Finite dimensional varieties on hypergroups

Let X be a hypergroup, K its compact subhypergroup and assume that (X, K) is a Gelfand pair. Connections between finite dimensional varieties and K-polynomials on X are discussed. It is shown that a K-variety on X is finite dimensional if and only if it is spanned by finitely many K-monomials. Next, finite dimensional varieties on affine groups over $${\mathbb {R}}^d$$ R d , where d is a positive integer are discussed. A complete description of those varieties using partial differential equations is given.

On the symmetric Gelfand pair (ℋn ×ℋn−1,diag(ℋn−1))

We show that the [Formula: see text]-conjugacy classes of [Formula: see text] where [Formula: see text] is the hyperoctahedral group on [Formula: see text] elements, are indexed by marked bipartitions of [Formula: see text] This will lead us to prove that [Formula: see text] is a symmetric Gelfand pair and that the induced representation [Formula: see text] is multiplicity free.

Gelfand Pairs Involving the Wreath Product of Finite Abelian Groups with Symmetric Groups

It is well known that the pair $(\mathcal {S}_n,\mathcal {S}_{n-1})$ is a Gelfand pair where $\mathcal {S}_n$ is the symmetric group on n elements. In this paper, we prove that if G is a finite group then $(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$ where $G\wr \mathcal {S}_n$ is the wreath product of G by $\mathcal {S}_n,$ is a Gelfand pair if and only if G is abelian.

Strong Gelfand pairs of symmetric groups

A strong Gelfand pair is a pair [Formula: see text], of finite groups such that the Schur ring determined by the [Formula: see text]-classes [Formula: see text], is a commutative ring. We find all strong Gelfand pairs [Formula: see text]. We also define an extra strong Gelfand pair [Formula: see text], this being a strong Gelfand pair of maximal dimension, and show that in this case [Formula: see text] must be abelian.

Corwin–Greenleaf multiplicity function for compact extensions of the Heisenberg group

Let [Formula: see text] be the [Formula: see text]-dimensional Heisenberg group and [Formula: see text] a closed subgroup of [Formula: see text] acting on [Formula: see text] by automorphisms such that [Formula: see text] is a Gelfand pair. Let [Formula: see text] be the semidirect product of [Formula: see text] and [Formula: see text]. Let [Formula: see text] be the respective Lie algebras of [Formula: see text] and [Formula: see text], and [Formula: see text] the natural projection. For coadjoint orbits [Formula: see text] and [Formula: see text], we denote by [Formula: see text] the number of [Formula: see text]-orbits in [Formula: see text], which is called the Corwin–Greenleaf multiplicity function. In this paper, we give two sufficient conditions on [Formula: see text] in order that [Formula: see text] For [Formula: see text], assuming furthermore that [Formula: see text] and [Formula: see text] are admissible and denoting respectively by [Formula: see text] and [Formula: see text] their corresponding irreducible unitary representations, we also discuss the relationship between [Formula: see text] and the multiplicity [Formula: see text] of [Formula: see text] in the restriction of [Formula: see text] to [Formula: see text]. Especially, we study in Theorem 4 the case where [Formula: see text]. This inequality is interesting because we expect the equality as the naming of the Corwin–Greenleaf multiplicity function suggests.

A Paley-Wiener Theorem for the Spherical Transform Associated with the Generalized Gelfand Pair $(U(p,q),H_{n})$, $p+q=n$

In this work we prove a Paley-Wiener theorem for the spherical transform associated to the generalized Gelfand pair $(H_n\ltimes U(p,q),H_n)$, where $H_n$ is the $2n+1$-dimensional Heisenberg group. In particular, by using the identification of the spectrum of $(U(p,q),H_n)$ with a subset $\Sigma$ of $\mathbb{R}^2$, we prove that the restrictions of the spherical transforms of functions in $C_{0}^{\infty}(H_n)$ to appropriated subsets of $\Sigma$, can be extended to holomorphic functions on $\mathbb{C}^2$. Also, we obtain a real variable characterizations of such transforms.