs, in particular we show that a Gelfand pair is an internally

Gelfand pairs. We also study the link between Gelfand pairs and internally Gelfand pairs. In section 4, we

define the notion of G(0)-spherical function associated to internally Gelfand pairs and study some properties

of these functions. We establish a connection between G(0)-spherical functions and internally irreducible

representations introduce by R. Bos in his paper [3]. In section 5, we give an extension of Bochner Theorem.

2. preliminaries

We use the notations and setup of this section in the rest of the paper without mentioning. For basic

notions on groupoids and Haar systems, we refer the reader to [20]. Throughout G will be a second countable

locally compact Hausdorff groupoid with unit space G(0) and left Haar system {λu, u ∈ G(0)}. G(2) will

denote the set of composable pairs. For x ∈ G, r(x) = xx−1 and d(x) = x−1x are respectively the range and

the domain of x. For u,v ∈ G(0), let us put Gu = r−1(u), Gv = d−1(v), Guv = Gu ∩Gv and for each unit

element u, G(u) = {x ∈ G : r(x) = d(x) = u} is the isotropy group at u. The set G′ = {x ∈ G : r(x) = d(x)}

is the isotropy group bundle of G. The relation on G(0) defined by: u,v ∈ G(0), u ∼ v iff Guv 6= ∅ is an

equivalence relation. The equivalence class of u is denoted by [u]G and is called the orbit of u. The graph

R = {(r(x),d(x)) : x ∈ G} of this equivalence relation is a groupoid with unit space G(0). The anchor map

θ=(r,d) is a continuous homomorphism of G into G(0) ×G(0) with image R. A groupoid is transitive if θ is

onto i.e. the range of θ is equal to G(0) ×G(0). Otherwise, a groupoid is transitive if it has a single orbit.

A groupoid is proper if θ is a proper map. For u ∈ G(0), λu will denote the image of λu by the inverse

map and {λu, u ∈ G(0)} is a right Haar system on G. Let µ be a quasi-invariant measure on G(0) for the

Haar system {λu,u ∈ G(0)}, ν =
∫
λudµ(u) be the induced measure by µ on G, ν−1 =

∫
λudµ(u) be the

inverse of ν, ν2 =
∫
λu ×λudµ(u) be the induced measure by µ on G(2) and ∆ the modular function of µ.

There is a decomposition of the left Haar system {λu, u ∈ G(0)} for G over R. Firstly, there is a measure

βuv concentrated on G
u
v for all (u,v) ∈ R such that βuu is a left Haar measure on G(u), and βuv is a translate

of βvv i.e. β
u
v =xβ

v
v if x ∈ Guv .

Notice that βuv is independent of the choice of x ∈ Guv . Then, there is a unique Borel Haar system α={αu :



Int. J. Anal. Appl. 17 (6) (2019) 930

u ∈ G(0)} for R with the property that for every u ∈ G(0), we have λu=
∫
βωv dα

u(ω,v). Cc(G) will denote

the space of complex-valued continuous functions on G with compact support, endowed with the inductive

limit topology and L1(G,ν) the space of ν− integrable functions on G. In [12], P. Hahn defines the following

norm on L1(G,ν): ‖ f ‖I = max(‖ f ‖I,r;‖ f ‖I,d) where ‖ f ‖I,r= sup{
∫
Gu
|f(x)|dλu(x),u ∈ G(0)},

‖ f ‖I,d= sup{
∫
Gu
|f(x)|dλu(x),u ∈ G(0)} and introduce the following groupoid algebra,

I(G,λ,µ) = {f ∈ L1(G,ν) :‖ f ‖I< ∞}

Under the convolution product defined by: for all f,g ∈ I(G,λ,µ),

f ∗g(x) =
∫
Gr(x)

f(y)g(y−1x)dλr(x)(y).

and the involution defined by: for f ∈ I(G,λ,µ),

f∗(x) = ∆(x−1)f(x−1) = ∆(x−1)f̌(x).

I(G,λ,µ) is a Banach ∗-algebra. Let K be a proper subgroupoid of G with unit space G(0) and equipped

with a Haar system {γu, u ∈ G(0)}. As it is explain above, {γu, u ∈ G(0)} has a decomposition

{(γuv )(u,v)∈RK, (ρ
u)u∈G(0)}, where RK is the graph of the equivalence relation on G(0) seen as unit space

of K, such that γu =
∫
γωv dρ

u(ω,v). We put

I(G\\K) = {f ∈ I(G,λ,µ) : f(kxk′) = f(x)∀x ∈ G,∀k ∈ Kr(x),∀k′ ∈ Kd(x)};

the space of bi-K− invariant integrable functions which is a Banach ∗-subalgebra of I(G,λ,µ). For any

f ∈ I(G,λ,µ), let us denote by f\ the bi-K-invariant function defined by: for all x ∈ G,

f\(x) =

∫ ∫
f(kxk′)dγr(x)(k)dγ

d(x)(k′).

If I(G\\K) is commutative for convolution product, we say that (G,K) is a Gelfand pair. This notion in

groupoids case has been studied by authors in [21, 22]. Let H=(Hu)u∈G(0) be a Hilbert bundle over G(0)

and U(H) the unitary groupoid of the bundle H. (π,H) is a unitary continuous representation of G if π

is a groupoid morphism of G into U(H) such that for all square integrable sections ξ and η of H, the map

x 7→< π(x)ξ(d(x)),η(r(x)) > is continuous. A closed nonzero subbundle M of H (i.e. Mu is a nonzero closed

subspace of Hu for each u ∈ G(0)) is invariant under π if π(x)Md(x) ⊂ Mr(x), for each x ∈ G. If π admits a non

trivial closed invariant subbundle M, it is called reducible. Otherwise it is called irreducible. If ξ is a section

of H, the subbundle Mξ whose leaf at u ∈ G(0) is the closed linear span of the set {π(x)ξ(d(x)) : x ∈ Gu} is

called the cyclic subbundle generated by ξ. We say that ξ is cyclic if (Mξ)u is dense in Hu, for each u ∈ G(0).

We denote by Γµ(H), the Hilbert space of square integrable section of H. In [20], J. Renault associates to

any unitary representation (π,H) a representation L of Cc(G) on Γµ(H) defined by:

(L(f)ξ,η) =

∫
f(x) < π(x)ξ(d(x)),η(r(x)) > dν0(x),



Int. J. Anal. Appl. 17 (6) (2019) 931

for all f ∈ Cc(G), ξ, η ∈ Γµ(H), where ν0=∆
−1
2 ν. L is a bounded non-degenerate ∗-representation

of Cc(G) where Cc(G) is equipped with the norm ‖ · ‖I. We may also define L by: L(f)ξ(u) =∫
Gu

f(x)π(x)ξ(d(x))∆
−1
2 (x)dλu(x). In [18], the authors extend the notion of positive definite function to

groupoids. In fact, a bounded continuous function p : G → C is positive definite if for each u ∈ G(0) and for

each f ∈Cc(G) we have ∫ ∫
f(x)f̄(y)p(y−1x)dλu(x)dλu(y) ≥ 0.

Ramsay and Walter establish for groupoids the well-known correspondence between positive definite functions

and representations. In fact, for any bounded continuous positive definite function p : G → C, there exists a

unitary representation π of G on a Hilbert bundle H, and a bounded continuous cyclic section ξ of H such

that for each x ∈ G, p(x) =< π(x)ξ(d(x)),ξ(r(x)) >.

3. Internally Gelfand Pairs

Let G be a locally compact, Hausdorff and second countable groupoid and K a proper subgroupoid of G.

Definition 3.1. (G,K) is an internally Gelfand pair if for any u ∈ G(0), (G(u),K(u)) is a Gelfand pair.

The first example is given by (G,G(0)), where G is an abelian groupoid with unit space G(0) which is seen

here as a cotrivial groupoid. Let us notice that a groupoid G is said abelian if for any u ∈ G(0) the isotropy

group G(u) is abelian.

The following results give some necessary conditions for internally Gelfand pairs.

Theorem 3.1. Let L be a locally compact group acting continuously on a compact space S and let T be

a compact subgroup of L acting trivially on S. If (L,T) is a Gelfand pair then the pair of transformation

groupoids (L ∝ S,T ∝ S) is an internally Gelfand pair.

Proof. We have (T ∝ S)(u) = T ⊂ (L ∝ S)(u) ⊂ L. Since (L,T) is a Gelfand pair, then ((L ∝ S)(u), (T ∝

S)(u)) is a Gelfand pair. (see [1]) �

Consider the relation on G defined by x ∼ y iff r(x) = r(y) and y−1x ∈ K. It is an equivalence relation

and the quotient space G/K, equipped with the quotient topology, is Hausdorff and locally compact. The

range map r induces a continuous, open surjection p : ġ 7→ r(g) from G/K to G(0) (see [20]). The groupoid

G acts on G/K, that is , the map (g,s) 7→ g.s from G∗G/K := {(g,s) ∈ G×G/K : d(g) = p(s)} to G/K,

is continuous and satisfy

(a) p(g.s) = r(g)

(b) (g1g2).s = g1.(g2.s) if (g1,g2) ∈ G(2) and



Int. J. Anal. Appl. 17 (6) (2019) 932

(c) p(s).s = s for all s ∈ G/K.

For action groupoids, see [14]. Let us put

G/K ×p G/K = {(ġ1, ġ2) ∈ G/K ×G/K : p(ġ1) = p(ġ2)}.

There is an action of G on G/K ×p G/K defined by the relation: g.(s,t) = (g.s,g.t) for g ∈ G and s,t ∈

G/K.

Theorem 3.2. If for any s,t ∈ G/K, (s,t) ∼ (t,s) then (G,K) is an internally Gelfand pair.

Proof. Let u ∈ G(0). For s = uK and x ∈ G(u) we have

(s,x−1.s) = x−1(x.s,s) ∼ (x.s,s) ∼ (s,x.s)

Thus there exists y ∈ Gu such that y.s = s and yx−1.s = x.s. The first relation shows that y ∈ K(u) and

the second one implies that x−1yx−1.s = s. So we have t ∈ K(u) and x−1yx−1 ∈ K(u). It follows that

x−1 ∈ K(u)xK(u) and consequently (G(u),K(u)) is a Gelfand pair thanks to Proposition I.2 in [8]. . �

Theorem 3.3. Let G be a locally compact, Hausdorff and second countable groupoid and K a proper sub-

groupoid of G. If (G,K) is a Gelfand pair then (G,K) is an internally Gelfand pair.

Proof. Suppose that (G,K) is a Gelfand pair. We consider the map ψ from L1(G(u)\\K(u)) to I(G\\K)

defined by f 7→ F where

F(x) =
∫
f(kxk′)dγu

r(x)
(k)dγ

d(x)
u (k

′) if K
d(x)
u 6= ∅ and Kur(x) 6= ∅ and F(x) = 0 in other case . Let’s show

that F is actually in I(G\\K). For l ∈ Kr(x), l′ ∈ Kd(x), we assume first that K
d(l′)
u 6= ∅ and Kur(l) 6= ∅. If

k ∈ Kd(l
′)

u then d(k) = u,r(k) = d(l
′). Thus l′k is defined and l′k ∈ Kr(l

′)
u = K

d(x)
u that is K

d(x)
u 6= ∅. We

show in the same way that if Ku
r(l)
6= ∅ then Ku

r(x)
6= ∅. So,

F(lxl′) =

∫
f(klxl′k′)dγur(l)(k)dγ

d(l′)
u (k

′) =

∫
f(kxlk′)dγud(l)(k)dγ

r(l′)
u (k

′) = F(x)

Now, if we assume that K
d(l′)
u = ∅ or Kur(l) = ∅ then F(lxl

′) = 0. If there exists k ∈ Kd(x)u then r(k) =

d(x) = r(k′) and l′−1k is defined. We have l′−1k ∈ Kd(l
′)

u . So, K
d(x)
u = ∅ and F(x) = 0.



Int. J. Anal. Appl. 17 (6) (2019) 933

ψ is clearly linear. Now for f,h ∈ L1(G(u)\\K(u)), we set F = ψ(f) and H = ψ(h), we have

F ∗H(x) =
∫
F(y)H(y−1x)dλr(x)(y)

=

∫
f(kyk′)h(ly−1xl′)dγur(y)(k)dγ

d(y)
u (k

′)dγud(y)(l)dγ
d(x)
u (l

′)dλr(x)(y)

=

∫
f(kyk′)h(ly−1xl′)dγur(x)(k)dγ

d(y)
u (k

′)dγud(y)(l)dγ
d(x)
u (l

′)dλd(k)v (y)dµ(v)

=

∫
f(yk′)h(ly−1kxl′)dγur(x)(k)dγ

d(y)
u (k

′)dγud(y)(l)dγ
d(x)
u (l

′)dλr(k)v (y)dµ(v)

=

∫
f(yk′)h(ly−1kxl′)dγur(x)(k)dγ

v
u(k
′)dγuv (l)dγ

d(x)
u (l

′)dλur(k′)(y)dµ(v)

=

∫
f(y)h(lk′y−1kxl′)dγur(x)(k)dγ

v
u(k
′)dγuv (l)dγ

d(x)
u (l

′)dλud(k′)(y)dµ(v)

=

∫
f(y)h(y−1kxl′)dγur(x)(k)dγ

v
u(k
′)dγuv (l)dγ

d(x)
u (l

′)dλuu(y)dµ(v)

=

∫
f(y)h(y−1kxl′)dγur(x)(k)dγ

d(x)
u (l

′)dλuu(y)

=

∫
f ∗h(kxl′)dγur(x)(k)dγ

d(x)
u (l

′) = ψ(f ∗h)

where the line 7 is due to K(u)-biinvariance of h.

Thus ψ is a morphism of convolution algebras. Moreover ψ is injective. In fact, let us notice first that if

x ∈ G(u) then F(x) =
∫
f(kxk′)dγuu(k)dγ

u
u(k
′) = f(x). Thus ψ(f) = ψ(h) implies that F = H and in

particular, F|G(u) = H|G(u) that is f = h. �

The converse is not generally true. For instance if G is abelian, the pair (G,G0) is an internally Gelfand

pair but not a Gelfand pair. For Transitive groupoids, the converse is true (see [21]). In the following result,

we give a condition for the converse.

Theorem 3.4. Let G be a locally compact Hausdorf groupoid with unit space G(0) and let K be a proper

subgroupoid of G with unit space G(0) such that (G,K) is an internally Gelfand pair. If there exists a unit

u ∈ G(0) such that its orbit in K, [u]K, is dense in G(0) then (G,K) is a Gelfand pair.

Proof. Since (G,K) is an internally Gelfand pair then (G(u),K(u)) is a Gelfand pair. It follows thanks to

theorem 3.3 of [21] that (G|[u]K,K|[u]K ) is a Gelfand pair. So if f,g ∈ I
1(G||K) then f|G|[u]K ∗ g|G|[u]K =

g|G|[u]K ∗f|G|[u]K and consequently we have (f∗g)|G|[u]K = (g∗f)|G|[u]K . Now it suffices to prove that G|[u]K
is dense in G to have f ∗ g = g ∗ f. In fact, since [u]K is dense in G(0), for x ∈ G there exists a sequence

{un}n∈N of elements of [u]K which converges to d(x) in G(0). The map d is continuous surjective and open,

so thanks to ( [11], proposition I.25, page 20) there exists a subsequence {unj}j∈J of {un}n∈N and a sequence

{xj} of G such that d(xj) = unj for any j ∈ J. Since an orbit is an invariant subset of G(0) and d(xj) ∈ [u]K

then r(xj) are in [u]K. We conclude that {xj} is a sequence of elements of G|[u]K converging to x ∈ G. �



Int. J. Anal. Appl. 17 (6) (2019) 934

Let H = (Hu)u∈G(0) be a continuous Hilbert bundle over G(0). π is an internally irreducible representation

on H if the restriction of π to G(u) is irreducible for any u ∈ G(0). We denote, as in [3], by IRepi(G) the set

of equivalence classes of internally irreducible unitary continuous representations of G and by Ĝ(u) the set of

equivalence classes of irreducible unitary continuous representations of G(u). The map Resu : IRepi(G) 7→

Ĝ(u), designates the restriction map. Let us denote by H
K(u)
u the subspace of K(u)-invariant vectors defined

by H
K(u)
u = {h ∈ Hu : π(k)h = h,∀k ∈ K(u)}.

Theorem 3.5. If (G,K) is an internally Gelfand pair then for any internally irreducible unitary represen-

tation π on H, dimHK(u)u ≤ 1 for all u ∈ G(0).

If for any u ∈ G(0), Resu is surjective then the converse holds.

Proof. Since (G(u),K(u)) is a Gelfand pair and π|G(u) is irreducible then by classical properties of Gelfand

pairs dimH
K(u)
u ≤ 1. For the converse, if πu is a unitary irreducible representation of G(u) then, since Resu

is surjective, there exists an internally unitary irreducible representation π of G such that πu = π|G(u).

So H
K(u)
u is the space of K(u)− invariant vector corresponding to πu and it follows that (G(u),K(u)) is a

Gelfand pair. �

Each H
K(u)
u is a closed subspace of Hu so HK = (H

K(u)
u )u∈G(0) is a continuous Hilbert subbundle of

H = (Hu)u∈G(0) . We set A = {u ∈ G(0) : H
K(u)
u 6= {0}}

Remark 3.1. The set A is an invariant open subset of G.

In fact, let us suppose that for x ∈ G, HK(d(x))
d(x)

6= {0}. If ξd(x) is a nonzero vector of H
K(d(x))
d(x)

we set

ηr(x) =
∫
π(k)ξd(x)dγ

r(x)
d(x)

(k). We have

‖ηr(x)‖ =
∫ ∫

< π(k)ξd(x),π(k
′)ξd(x) > dγ

r(x)
d(x)

(k)dγ
r(x)
d(x)

(k′)

=

∫ ∫
< π(k′−1k)ξd(x),ξd(x) > dγ

r(x)
d(x)

(k)dγ
r(x)
d(x)

(k′)

=

∫ ∫
< ξd(x),ξd(x) > dγ

r(x)
d(x)

(k)dγ
r(x)
d(x)

(k′)

= < ξd(x),ξd(x) >= ‖ξd(x)‖

and for k0 ∈ K
r(x)
r(x)

π(k0)ηr(x) =

∫
π(k0k)ξd(x)dγ

r(x)
d(x)

(k) =

∫
π(k)ξd(x)dγ

r(x)
d(x)

(k) = ηr(x)

Thus ηr(x) ∈ H
K(r(x))
r(x)

and is nonzero. So H
K(r(x))
r(x)

6= {0}. Now A is open as the support of a continuous

field of Hilbert space.

We end this section with some examples



Int. J. Anal. Appl. 17 (6) (2019) 935

(1) G = A ∝ S a transformation groupoid where A is a locally compact abelian group and S a topological

space. Let L be a subgroup of A acting continuously and properly on S, so (G = A ∝ S,K = L ∝ S)

is an internally Gelfand pair.

(2) P(M,G,π) a principal fiber. K a compact subgroup of G. (G,K) is a Gelfand pair if and only if

(P×P
G

, P×P
K

) is an internally Gelfand pair. In fact, (P×P
G

)(m) = G and (P×P
K

)(m) = K.

(3) Let G be a proper groupoid with unit space G(0). Let us consider a groupoid

G̃ = G×(r,d) G = {(x,y) ∈ G×G : (r,d)(x) = (r,d)(y)}

with groupoid structure defined in the following way: d(x,y) = d(x) = d(y), r(x,y) = r(x) =

r(y); (x,y)(x′,y′) = (xx′,yy′) if d(y) = r(x′) and (x,y)−1 = (x−1,y−1). The set K̃ = {(x,x) :

x ∈ G} is a closed subgroupoid of G̃. We have G̃(u) = G(u) × G(u) the cartesian product of

G(u) by G(u) and K̃(u) = Diag(G(u) × G(u)) = {(x,x) : x ∈ G(u)}. We know (see [7]) that

(G(u) × G(u),Diag(G(u) × G(u))) is a Gelfand pair. So, the pair (G̃,K̃) is an internally Gelfand

pair.

4. Harmonic Analysis on pairs (G,K)

In this section, (G,K) is an internally Gelfand pair.

Definition 4.1. Let ϕ be a bi-K−invariant continuous function on G. ϕ is G(0)-spherical if for any u ∈ G(0),

ϕ|G(u), the restriction of ϕ to G(u) is a K(u)- spherical function.

Let’s set that ϕ |G(u)= ϕu and f |G(u)= fu.

Theorem 4.1. Let ϕ be a bi-K−invariant continuous function on G such that ϕu 6= 0 for all u ∈ G(0).

Then ϕ is G(0)-spherical if and only if for all x,y ∈ G,

∫
ϕ(xky)dγ

d(x)
r(y)

(k) = ϕ(x)ϕ(y)

Proof. Let ϕ be G(0)-spherical. Then for any u ∈ G(0) and s,z ∈ G(u) we have

∫
ϕu(skz)dγ

u
u(k) = ϕ(s)ϕ(z)



Int. J. Anal. Appl. 17 (6) (2019) 936

Now for x,y ∈ G∫
ϕ(xky)dγ

d(x)
r(y)

(k) =

∫
ϕ((txl)(l−1kk1)(k

−1
1 yl1))dγ

v
r(x)(t)dγ

d(x)
v (l)dγ

r(y)
v (k1)dγ

d(y)
v (l1)dγ

d(x)
r(y)

(k)

=

∫
ϕ(txl)(l−1kk1)(k

−1
1 yl1)dγ

v
r(x)(t)dγ

d(x)
v (l)dγ

r(y)
v (k1)dγ

d(y)
v (l1)dγ

r(l)
r(k1)

(k)

=

∫
ϕ(txl)k(k−11 yl1)dγ

v
r(x)(t)dγ

d(x)
v (l)dγ

r(y)
v (k1)dγ

d(y)
v (l1)dγ

d(l)
d(k1)

(k)

=

∫ ∫
(ϕ(txl)k(k−11 yl1)dγ

v
v (k))dγ

v
r(x)(t)dγ

d(x)
v (l)dγ

r(y)
v (k1)dγ

d(y)
v (l1)

=

∫
ϕ(txl)ϕ(k−11 yl1)dγ

v
r(x)(t)dγ

d(x)
v (l)dγ

r(y)
v (k1)dγ

d(y)
v (l1)

= ϕ(x)ϕ(y)

For the converse it suffices to write for a fixed u ∈ G(0) the equality for x,y ∈ G(u) and apply the Proposition

6.1.5 of [5]. �

Theorem 4.2. Let ϕ be a bi-K-invariant continuous function non identically zero on each G(u). ϕ is

G(0)-spherical if and only if for all f ∈ I(G\\K) there exists a continuous map χf on G(0) such that for all

u ∈ G(0),ϕu ∗fu = χf (u)ϕu. In particular if there exists a dense orbit [u]K in G(0) then χf is constant on

G(0).

Proof. For any u ∈ G(0), ϕu is a spherical function on G(u), so for all f ∈ I(G\\K) there exists a complex

number χf (u) such that ϕu∗fu = χf (u)ϕu. Since ϕu is spherical then ϕu(u) = 1 and it follows that χf (u) =∫
G(u)

f(x)ϕ(x)dβuu(x). Thus the continuity of χf is due to the continuity of the map u 7→
∫
G(u)

f(x)dβuu(x)

for any f ∈ Cc(G). The converse is trivial. Now if u ∼K v then χf (u) = χf (v). In fact, for t ∈ Kuv let us

consider the map Lt from G(v) to G(u) defined by Lt(x) = txt
−1. Lt is a homeomorphism. If β

v
v is the Haar

measure on G(v) then it is straightforward to see that the image measure βuu = Lt(β
v
v ) is a Haar measure

on G(u) and equal to βuu since (G(u),K(u)) being a Gelfand pair, G(u) is unimodular. Thus

χf (u) =

∫
G(u)

f(x)ϕ(x−1)dβuu(x)

=

∫
G(v)

f(txt−1)ϕ(tx−1t−1)dβvv (x)

=

∫
G(v)

f(x)ϕ(x−1)dβvv (x) = χf (v)

So since [u]K is dense in G
(0) and χf is continuous then there exists c ∈ C such that χf (w) = c for all

w ∈ G(0). �

Theorem 4.3. Let π be an internally irreducible unitary representation on H. ξ a continuous K-invariant

section such that || ξ(u) ||= 1 for any u ∈ G(0). Then the map ϕ : x 7→ ϕ(x) =< π(x)ξ(d(x)),ξ(r(x)) > is a

positive definite G(0)-spherical function.



Int. J. Anal. Appl. 17 (6) (2019) 937

The proof is trivial.

Theorem 4.4. Let π be a unitary representation on H admitting a continuous K-invariant section on G(0).

If dimH
K(u)
u = 1 for all u ∈ G(0), then π is internally irreducible.

Proof. For all u ∈ G(0), π | G(u) is a unitary continuous representation of G(u) on Hu. Let ξ be a continuous

K-invariant section for π on G0. Then for all u ∈ G(0) ξ(u) is a K(u)− invariant vector for π | G(u) and since

dimH
K(u)
u = 1 then, thanks to Lemme 6.2.3. of [5] (or Proposition 2.6 of [8]), π | G(u) is irreducible. �

A positive definite function ϕ is said G(0)− elementary if the unitary continuous representation associated

to it is internally irreducible.

Theorem 4.5. Let ϕ be a bi-K-invariant, continuous, positive definite function such that ϕ(u) = 1 for all

u ∈ G(0). Then ϕ is G(0)− spherical if and only if ϕ is G0− elementary.

Proof. Let’s suppose that ϕ is G(0)− spherical and let πϕ be the unitary representation associated to ϕ. We

have ϕ(x) =< π(x)ξ(d(x)),ξ(r(x)) > where ξ is a continuous K-invariant section such that || ξ(u) ||= 1

for any u ∈ G(0). Let’s set ϕu = ϕ|G(u), the restriction of ϕ to G(u). ϕu is K(u)- invariant, continuous,

positive definite function such that ϕ(u) = 1. Since ϕu is spherical then the representation associated to it,

is irreducible and unitarily equivalent to πϕ |G(u). So πϕ is internally irreducible. Conversely, ϕ is G(0)−

elementary implies that the associated representation is internally irreducible. Thus, the positive definite

function φ associated to π |G(u) is spherical. But φ = ϕu = ϕ|G(u). So ϕ is G(0)− spherical. �

Denote by PG(0) the set of positive definite G
(0)- spherical functions on G and Pu the set of positive

definite spherical functions on G(u). We know by classical theory (see [5, 8]) that Pu equipped with the

topology σ(L∞,L1) is locally compact. For u ∈ G(0), let’s consider Resu : PG(0) → Pu the restriction map.

If we equip PG(0) with the coarsest topology making continuous the map Resu, then it is locally compact.

In this section, we shall suppose that Resu is bijective. The choice of the topology of PG(0) makes Resu a

continuous open bijection and therefore an homeomorphism. We start by given a definition of the Fourier

transform appropriated to our context.

Definition 4.2. For a function f ∈ I(G\\K), the Fourier transform, noted by F(f), is defined by:

F(f)(ϕ) =
∫
G(0)

∫
G(u)

f(x)ϕ(x−1)dβuu(x)dµ(u) for all ϕ ∈ PG(0) .

We have the following results known in classical case.

Theorem 4.6. (i) For f,g ∈ I(G\\K), F(f ∗g) = F(f)F(g)

(ii) F(f) is continuous on PG(0) and vanishing at infinity

(iii) The map f 7→F(f) is a linear transformation.



Int. J. Anal. Appl. 17 (6) (2019) 938

Proof. (i)For f,g ∈ I(G\\K)

F(f ∗g)(ϕ) =
∫

(f ∗g)(x)ϕ(x−1)dβuu(x)dµ(u)

=

∫
f(y)g(y−1x)ϕ(x−1)dλu(y)dβuu(x)dµ(u)

=

∫
f(y)g(ky−1x)ϕ(x−1)dγud(y)(k)dλ

u
m(y)dβ

u
u(x)dµ(m)dµ(u)

=

∫
f(yk)g(y−1x)ϕ(x−1)dγum(k)dλ

u
r(k)(y)dβ

u
u(x)dµ(m)dµ(u)

=

∫
f(yk)g(x)ϕ(x−1y−1)dγum(k)dλ

u
u(y)dβ

u
u(x)dµ(m)dµ(u)

=

∫
f(k′yk)g(x)ϕ(x−1y−1)dγmu (k

′)dγum(k)dλ
d(k′)
r(k)

(y)dβuu(x)dµ(m)dµ(u)

=

∫
f(y)g(x)(

∫
ϕ(x−1ky−1)dγ

d(x)
r(y)

(k))dγmu (k
′)dλ

r(k′)
d(k)

(y)dβuu(x)dµ(m)dµ(u)

=

∫
f(y)g(x)ϕ(x−1)ϕ(y−1)dγmu (k

′)dλ
r(k′)
d(k)

(y)dβuu(x)dµ(m)dµ(u)

= F(f)F(g)

(ii) F(f)(ϕ) =
∫
G(0)

φ(u,ϕ)dµ(u) where φ(u,ϕ) =
∫
G(u)

f(x)ϕ(x−1)dβuu(x). The map ϕ 7→ φ(u,ϕ) is con-

tinuous as the composition of continuous functions ϕ 7→ ϕ|G(u) and F(fu). Then, we have |φ(u,ϕ)| ≤

supu∈G(0) (
∫
G(u)
|f(x)|dβuu(x)) ≤ ||f||I. So F(f) is continuous. Since F(f)(ϕ) =

∫
G(0)
F(fu)(ϕu)dµ(u) and

F(fu) vanishing at infinity then F(f) is vanishing at infinity.

(iii) The proof is trivial. �

Theorem 4.7. Let φ be a bi-K-invariant continuous positive definite function on G such that φ(u) ≤ 1 for

all u ∈ G(0). Then there exists a unique measure Γ on M0(PG(0) ) such that for all x ∈ G,

φ(x) =

∫
P

G(0)

ω(x)dΓ(ω)

Proof. For any u ∈ G(0), φu the restriction of φ on G(u) is bi-K(u)-invariant continuous positive definite

function on G(u). Thus thanks to Bochner theorem for Gelfand pairs, there exists a unique measure θu on

M0(Pu) such that for all x ∈ G(0), φu(x) =
∫
Pu
wu(x)dθu(wu). We consider then the measure image Γ

u of

θu by Res
−1
u . So we obtain a family of measure {Γu : u ∈ G(0)} on PG(0) . We put Γ =

∫
G(0)

Γudµ(u). For all

x ∈ G we have

φ(x) =

∫
φ(kxk′)dγur(x)(k)dγ

d(x)
u (k

′)dµ(u)

=

∫
ωu(kxk

′)dθu(ωu)dγ
u
r(x)(k)dγ

d(x)
u (k

′)dµ(u)



Int. J. Anal. Appl. 17 (6) (2019) 939

=

∫
Resu(ω)(kxk

′)dΓ(ω)dγur(x)(k)dγ
d(x)
u (k

′)dµ(u)

=

∫
ω(x)dΓu(ω)dµ(u)

=

∫
ω(x)dΓ(ω)

�

References

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[3] R. Bos, Groupoids in Geometric Quantization, PhD. Thesis, Radboud University Nijmegen, 2007.

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Gruyter And Co, Berlin, 2009.

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[12] P. Hahn, The regular representations of measure groupoids, Trans. Amer. Math. Soc., 242 (1978), 34-72.

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University Press, 2005.

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3621-3641.

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	1. Introduction
	2. preliminaries
	3. Internally Gelfand Pairs
	4. Harmonic Analysis on pairs (G,K)
	References