Hypergroups Related to a Pair of Compact Hypergroups

The purpose of the present paper is to investigate a hypergroup associated with irreducible characters of a compact hypergroup $H$ and a closed subhypergroup $H_0$ of $H$ with $|H/H_0|<+\infty$. The convolution of this hypergroup is introduced by inducing irreducible characters of $H_0$ to $H$ and by restricting irreducible characters of $H$ to $H_0$. The method of proof relies on the notion of an induced character and an admissible hypergroup pair.


Introduction
The aim of the present paper is to contribute to the largely open problem of establishing a structure theory of hypergroups. Hypergroups are locally compact spaces on which the bounded measures convolve similar to the group case. The origin of the notion of hypergroup or generalized translation structure goes back to J. Delsarte and B.M. Levitan, the special class of double coset hypergroups appears already in the work of G. Frobenius.
There exists an axiomatic approach to hypergroups initiated by Charles F. Dunkl [2,3], R.I. Jewett [13], and R. Spector [15], which lead to an extensive harmonic analysis of hypergroups. For the historical background of the theory we just refer to R.I. Jewett's fundamental paper [13] and the monograph [1] by W.R. Bloom and H. Heyer. In fact, hypergroups arose in the theory of second order differential equations and developed to be of significant applicability in probability theory where the hypergroup convolution of measure reflects a stochastic operation in the basic space of the hypergroup. Nowadays hypergroup structures are studied within various frameworks from non-commutative duality of groups to quantum groups and bimodules.
Since every investigation of the structures of hypergroups is oriented on the search of new, probably large examples, aspects of a partial solution to the structure problem are extension of hypergroups [4,6], a cohomology theory for hypergroups [7] and imprimitivity of representations of hypergroups [8]. There are interesting results on hypergroup structures arising from dual objects of a hypergroup including the group case [5]. Recent research on the structure of hypergroups relies on the application of induced characters [11,12], hyperfields [9] and compact hypergroup pairs [10]. At this point we can outline our new results.
Let H be a strong compact hypergroup satisfying the second axiom of countability, and let H 0 be a subhypergroup of H with |H/H 0 | < +∞. By Z q (2) we denote the q-deformation of Z(2), and the hats on H and H 0 signify their duals. In [11] the notion of an induced character of a finite-dimensional representation of H was introduced and studied in detail. The results obtained in that paper enable us in the present work to discuss character hypergroups of the type K Ĥ ∪ H 0 , Z q (2) which generalize those introduced in [10]. The admissible group pair of [10] will now be replaced by an admissible hypergroup pair, and the hypergroup structure of K Ĥ ∪ H 0 , Z q (2) will be characterized by the hypergroup pair (H, H 0 ) (Theorem 3.8). Applications to semi-direct product hypergroups follow (Theorem 4.7), and a list of new hypergroups appears in Section 5.

Preliminaries
In order to facilitate the reader's access to the problem discussed in this paper we recapitulate the notion of a hypergroup and of a few often applied facts. Details of the theory of hypergroups and standard examples can be found in the seminal paper [13] of R.I. Jewett and in the monograph [1] of W.R. Bloom and H. Heyer.
For a given locally compact (Hausdorff) space X we denote by C b (X) the space of bounded continuous functions on X, and by C c (X) and C 0 (X) its subspaces of functions with compact support or of functions vanishing at infinity respectively. For each compact subset K of X let C K (X) be the subset of functions f ∈ C c (X) with supp(f )⊂ K. By M (X) we denote the set of Radon measures on X defined as linear functionals on C c (X) whose restriction to each C K (X) is continuous with respect to the topology of uniform convergence. M b (X) symbolizes the set of bounded measures on X. In fact, M b (X) is the dual of the Banach space C 0 (X), and it is furnished with the norm Moreover, we shall refer to the subspaces M c (X) and M 1 (X) of measures with compact support or probability measures on X respectively.
Finally, M 1 c (X) := M 1 (X) ∩ M c (X). We denote the Dirac measure in x ∈ X by ε x . A hypergroup is a locally compact (Hausdorff) space H together with a weakly continuous associative and bilinear convolution * in the Banach space M b (H) satisfying the following axioms: (HG1) For all x, y ∈ H, ε x * ε y belongs to M 1 c (H). (HG2) There exist a neutral element e ∈ H such that ε x * ε e = ε e * ε x = ε x for all x ∈ H, and a continuous involution from H × H into the space of compact subsets of H equipped with the Michael topology is continuous.
As a consequence of the weak continuity and bilinearity the convolution of arbitrary bounded measures on H is uniquely determined by the convolution of Dirac measures. In other words A hypergroup H is called commutative if its convolution is commutative. Clearly, locally compact groups are hypergroups. Also double coset spaces G//L arising from Gelfand pairs (G, L) are (commutative) hypergroups. Given a hypergroup H one can introduce subhypergroups, quotient hypergroups, direct and semi-direct product hypergroups (for the latter notion see [8,18]), and hypergroup joins.
Every compact hypergroup H has the normalized Haar measure ω H ∈ M (H) which is invariant with respect to the translation for all y ∈ H. Let (H, * ) and (L, •) be two hypergroups with convolutions * and • as well with neutral elements ε H and ε L respectively. A continuous mapping ϕ : H → L is called a hypergroup homomorphism if ϕ(ε H ) = ε L and if ϕ is the unique linear weakly continuous extension from M b (H) to M b (L) satisfying the following conditions: If, in addition, ϕ is a homeomorphism from H onto L, it is called an isomorphism from H onto L, and in the case L = H it is called an automorphism of H. The set Aut(H) of all automorphisms of H becomes a topological group furnished with the weak topology of M b (H).
An action of a locally compact group G on a hypergroup H is a continuous homomorphism from G into Aut(H).
Given an action α of G on H there is the notion of a semi-direct product hypergroup K = H α G which in general is a non-commutative hypergroup, efficiently applied all over in our work.
Let H be a hypergroup, and let H be a (separable) Hilbert space with inner product ·, · . By B(H) we denote the Banach * -algebra of bounded linear operator on H. A * -homomorphism and for all u, v ∈ H the mapping In the sequel we shall deal with classes of representations and of irreducible representations of H under unitary equivalence.
Now let H be a compact hypergroup with a countable basis of its topology.Ĥ will denote the set of all equivalence classes of irreducible representations of H. H is said to be of strong type ifĤ carries a hypergroup structure. If H is commutative, more structure is available. In this caseĤ consists of characters of H which are defined as nonvanishing functions χ ∈ C b (H) satisfying the equality valid for all x, y ∈ H. OnceĤ is a hypergroup, the double dualĤ can be formed, and the identificationĤ ∼ = H defines Pontryagin hypergroups.
Returning to an arbitrary compact hypergroup H and a closed subhypergroup H 0 of H, for a representation π 0 of H 0 with representing Hilbert space H(π 0 ) one introduces the representation π := ind H H 0 π 0 induced by π 0 from H 0 to H as follows: For further details on induced representations, see [8,11].

Hypergroups related to admissible pairs
Let H be a strong compact hypergroup which satisfies the second axiom of countability, andĤ its dual. Then is a countable discrete commutative hypergroup, where ch(π)(h) = 1 dim π tr(π(h)).
for all π ∈Ĥ, h ∈ H. Now, let H 0 be a subhypergroup of H which is assumed to be also of strong type and such that |H/H 0 | < +∞. For τ ∈ H 0 the induced representation ind H H 0 τ of τ from H 0 to H is finite-dimensional and decomposes as where π 1 , . . . , π m ∈Ĥ (m ≥ 1). The induced character of ch(τ ) is defined as where d(π j ) for j = 1, . . . , m is the hyperdimension of π j in the sense of Vrem [17] and with a k > 0 (k = 1, . . . , ) and a 1 + · · · + a = 1. Concerning characters induced from H 0 to H and the following definition see [11].
Our main objective of study will be formulated in the subsequent Definition 3.2. Let Z q (2) be a hypergroup of order 2 with parameter q ∈ (0, 1]. The twisted convolution * = * q on the space associated with Z q (2) is given as follows: For details on deformation of hypergroups see [14]. (1) for π ∈Ĥ and τ ∈ H 0 where τ 0 is the trivial representation of H 0 .
Remark 3.4. If a pair (G, G 0 ) consisting of a compact group and a closed subgroup G 0 is admissible in the sense of [10], then it is an admissible hypergroup pair.
Lemma 3.5. If (H, H 0 ) is an admissible hypergroup pair, the following formulae hold: (1) for π ∈Ĥ and τ i , Proof . It is easy to see the desired formulae by the definition of ind H H 0 (ch(τ i ) ch(τ j )).
(2) By the associativity (A4) Hence we obtain the admissibility condition (2) res Proof . If (H, H 0 ) is an admissible hypergroup pair, then the associativity relations (A1), (A2), (A3) and (A4) are a consequence of Proposition 3.6. It is easy to check the remaining axioms of a hypergroup for K Ĥ ∪ H 0 , Z q (2) . The converse statement follows from Proposition 3.7.
(1) The above K Ĥ ∪ H 0 , Z q (2) is a discrete commutative (at most countable) hypergroup such that the sequence (4) If H is a compact commutative hypergroup of strong type and H 0 is a closed subhypergroup of H with |H/H 0 | < +∞. Then (H, H 0 ) is always an admissible hypergroup pair and K Ĥ ∪ H 0 , Z q (2) is a hypergroup. For more details see [9].

Semi-direct product hypergroups
We consider a non-commutative compact hypergroup, namely the semi-direct product hypergroup K := H α G, where α is an action of a compact group G on a compact commutative hypergroup H of strong type. For a representation π of K we denote the restrictions of π to H and G by ρ and τ respectively. We shall write π = ρ τ expressing π(h, g) = ρ(h)τ (g) for all h ∈ H, g ∈ G. The actionα of G onĤ induced by α is given bŷ be the stabilizer of χ ∈Ĥ under the actionα of G onĤ.
Proof . This statement is obtained by an application of the Mackey machine as stated in Theorem 7.1 of [8].
Proof . By an application of the character formulae as proved in Proposition 4.3 and Theorem 4.5 of [11], we obtain the desired formulae. Then the induced representation π = ind H αG G τ of an irreducible representation τ of G to H α G is finite-dimensional, and it is decomposed as The character ch(π) of π is ch π (χ,τ ) .

Definition 4.4 ([5]
). The action α of G on H is said to satisfy the regularity condition (or is called regular) provided for all χ k ∈Ĥ such that χ k ∈ supp(δ χ i * δ χ j ) whenever χ i , χ j ∈Ĥ, k, i, j ∈ {0, 1, . . . , n} and * symbolizes the convolution onĤ. . If the action α satisfies the regularity condition, then the character set K H α G of the semi-direct product hypergroup H α G is a commutative hypergroup.
Proposition 4.6. Let H be a finite commutative hypergroup of strong type and G a compact group. Assume that the action α of G on H satisfies the regularity condition. Then the followings hold: (1)Ĥ(g) is a subhypergroup ofĤ, where ωĤ (g) is the normalized Haar measure ofĤ(g), Proof .
Theorem 4.7. Let H be a finite commutative hypergroup of strong type and G a compact group. Suppose that the action α of G on H satisfies the regularity condition. Then the pair (H α G, G) is an admissible hypergroup pair and K H α G∪Ĝ, Z q (2) is a discrete commutative hypergroup.
Proof . By the Mackey machine an irreducible representation π of K = H α G is given by where χ ∈Ĥ, τ 1 ∈ G(χ) and by Propositions 4.1 and 4.2.
Hence we see the admissibility condition (2). By Theorem 3.8 we see that K H α G∪Ĝ, Z q (2) is a discrete commutative hypergroup.