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 Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d$$\end{document}, where d is a positive integer are discussed. A complete description of those varieties using partial differential equations is given.


Introduction
In this paper we investigate finite dimensional varieties and their applications in hypergroup settings. This study is motivated by results on spectral analysis and synthesis on vector modules discussed in Chapter 11 of [6] and also results in [7]. In the vector module settings a variety is a closed vector submodule. Spectral analysis for a variety means that there are nonzero finite dimensional subvarieties in every nonzero variety. On the other hand, spectral synthesis means that there are sufficiently many nonzero finite dimensional varieties in every nonzero subvariety. In this paper we are going to investigate finite dimensional varieties invariant with respect to a compact subhypergroup K of a hypergroup X such that (X, K) is a Gelfand pair, which means that the measure algebra of K-invariant measures is commutative.

Finite dimensional varieties
The terminology in this paper is in accordance with the monograph [1]. Let X = (X, * ,ˇ, e) be a hypergroup. Let C(X) denote the locally convex topological For each f in C(X) the function defined by (l) for each x in X is called the projection of f . The projection f → f # is a continuous linear mapping on C(X) onto C K (X). Moreover, f ## = f # and The projection μ # of the measure μ in M c (X) is defined by As a special case, the projection of the point mass δ y is defined by We define the (left) K-translate of a function f by y in X in the following way: for each x in X. In particular, for each K-invariant function f we have for each x and y in X. Similarly, for any μ in M c,K (X) we define τ # y μ = δ # y * μ. From now on if we say that "Let (X, K) be a Gelfand pair", then we mean that X is a hypergroup, K ⊆ X is a compact subhypergroup, and (X, K) is a Gelfand pair, i.e. the algebra M c,K (X) is commutative.
For every f in C K (X) and for every y in X the K-invariant measure δ e is called the modified K-spherical difference, or simply modified K-difference of f by increment y. The higher order modified differences are defined in the following way: for any natural number n and for each y 1 , . . . , y n+1 in X. On the right hand side the product is meant as a convolution product. The non-zero K-invariant function s : for each x and y in X. This is equivalent to the requirement that s satisfies (2.1) and s(e) = 1. K-spherical functions are exactly the common normalized eigenfunctions of all convolution operators corresponding to K-invariant measures, that is, s(e) = 1, and for each K-invariant measure μ there exists a complex number λ μ such that For a K-spherical function s : X → C and a K-invariant measure μ for which the representation for f is in C K (X), we define the generalized difference operator as follows: The intersection of any family of K-varieties is a K-variety. The intersection of all K-varieties including the K-invariant function f is called the K-variety generated by f and is denoted by τ K (f ). This is the closure of the linear space spanned by all K-translates of f .
A function f in C K (X) is called a generalized K-monomial, if there exists a spherical function s and a natural number d such that for each x, y 1 , . . . , y n+1 in X. If f is non-zero, then the spherical function s is unique and we call f a generalized spherical s-monomial, or simply generalized s-monomial, and the smallest number n with the above property we call the degree of f . For f = 0 we do not define the degree. A generalized smonomial is simply called an s-monomial, if its K-variety is finite dimensional. A linear combination of (generalized) K-monomials are called (generalized) Kpolynomials.
A K-variety in C K (X) is called decomposable if it is the sum of two proper K-subvarieties. Otherwise the K-variety is called indecomposable. The dual concept is the following: the ideal I in M c,K (X) is called decomposable, if it is the intersection of two ideals which are different from I. Otherwise the ideal is said to be indecomposable.
In this paper we are interested in finite dimensional K-varieties on different hypergroups. We know that, by definition, every spherical monomial spans a finite dimensional K-variety. Theorem 2.1. Let (X, K) be a Gelfand pair. Every finite dimensional Kvariety can be decomposed into a finite sum of indecomposable K-varieties.
V , hence the dimension of V 1 and V 2 is smaller than the dimension of V . If both are indecomposable, then we are ready. If not, then we continue this process which must terminate as the dimensions are strictly decreasing.
for each x in X. Obviously, the left hand side of this equation can be written as (ν * μ) * f k (x), and we obtain By the linear independence of the f i 's we infer which can also be written, by the commutativity of M c,K (X), as Let M(C d ) denote the algebra of complex d×d matrices. We define the mapping Λ : then clearly Λ(δ e ) = I, the d × d identity matrix, and Λ is an algebra homomorphism of M c,K (X) into M(C d ): holds for each μ, ν in M c,K (X). We show that the matrix elements of Λ restricted to X, that is the functions For the proof we shall need the following theorem (see [3,4]): In other words, there exist positive integers k, n 1 , n 2 , . . . , n k with the property n 1 + n 2 + · · ·+ n k = n, and there exists a regular matrix S such that every matrix L in S has the form . . , L k along the main diagonal, and all diagonal elements of the block L j are the same. As a consequence the following theorem holds true.
in which all diagonal elements are equal, and it satisfies (2.4) for each μ, ν in M c,K (X) and for every j = 1, 2, . . . , k.
Proof. First we apply Theorem 2.3 to diagonalize L. For the sake of simplicity we suppose that Λ(μ) itself has the properties of the Λ j (μ)'s in Theorem 2.3, that is, holds for i = 1, 2, . . . , j and for each x, y in X. We have for i = 1, 2, . . . , j and for each x, y in X. If we put j = i in (2.6) we get for i = 1, 2, . . . , d and for each x, y in X. Hence we infer which means that the functions λ # i,i (i = 1, 2, . . . , d) are K-spherical functions. By assumption, all λ i,i 's (i = 1, 2, . . . , d) coincide, and we write s = λ # i,i for i = 1, 2, . . . , d. We show by induction on j − i that λ # i,j is an s-monomial of degree at most j − i. First we show that D s;y1,y2,...,yj−i+1 λ # i,j (x) = 0. Clearly, the statement holds for j − i = 0. Suppose that we have proved it for j − i ≤ l and let j = i + l + 1. Then we have

Affine groups
In the previous sections we have seen that Gelfand pairs play an eminent role in our investigation. In fact, in the case of Gelfand pairs the commutativity of the basic structure, which may be a group, or hypergroup, can be relaxed to the commutativity of the measure algebra. It is obvious that if the hypergroup X is commutative, then so is its measure algebra M c (X), and so are all of its subalgebras. The converse is also obvious: if the measure algebra M c (X) is commutative, then the point masses commute with respect to convolution, but the commutativity of the convolution of point masses means exactly the commutativity of X. Nevertheless, in the case of Gelfand pairs we do not require the commutativity of the whole measure algebra M c (X), but only its subalgebra M c,K (X) of K-invariant measures. Typically, point masses are not K-invariant -apart from trivial cases. On the other hand, in general, semidirect products of groups are non-commutative. Still, a large class of examples for Gelfand pairs is served by semidirect product constructions -namely, by affine groups, as we shall see below. The point is that if we start with a commutative group X and a compact group K of automorphisms of X, then the pair (X K, K) is always a Gelfand pair. Here for X a commutative semigroup can be taken as well.
In this section we shall consider finite dimensional varieties on affine groups over R d , where d is a positive integer. We shall give a complete description of those varieties using partial differential equations.
Let GL(R d ) denote the general linear group, that is, the topological group of all linear bijections of the linear space R d . This is a locally compact group with the topology inherited from R d 2 and with the group operation defined as the composition of linear mappings. Given an arbitrary closed subgroup K ⊆ GL(R d ) the semidirect product R d K is called the affine group of K over R d , and it is denoted by Aff K. The group Aff K can be identified with the group of all affine mappings of the form x → kx + u, where k is in K and u is in R d , and the group operation is the composition. We identify K with the closed subgroup of Aff K formed by all elements (0, k) with k in K, and R d with the closed normal subgroup of Aff K formed by all elements (u, id) with u in R d , where id stands for the identity mapping.
If K is compact with normalized Haar measure ω K , then we equip the orbit space X = Aff K/K with the hypergroup structure given by for each x, y in R d and K-invariant continuous function f : holds for each x in R d and k in K. We have: by the inversion invariance of the Haar measure and the K-invariance of f . It follows that the hypergroup X is commutative. The following theorem is fundamental.
Proof. By Corollary 2.6, it is enough to show that every K-monomial is infinitely many times differentiable. Let s be a K-spherical function. We choose a compactly supported continuous K-invariant function g : This is possible, as compactly supported functions form a dense set in C K (R d ). We define the linear functional μ g on C K (R d ) by Clearly, μ g is a K-invariant measure. Now let f be a generalized s-monomial of degree at most n. Then we have where the power on the left is a convolution product. Expanding the power we have an equation of the form As the coefficient of f on the right is nonzero, f is infinitely differentiable.
We shall use multi-index notation: if α, β are multi-indices in N r , then we write We consider C ∞ (R d ) equipped with the Schwartz topology: a net (f i ) in It is known (see [2, Theorem 1.4]) that every differential operator has the form where the a α 's are infinitely differentiable functions. The effect of D on a function f in C ∞ (R d ) is obvious. If K ⊆ GL(R d ) is a closed subgroup, then a differential operator D is called K-invariant, if holds for each f in C ∞ (R d ) and k in K. All K-invariant differential operators form a unital algebra over C, which we denote by D K (R d ). The space . From now on we assume that K ⊆ GL(R d ) is a compact subgroup with normalized Haar measure ω K , then the K-invariant differential operators D 1 , D 2 , . . . , D r form a generating set of the commutative algebra D K (R d ) of K-invariant differential operators (see [2,Chapter IV,§2]). Here r ≤ d. For each λ in C r , let s λ denote the unique K-spherical function such that P (D 1 , D 2 , . . . , D r )s λ = P (λ)s λ whenever P is a complex polynomial in r variables. It means that every Kinvariant differential operator has the form P (D 1 , D 2 , . . . , D r ) with some complex polynomial P . For each vector λ = (λ 1 , λ 2 , . . . , λ r ) in C r we denote Vol. 95 (2021) Finite dimensional varieties on hypergroups 561 by s λ the unique K-spherical function satisfying D j s λ = λ j · s λ for each j = 1, 2, . . . , r. In other words, for every λ = (λ 1 , λ 2 , . . . , λ r ) in C r , s λ is the unique solution of the system of partial differential equations (D j − λ j )s λ = 0 for j = 1, 2, . . . , r (3.1) with s λ (0) = 1. From the theory of partial differential equations it follows that λ → s λ is infinitely differentiable in each coordinate of the variable λ, hence it is infinitely differentiable; in fact, it is analytic. We denote by ∂ i s λ the partial derivative of s λ with respect to λ i . In this section we use the symbol ∂ exclusively to denote differentiation with respect to λ. We show that for each multi-index α in N r the function ∂ α s λ is an s λmonomial of degree at most |α|. We keep our notation in the subsequent statements.

Lemma 3.2.
Let α be a multi-index in N r . Then for each j = 1, 2, . . . , r we have Proof. We have D i s λ = λ i s λ for i = 1, 2, . . . , r. Applying ∂ j on both sides we have Repeating this process we get In particular, Proof. Here we use the notation The statement follows immediately from the previous lemma. If β = α but |β| = |α|, then there exists an i with 1 ≤ i ≤ d such that β i > α i . Hence the previous equation implies consequently c β = 0. Repeating this argument we get the statement. Proof. The statement follows from the previous lemma. Proof. We prove the statement by induction on N = |α|, and it clearly holds for N = 0. Now we suppose that we have proved the statement for every k = 0, 1, . . . , N, and we prove it for k = N + 1. Let D i1 , D i2 , . . . , D iN+1 , D iN+2 be given; then we have for |α| = N + 1: and, by assumption, the right hand side is an s λ -monomial of degree at most ) on both sides, the statement follows.
It turns out that all s λ -monomials of the form ∂ α s with |α| ≤ N span the space of K-monomials of degree at most N , as the following theorem shows. Proof. We prove the statement by induction on N , and it clearly holds for N = 0. We assume that we have proved it for N . Let f = 0 be an s λmonomial of degree at most N + 1; it follows that the functions (D i − λ i )f are s λ -monomials of degree at most N , hence, by our assumption, we have the representations with some complex numbers a i,α for i = 1, 2, . . . , r. We define the polynomials for z in C r and i = 1, 2, . . . , r. Then and, by Lemma 3.2 Similarly, we have Consequently, we have (∂ j Q i )(∂)s λ = (∂ i Q j )(∂)s λ .
By Lemma 3.5, we have ∂ i Q j = ∂ j Q i for each i, j = 1, 2, . . . , r. We infer that there exists a complex polynomial P in r variables such that ∂ i P = Q i for i = 1, 2, . . . , r. Clearly, the degree of P is at most N + 1. We define ϕ = P (∂)s λ . We conclude that f − ϕ is a joint eigenfunction of the generators of D K (R n ) with the same eigenvalues as s λ , hence, by the uniqueness of the K-spherical function s λ , it is a constant multiple of s λ : f − ϕ = cs λ with some complex number c. As ϕ is a linear combination of the partial derivatives ∂ α s λ with |α| ≤ N + 1, our theorem is proved.