Finite dimensional varieties on hypergroups

Abstract

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 ${\mathbb {R}}^d$, 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 $X=(X,*,\check{}, e)$ be a hypergroup. Let ${\mathcal {C}}(X)$ denote the locally convex topological vector space of all continuous complex-valued functions defined on X equipped with pointwise linear operations and the topology of compact convergence. The dual of ${\mathcal {C}}(X)$ can be identified with ${\mathcal {M}}_c(X)$, the space of all compactly supported complex measures on X and the pairing between ${\mathcal {C}}(X)$ and ${\mathcal {M}}_c(X)$ is given by

$$\begin{aligned} \left\langle \mu , f\right\rangle =\int _X f d\mu \end{aligned}$$

for each $\mu$ in ${\mathcal {M}}_c(X)$ and f in ${\mathcal {C}}(X)$. For any function $f: X \rightarrow {\mathbb {C}}$ we define ${\check{f}}(x):=f({\check{x}})$ for each x in X.

Convolution on ${\mathcal {M}}_c(X)$ is given by

$$\begin{aligned} \left\langle \mu * \nu ,f\right\rangle =\int _X \int _X f(x*y)d\mu (x)d\nu (y) \end{aligned}$$

for any $\mu ,\nu$ in ${\mathcal {M}}_c(X)$ and f in ${\mathcal {C}} (X)$. The explanation and the detailed discussion of the proper interpretation of the notation $f(x*y)$ can be found in Chapter 1 of [5]. The space ${\mathcal {M}}_c(X)$ with convolution is a unital algebra with the unit $\delta _e$, where e denotes the unit of the hypergroup X. In general, for an arbitrary x in X the symbol $\delta _x$ denotes the point mass with support $\left\{ x \right\}$.

Convolution of measures from ${\mathcal {M}}_c(X)$ and functions from ${\mathcal {C}}(X)$ is defined by

$$\begin{aligned} \mu * f(x) =\int _X f(x*{\check{y}})d\mu (y) \end{aligned}$$

for each $\mu$ in ${\mathcal {M}}_c(X)$, f in ${\mathcal {C}}(X)$ and $x\in X$. The convolution operator $\mu * f$ is continuous. With this action of ${\mathcal {M}}_c(X)$ on ${\mathcal {C}}(X)$ the space ${\mathcal {C}}(X)$ is a topological left module.

For any y in X and a continuous function $f:X \rightarrow {\mathbb {C}}$ we define the function $\tau _y f:X \rightarrow {\mathbb {C}}$ by the formula

$$\begin{aligned} \tau _yf(x):=f(x*y):=\int _X f(t) d(\delta _x*\delta _y)(t) \end{aligned}$$

and call it the left translation of f by y. In a similar way one can define the right translation of f by y. A subset H of ${\mathcal {C}} (X)$ is called left-translation invariant, if for any f in H and any y in X the function $\tau _yf$ belongs to H. A closed, left invariant subspace of ${\mathcal {C}} (X)$ is called a left variety.

Let K be a compact subhypergroup of the hypergroup X. The function f in ${\mathcal {C}} (X)$ is called K -invariant, if it satisfies

$$\begin{aligned} f(k*x*l)=f(x) \end{aligned}$$

for all x in X and k, l in K. The set of all K-invariant functions form a closed subspace of ${\mathcal {C}} (X)$ and it is denoted by ${\mathcal {C}}_K(X)$. Observe that f is K-invariant if and only if ${\check{f}}$ is K-invariant.

For each f in ${\mathcal {C}} (X)$ the function defined by

$$\begin{aligned} f^{\#}(x)=\int _K\int _K f(k*x*l)d\omega (k)d\omega (l) \end{aligned}$$

for each x in X is called the projection of f. The projection $f\mapsto f^{\#}$ is a continuous linear mapping on ${\mathcal {C}} (X)$ onto ${\mathcal {C}}_K(X)$. Moreover, $f^{\#\#}=f^{\#}$ and $\left( f^{\#} \right) \check{}=\left( {\check{f}} \right) ^{\#}$ for each f in ${\mathcal {C}} (X)$. Further, f is K-invariant if and only if $f^{\#}=f$.

The projection $\mu ^{\#}$ of the measure $\mu$ in ${\mathcal {M}}_c(X)$ is defined by

$$\begin{aligned} \langle \mu ^{\#},f\rangle = \langle \mu ,f^{\#}\rangle =\int _X\int _K\int _K f(k*x*l)d\omega (k)d\omega (l)d\mu (x) \end{aligned}$$

for each f in ${\mathcal {C}} (X)$. Clearly $\mu ^{\#}$ is a measure. A measure $\mu$ in ${\mathcal {M}}_c(X)$ is called K-invariant if $\mu ^{\#}=\mu$. The projection $\mu \mapsto \mu ^{\#}$ is the adjoint of the projection $f\mapsto f^{\#}$, hence it is a continuous linear mapping on ${\mathcal {M}}_c (X)$ onto the set $M_{c,K}(X)$ of all K-invariant measures. Moreover, $\mu ^{\#\#}=\mu ^{\#}$ and $\left( \mu ^{\#} \right) \check{}=\left( {\check{\mu }} \right) ^{\#}$ for each $\mu$ in ${\mathcal {M}}_c (X)$. Further, $\mu$ is K-invariant if and only if $\mu ^{\#}=\mu$.

As a special case, the projection of the point mass $\delta _y$ is defined by

$$\begin{aligned} \langle \delta _{y}^{\#},f\rangle = f^{\#}(y)=\int _K\int _K f(k*y*l)d\omega (k)d\omega (l). \end{aligned}$$

We define the (left) K-translate of a function f by y in X in the following way:

$$\begin{aligned} \tau _y^{\#} f(x)=\delta ^{\#}_{{\check{y}}}*f(x)=\int _K\int _K f(k*y*l*x) d\omega (k)d\omega (l) \end{aligned}$$

for each x in X. In particular, for each K-invariant function f we have

$$\begin{aligned} \tau _y^{\#} f(x)=\int _K\int _K f(k*y*x) d\omega (k) \end{aligned}$$

for each x and y in X. Similarly, for any $\mu$ in ${\mathcal {M}}_{c,K}(X)$ we define

$$\begin{aligned} \tau _y^{\#}\mu = \delta ^{\#}_{{\check{y}}}*\mu . \end{aligned}$$

From now on if we say that “Let (X, K) be a Gelfand pair”, then we mean that X is a hypergroup, $K\subseteq X$ is a compact subhypergroup, and (X, K) is a Gelfand pair, i.e. the algebra ${\mathcal {M}}_{c,K}(X)$ is commutative.

For every f in ${\mathcal {C}}_K(X)$ and for every y in X the K-invariant measure

$$\begin{aligned} D_{f;y}=\delta ^{\#}_{{\check{y}}}-f(y)\delta _e \end{aligned}$$

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:

$$\begin{aligned} D_{f;y_1,\ldots , y_{n+1}}:=\prod _{j=1}^{n+1} D_{f;y_j} \end{aligned}$$

for any natural number n and for each $y_1,\ldots , y_{n+1}$ in X. On the right hand side the product is meant as a convolution product.

The non-zero K-invariant function $s:X\rightarrow {\mathbb {C}}$ is called a K-spherical function, if it satisfies

$$\begin{aligned} \int _Ks(x*k*y) d\omega (k)=s(x)s(y) \end{aligned}$$

(2.1)

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 $\mu$ there exists a complex number $\lambda _{\mu }$ such that

$$\begin{aligned} \mu *s=\lambda _{\mu } s \end{aligned}$$

holds.

For a K-spherical function $s:X\rightarrow {\mathbb {C}}$ and a K-invariant measure $\mu$ for which the representation

$$\begin{aligned} \langle \mu ,f\rangle =\int _{X} f(x) \,d\mu (x) \end{aligned}$$

for f is in ${\mathcal {C}}_K(X)$, we define the generalized difference operator as follows:

$$\begin{aligned} D_{s;{\check{\mu }}}=\mu -\langle \mu , {\check{s}} \rangle \delta _0. \end{aligned}$$

A subset H of ${\mathcal {C}}_K(X)$ is K-invariant, if for each f in H and y in K the function $\tau _y^{\#}f$ is in H. A closed K-invariant linear subspace of ${\mathcal {C}}_K(X)$ is a K-variety. 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 $\tau _K(f)$. This is the closure of the linear space spanned by all K-translates of f.

A function f in ${\mathcal {C}}_K(X)$ is called a generalized K-monomial, if there exists a spherical function s and a natural number d such that

$$\begin{aligned} D_{s;y_1,\ldots , y_{n+1}}*f(x)=\bigl (\prod _{j=1}^{n+1} D_{s;y_j}\bigr )*f(x)=0 \end{aligned}$$

for each $x,y_1,\ldots , y_{n+1}$ 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 s-monomial is simply called an s-monomial, if its K-variety is finite dimensional. A linear combination of (generalized) K-monomials are called (generalized) K-polynomials.

A K-variety in ${\mathcal {C}}_K(X)$ 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 ${\mathcal {M}}_{c,K}(X)$ 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 K-variety can be decomposed into a finite sum of indecomposable K-varieties.

Proof

If V is a decomposable K-variety in ${\mathcal {C}}_K(X)$, then we have $V=V_1+V_2$, where $V_1, V_2\subsetneq V$, 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. $\square$

Let V be a finite dimensional K-variety in ${\mathcal {C}}_K(X)$ and let $f_1,f_2,\ldots ,f_d$ form a basis in V. Then we have

$$\begin{aligned} \mu *f_k(x)=\sum _{j=1}^d \lambda _{k,j}(\mu )f_j(x) \end{aligned}$$

(2.2)

for each $\mu$ in $M_{c,K}(X)$ and x, y in X, where $\lambda _{k,j}:{\mathcal {M}}_{c,K}(X)\rightarrow {\mathbb {C}}$ are some functions $(j,k=1,2,\ldots ,d)$. For $\nu$ in $M_{c,K}(X)$ we have

$$\begin{aligned} (\nu *(\mu *f_k))(x)=\sum _{j=1}^d \lambda _{k,j}(\mu )(\nu *f_j)(x)= \sum _{j=1}^d \sum _{i=1}^d \lambda _{k,j}(\mu )\lambda _{j,i}(\nu )f_i(x) \end{aligned}$$

for each x in X. Obviously, the left hand side of this equation can be written as $\bigl ((\nu *\mu )*f_k\bigr )(x)$, and we obtain

$$\begin{aligned} \sum _{i=1}^d \sum _{j=1}^d \lambda _{k,j}(\mu )\lambda _{j,i}(\nu )f_i(x)=\bigl ((\nu *\mu )*f_k\bigr )(x)=\sum _{i=1}^d \lambda _{k,i}(\nu *\mu )f_i(x). \end{aligned}$$

By the linear independence of the $f_i$’s we infer

$$\begin{aligned} \lambda _{k,i}(\nu *\mu )=\sum _{j=1}^d \lambda _{k,j}(\mu )\lambda _{j,i}(\nu ), \end{aligned}$$

which can also be written, by the commutativity of ${\mathcal {M}}_{c,K}(X)$, as

$$\begin{aligned} \lambda _{k,i}(\nu *\mu )=\sum _{j=1}^d \lambda _{k,j}(\mu )\lambda _{j,i}(\nu ). \end{aligned}$$

(2.3)

Let ${{\mathrm{M}}}({\mathbb {C}}^d)$ denote the algebra of complex $d\times d$ matrices. We define the mapping $\Lambda :{\mathcal {M}}_{c,K}(X)\rightarrow {{\mathrm{M}}}({\mathbb {C}}^d)$ as

$$\begin{aligned} \Lambda (\mu )=\bigl (\Lambda _{i,j}(\mu )\bigr )_{i,j=1}^d \text{ with } \Lambda _{i,j}(\mu )=\lambda _{i,j}(\mu ), \end{aligned}$$

then clearly $\Lambda (\delta _e)=I$, the $d\times d$ identity matrix, and $\Lambda$ is an algebra homomorphism of ${\mathcal {M}}_{c,K}(X)$ into ${{\mathrm{M}}}({\mathbb {C}}^d)$:

$$\begin{aligned} \Lambda (\mu *\nu )=\Lambda (\mu ) \Lambda (\nu ) \end{aligned}$$

(2.4)

holds for each $\mu ,\nu$ in ${\mathcal {M}}_{c,K}(X)$. We show that the matrix elements of $\Lambda$ restricted to X, that is the functions $x\mapsto \lambda _{i,j}(\delta _x^{\#})$ are K-polynomials. For the proof we shall need the following theorem (see [3, 4]):

Theorem 2.2

Let d be a positive integer and ${\mathcal {S}}$ a family of commuting linear operators in ${{\mathrm{M}}}({\mathbb {C}}^d)$. Then ${\mathbb {C}}^d$ decomposes into a direct sum of linear subspaces $A_j$ such that each $A_j$ is a minimal invariant subspace under the operators in ${\mathcal {S}}$. Further, ${\mathbb {C}}^d$ has a basis in which every operator in ${\mathcal {S}}$ is represented by an upper triangular matrix.

In other words, there exist positive integers $k, n_1,n_2,\ldots ,n_k$ with the property $n_1+n_2+\cdots +n_k=n$, and there exists a regular matrix S such that every matrix L in ${\mathcal {S}}$ has the form

$$\begin{aligned} L= S^{-1} \mathrm {diag\,}\{L_1,L_2,\ldots ,L_k\} S \end{aligned}$$

where $L_j$ is upper triangular for $j=1,2,\ldots ,k$. Here $\mathrm {diag\,}\{L_1,L_2,\ldots ,L_k\}$ denotes the block matrix with blocks $L_1,L_2,\ldots ,L_k$ along the main diagonal, and all diagonal elements of the block $L_j$ are the same. As a consequence the following theorem holds true.

Theorem 2.3

Let (X, K) be a Gelfand pair, d a positive integer, and let $\Lambda :{\mathcal {M}}_{c,K}(X)\rightarrow {{{\mathrm{M}}}}({\mathbb {C}}^d)$ be a continuous mapping satisfying (2.4) for each $\mu ,\nu$ in ${\mathcal {M}}_{c,K}(X)$. Then there exist positive integers $k, d_1,d_2,\ldots ,d_k$ with the property $d_1+d_2+\cdots +d_k=d$, and there exists a regular matrix S such that

$$\begin{aligned} \Lambda (\mu )=S^{-1} \mathrm {diag\,}\{\Lambda _1(\mu ),\Lambda _2(\mu ),\ldots ,\Lambda _k(\mu )\} S \end{aligned}$$

(2.5)

for each $\mu$ in ${\mathcal {M}}_{c,K}(X)$, where $\Lambda _j(\mu )$ is an upper triangular $d_j\times d_j$ matrix in which all diagonal elements are equal, and it satisfies (2.4) for each $\mu ,\nu$ in ${\mathcal {M}}_{c,K}(X)$ and for every $j=1,2,\ldots ,k$.

Theorem 2.4

Let (X, K) be a Gelfand pair, d a positive integer. Suppose that $\Lambda :{\mathcal {M}}_{c,K}(X)\rightarrow {\mathrm{M}}({\mathbb {C}}^d)$ is an algebra homomorphism. Then the matrix elements $x\mapsto \Lambda _{i,j}(\delta _x^{\#})$ are K-polynomials of degree at most d.

Proof

First we apply Theorem 2.3 to diagonalize L. For the sake of simplicity we suppose that $\Lambda (\mu )$ itself has the properties of the $\Lambda _j(\mu )$’s in Theorem 2.3, that is, $\Lambda (\mu )=\bigl (\lambda _{i,j}(\mu )\bigr )_{i,j=1}^d$ is a $d\times d$ upper triangular matrix in which all diagonal elements are equal. We note that

$$\begin{aligned} \lambda _{i,j}^{\#}(x)=\lambda _{i,j}^{\#}(\delta _x)=\int _X \lambda _{i,j}^{\#}(t)\,d\delta _x(t)=\int _X \lambda _{i,j}(t)\,d\delta _x^{\#}(t)=\lambda _{i,j}(\delta _x^{\#}) \end{aligned}$$

holds for each $i,j=1,2\ldots ,d$ and x in X. This means that $\lambda _{i,j}=0$ for $i>j$, it satisfies equation (2.3), and all diagonal elements in $\Lambda (\mu )$ are the same: $\lambda _{i, i}=\lambda _{j, j}$ for $i,j=1,2,\ldots ,d$. Then

$$\begin{aligned} \lambda _{i,j}(\delta _x^{\#}*\delta _y^{\#})=\sum _{k=i}^j \lambda _{i,k}(\delta _x^{\#})\cdot \lambda _{k,j}(\delta _y^{\#}) \end{aligned}$$

(2.6)

holds for $i=1,2,\ldots ,j$ and for each x, y in X. We have

$$\begin{aligned} \lambda _{i,j}(\delta _x^{\#}*\delta _y^{\#})&=\langle \delta _x^{\#}*\delta _y^{\#},\lambda _{i,j}\rangle =\int _X \int _X \lambda _{i,j}(u*v)\,d\delta _x^{\#}(u)\,d\delta _y^{\#}(v)\\&=\int _K \int _K \int _K \lambda _{i,j}(k_1*x*k*y*l_1)\,d\omega _K(k_1)\,d\omega _K(k)\,d\omega _K(l_1)\\&= \int _K \lambda _{i,j}^{\#}(x*k*y)\,d\omega _K(k). \end{aligned}$$

Substitution into (2.6) gives

$$\begin{aligned} \int _K \lambda _{i,j}^{\#}(x*k*y)\,d\omega _K(k)=\sum _{k=i}^j \lambda _{i,k}(\delta _x^{\#})\cdot \lambda _{k,j}(\delta _y^{\#}) \end{aligned}$$

(2.7)

for $i=1,2,\ldots ,j$ and for each x, y in X. If we put $j=i$ in (2.6) we get

$$\begin{aligned} \lambda _{i,i}(\delta _x^{\#}*\delta _y^{\#})=\lambda _{i,i}(\delta _x^{\#})\cdot \lambda _{i,i}(\delta _y^{\#}) \end{aligned}$$

(2.8)

for $i=1,2,\ldots ,d$ and for each x, y in X. Hence we infer

$$\begin{aligned} \int _K \lambda _{i,i}^{\#}(x*k*y)\,d\omega _K(k)=\lambda _{i,i}(\delta _x^{\#})\cdot \lambda _{i,i}(\delta _y^{\#})=\lambda _{i,i}^{\#}(x)\cdot \lambda _{i,i}^{\#}(y) \end{aligned}$$

which means that the functions $\lambda _{i,i}^{\#}$ $(i=1,2,\ldots ,d)$ are K-spherical functions. By assumption, all $\lambda _{i,i}$’s $(i=1,2,\ldots ,d)$ coincide, and we write $s=\lambda _{i,i}^{\#}$ for $i=1,2,\ldots ,d$. We show by induction on $j-i$ that $\lambda _{i,j}^{\#}$ is an s-monomial of degree at most $j-i$. First we show that

$$\begin{aligned} \mathrm {D}_{s;y_1,y_2,\ldots ,y_{j-i+1}} \lambda _{i,j}^{\#}(x)=0. \end{aligned}$$

Clearly, the statement holds for $j-i=0$. Suppose that we have proved it for $j-i\le l$ and let $j=i+l+1$. Then we have

$$\begin{aligned}&\mathrm {D}_{s;y_1,y_2,\ldots ,y_{l+1},y_{l+2}} \lambda _{i,i+l+1}^{\#}(x)\\&\quad =\mathrm {D}_{s;y_1,y_2,\ldots ,y_{l+1}} \Bigl [\int _K\lambda _{i,i+l+1}^{\#}(y_{l+2}*k*x)\,d\omega _K(k)-s(y_{l+2}) \lambda _{i,i+l+1}^{\#}(x)\Bigr ]\\&\quad =\mathrm {D}_{s;y_1,\ldots ,y_{l+1}}\Bigl [\sum _{k=i}^{i+l+1} \lambda _{i,k}^{\#}(x) \lambda _{k,i+l+1}^{\#}(y_{l+2})\Bigr ]- s(y_{l+2}) \mathrm {D}_{m;y_1,\ldots ,y_{l+1}} \lambda _{i,i+l+1}^{\#}(x)\\&\quad =\mathrm {D}_{m;y_1,\ldots ,y_{l+1}}\Bigl [\lambda _{i,i+l+1}^{\#}(x) s(y_{l+2})\Bigr ] - s(y_{l+2}) \mathrm {D}_{s;y_1,\ldots ,y_{l+1}} \lambda _{i,i+l+1}^{\#}(x)=0. \end{aligned}$$

This shows that the functions $\lambda _{i,j}^{\#}$ are all generalized exponential monomials. By (2.6), the K-variety of $\lambda _{i,j}^{\#}$ is spanned by the functions $\lambda _{i,k}^{\#}$ for every $k=i,i+1,\ldots ,j$, hence it is finite dimensional. The proof is complete. $\square$

Corollary 2.5

Let (X, K) be a Gelfand pair. An indecomposable K-variety on X consists of s-monomials for some K-spherical function s.

Corollary 2.6

Let (X, K) be a Gelfand pair. A K-variety on X is finite dimensional if and only if it is spanned by finitely many K-monomials.

Proof

The statement follows from Theorem 2.1 and Corollary 2.5. $\square$

Now we have a characterization of K-monomials.

Corollary 2.7

Let (X, K) be a Gelfand pair. Then the K-polynomials are exactly those continuous K-invariant functions on X whose K-variety is finite dimensional.

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 ${\mathcal {M}}_c(X)$, and so are all of its subalgebras. The converse is also obvious: if the measure algebra ${\mathcal {M}}_c(X)$ 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 ${\mathcal {M}}_c(X)$, but only its subalgebra ${\mathcal {M}}_{c,K}(X)$ 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 $(X\rtimes K,K)$ 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 ${\mathbb {R}}^d$, where d is a positive integer. We shall give a complete description of those varieties using partial differential equations.

Let $GL({\mathbb {R}}^d)$ denote the general linear group, that is, the topological group of all linear bijections of the linear space ${\mathbb {R}}^d$. This is a locally compact group with the topology inherited from ${\mathbb {R}}^{d^2}$ and with the group operation defined as the composition of linear mappings. Given an arbitrary closed subgroup $K\subseteq GL({\mathbb {R}}^d)$ the semidirect product ${\mathbb {R}}^d\rtimes K$ is called the affine group of K over ${\mathbb {R}}^d$, and it is denoted by $\mathrm {Aff\,}K$. The group $\mathrm {Aff\,}K$ can be identified with the group of all affine mappings of the form $x\mapsto kx+u$, where k is in K and u is in ${\mathbb {R}}^d$, and the group operation is the composition. We identify K with the closed subgroup of $\mathrm {Aff\,}K$ formed by all elements (0, k) with k in K, and ${\mathbb {R}}^d$ with the closed normal subgroup of $\mathrm {Aff\,}K$ formed by all elements (u, id) with u in ${\mathbb {R}}^d$, where id stands for the identity mapping.

If K is compact with normalized Haar measure $\omega _K$, then we equip the orbit space $X=\mathrm {Aff\,}K/K$ with the hypergroup structure given by

$$\begin{aligned} \int _X f\,d(\delta _{Kx}*\delta _{Ky})=\int _K f(x+ky)\,d\omega _K(k) \end{aligned}$$

for each x, y in ${\mathbb {R}}^d$ and K-invariant continuous function $f:{\mathbb {R}}^d\rightarrow {\mathbb {C}}$, that is

$$\begin{aligned} f(kx)=f(x) \end{aligned}$$

holds for each x in ${\mathbb {R}}^d$ and k in K. We have:

$$\begin{aligned} \int _X f(x+ky)\,d\omega _K(k)&=\int _X f\bigl (k(k^{-1}x+y)\bigr )\,d\omega _K(k)=\\ \int _X f(k^{-1}x+y)\,d\omega _K(k)&=\int _X f(y+kx)\,d\omega _K(k), \end{aligned}$$

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.

Theorem 3.1

Let $K\subseteq GL({\mathbb {R}}^d)$ be a compact subgroup and $V\subseteq {\mathcal {C}}_K({\mathbb {R}}^d)$ a finite dimensional K-variety. Then $V\subseteq {\mathcal {C}}^{\infty }({\mathbb {R}}^d)$.

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 $g:{\mathbb {R}}^d\rightarrow {\mathbb {C}}$ such that

$$\begin{aligned} \int _{{\mathbb {R}}^d} g(x) s(-x)\,dx\ne 0. \end{aligned}$$

This is possible, as compactly supported functions form a dense set in ${\mathcal {C}}_K({\mathbb {R}}^d)$. We define the linear functional $\mu _g$ on ${\mathcal {C}}_K({\mathbb {R}}^d)$ by

$$\begin{aligned} \langle \mu _g,f\rangle =\int _{{\mathbb {R}}^d} f(x) g(x)\,dx \end{aligned}$$

whenever f is in ${\mathcal {C}}_K({\mathbb {R}}^d)$. Clearly, $\mu _g$ is a K-invariant measure. Now let f be a generalized s-monomial of degree at most n. Then we have

$$\begin{aligned} 0= & {} \mathrm {D}_{s;{\check{\mu }}_g}^{n+1}*f=(\mu _g-\langle \mu _g,{\check{s}}\rangle \delta _0)^{n+1}* f\\= & {} (\mu _g-\bigl [\int _{{\mathbb {R}}^d} s(-x)g(x)\,dx\bigr ] \cdot \delta _0)^{n+1}* f, \end{aligned}$$

where the power on the left is a convolution product. Expanding the power we have an equation of the form

$$\begin{aligned} \psi *f=\bigl [\int _{{\mathbb {R}}^d} s(-x)g(x)\,dx\bigr ]^{n+1} f, \end{aligned}$$

where $\psi :{\mathbb {R}}^d\rightarrow {\mathbb {C}}$ is infinitely differentiable. As the coefficient of f on the right is nonzero, f is infinitely differentiable. $\square$

We shall use multi-index notation: if $\alpha , \beta$ are multi-indices in ${\mathbb {N}}^r$, then we write

$$\begin{aligned} |\alpha |= & {} \alpha _1+\alpha _2+\cdots +\alpha _r,\quad \partial ^{\alpha }=\partial _1^{\alpha _1}\partial _2^{\alpha _2}\cdots \partial _r^{\alpha _r},\quad \alpha !=\alpha _1! \alpha _2!\cdots \alpha _r!,\\ \alpha [j]= & {} (\alpha _1,\alpha _2,\ldots ,\alpha _j-1,\ldots ,\alpha _r),\quad \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) =\left( {\begin{array}{c}\alpha _1\\ \beta _1\end{array}}\right) \left( {\begin{array}{c}\alpha _2\\ \beta _2\end{array}}\right) \cdots \left( {\begin{array}{c}\alpha _r\\ \beta _r\end{array}}\right) . \end{aligned}$$

We consider ${\mathcal {C}}^{\infty }({\mathbb {R}}^d)$ equipped with the Schwartz topology: a net $(f_i)$ in ${\mathcal {C}}^{\infty }({\mathbb {R}}^d)$ is convergent to f in ${\mathcal {C}}^{\infty }({\mathbb {R}}^d)$ if and only if $\partial ^{\alpha }f_i$ uniformly converges to $\partial ^{\alpha }f$ on every compact subset of ${\mathbb {R}}^d$, for each multi-index $\alpha$. A linear operator D on ${\mathcal {C}}^{\infty }({\mathbb {R}}^d)$ is called a differential operator, if it is support-decreasing, that is

$$\begin{aligned} \mathrm {supp\,}Df\subseteq \mathrm {supp\,}f \end{aligned}$$

holds for each f in ${\mathcal {C}}^{\infty }({\mathbb {R}}^d)$. It is known (see [2, Theorem 1.4]) that every differential operator has the form

$$\begin{aligned} D=\sum _{|\alpha |\le N} a_{\alpha } \partial ^{\alpha }, \end{aligned}$$

where the $a_{\alpha }$’s are infinitely differentiable functions. The effect of D on a function f in ${\mathcal {C}}^{\infty }({\mathbb {R}}^d)$ is obvious. If $K\subseteq GL({\mathbb {R}}^d)$ is a closed subgroup, then a differential operator D is called K-invariant, if

$$\begin{aligned} D(f\circ k)=Df\circ k \end{aligned}$$

holds for each f in ${\mathcal {C}}^{\infty }({\mathbb {R}}^d)$ and k in K. All K-invariant differential operators form a unital algebra over ${\mathbb {C}}$, which we denote by ${\mathcal {D}}_K({\mathbb {R}}^d)$. The space ${\mathcal {C}}_K^{\infty }({\mathbb {R}}^d)$ of all infinitely differentiable K-invariant functions is a left module over ${\mathcal {D}}_K({\mathbb {R}}^d)$. From now on we assume that $K\subseteq GL({\mathbb {R}}^d)$ is a compact subgroup with normalized Haar measure $\omega _K$, then the K-invariant differential operators $D_1,D_2,\ldots ,D_r$ form a generating set of the commutative algebra ${\mathcal {D}}_K({\mathbb {R}}^d)$ of K-invariant differential operators (see [2, Chapter IV, §2]). Here $r\le d$. For each $\lambda$ in ${\mathbb {C}}^r$, let $s_{\lambda }$ denote the unique K-spherical function such that

$$\begin{aligned} P(D_1,D_2,\ldots ,D_r)s_{\lambda }=P(\lambda )s_{\lambda } \end{aligned}$$

whenever P is a complex polynomial in r variables. It means that every K-invariant differential operator has the form $P(D_1,D_2,\ldots ,D_r)$ with some complex polynomial P. For each vector $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _r)$ in ${\mathbb {C}}^r$ we denote by $s_{\lambda }$ the unique K-spherical function satisfying $D_js_{\lambda }=\lambda _j \cdot s_{\lambda }$ for each $j=1,2,\ldots ,r$. In other words, for every $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _r)$ in ${\mathbb {C}}^r$, $s_{\lambda }$ is the unique solution of the system of partial differential equations

$$\begin{aligned} (D_j-\lambda _j)s_{\lambda }=0\quad \text {for } j=1,2,\ldots ,r \end{aligned}$$

(3.1)

with $s_{\lambda }(0)=1$. From the theory of partial differential equations it follows that $\lambda \mapsto s_{\lambda }$ is infinitely differentiable in each coordinate of the variable $\lambda$, hence it is infinitely differentiable; in fact, it is analytic. We denote by $\partial _is_{\lambda }$ the partial derivative of $s_{\lambda }$ with respect to $\lambda _i$. In this section we use the symbol $\partial$ exclusively to denote differentiation with respect to $\lambda$.

We show that for each multi-index $\alpha$ in ${\mathbb {N}}^r$ the function $\partial ^{\alpha } s_{\lambda }$ is an $s_{\lambda }$-monomial of degree at most $|\alpha |$. We keep our notation in the subsequent statements.

Lemma 3.2

Let $\alpha$ be a multi-index in ${\mathbb {N}}^r$. Then for each $j=1,2,\ldots ,r$ we have

$$\begin{aligned} (D_j-\lambda _j)\partial ^{\alpha } s_{\lambda }=\alpha _j \partial ^{\alpha [j]}s_{\lambda }. \end{aligned}$$

Proof

We have $D_is_{\lambda }=\lambda _i s_{\lambda }$ for $i=1,2,\ldots ,r$. Applying $\partial _j$ on both sides we have

$$\begin{aligned} D_i \partial _j s_{\lambda }= {\left\{ \begin{array}{ll} \lambda _i \partial _js_{\lambda }&{} \text {if } i\ne j\\ s_{\lambda }+\lambda _j \partial _j s_{\lambda }&{} \text {if } i= j, \end{array}\right. } \end{aligned}$$

or

$$\begin{aligned} (D_i-\lambda _i) \partial _j s_{\lambda }= {\left\{ \begin{array}{ll} 0&{} \text {if } i\ne j\\ s_{\lambda }&{} \text {if } i= j. \end{array}\right. } \end{aligned}$$

Repeating this process we get

$$\begin{aligned} (D_i-\lambda _i) \partial ^l_j s_{\lambda }= {\left\{ \begin{array}{ll} 0&{} \text {if } i\ne j\\ l\,\partial ^{l-1}_is_{\lambda }&{} \text {if } i= j, \end{array}\right. } \end{aligned}$$

and the statement follows by iteration. $\square$

Lemma 3.3

Let $\alpha ,\beta$ be arbitrary multi-indices in ${\mathbb {N}}^r$. Then we have

$$\begin{aligned} (D-\lambda )^{\beta }\partial ^{\alpha }s_{\lambda }= {\left\{ \begin{array}{ll} \beta ! \left( {\begin{array}{c}\alpha \\ \beta \end{array}}\right) \partial ^{\alpha -\beta }s_{\lambda }&{} \text {if }\beta \le \alpha \\ 0&{} \text {if } \beta \nleq \alpha . \end{array}\right. } \end{aligned}$$

In particular, $(D-\lambda )^{\alpha }\partial ^{\alpha }s_{\lambda }=\alpha ! \,s_{\lambda }$.

Proof

Here we use the notation

$$\begin{aligned} (D-\lambda )^{\alpha }=(D_1-\lambda _1)^{\alpha _1}(D_2 -\lambda _2)^{\alpha _2}\cdots (D_1-\lambda _r)^{\alpha _r}. \end{aligned}$$

The statement follows immediately from the previous lemma. $\square$

Lemma 3.4

The functions $\partial ^{\alpha }s_{\lambda }$ are linearly independent for different multi-indexes $\alpha$ in ${\mathbb {N}}^r$.

Proof

Suppose that

$$\begin{aligned} \sum _{k=0}^N\sum _{|\alpha |= k} c_{\alpha } \partial ^{\alpha }s_{\lambda }=0 \end{aligned}$$

with some complex numbers $c_{\alpha }$. Let $\beta$ be arbitrary in ${\mathbb {N}}^d$ with $|\beta |=N$, then applying the operator $(D-\lambda )^{\beta }$ on both sides of the equation we have, by the previous lemma

$$\begin{aligned} \sum _{|\alpha |= N} c_{\alpha } (D-\lambda )^{\beta }\partial ^{\alpha }s_{\lambda }=0. \end{aligned}$$

If $\beta \ne \alpha$ but $|\beta |=|\alpha |$, then there exists an i with $1\le i\le d$ such that $\beta _i>\alpha _i$. Hence the previous equation implies

$$\begin{aligned} 0=c_{\beta } (D-\lambda )^{\beta }\partial ^{\beta }s_{\lambda }=c_{\beta } \beta ! s_{\lambda }, \end{aligned}$$

consequently $c_{\beta }=0$. Repeating this argument we get the statement. $\square$

Lemma 3.5

Let P be a complex polynomial in r variables. If

$$\begin{aligned} P(\partial _1,\partial _2,\ldots ,\partial _r)s_{\lambda }=0, \end{aligned}$$

then $P=0$.

Proof

The statement follows from the previous lemma. $\square$

Theorem 3.6

For each multi-index $\alpha$ in ${\mathbb {N}}^r$, $\partial ^{\alpha }s_{\lambda }$ is an $s_{\lambda }$-monomial of degree at most $|\alpha |$.

Proof

We prove the statement by induction on $N=|\alpha |$, and it clearly holds for $N=0$. Now we suppose that we have proved the statement for every $k=0,1,\ldots ,N$, and we prove it for $k=N+1$. Let $D_{i_1},D_{i_2},\ldots ,D_{i_{N+1}},D_{i_{N+2}}$ be given; then we have for $|\alpha |=N+1$:

$$\begin{aligned} (D_{i_{N+2}}-\lambda _{i_{N+2}}) \partial ^{\alpha } s_{\lambda }=\alpha _{i_{N+2}} \partial ^{\alpha [i_{N+2}]}s_{\lambda }, \end{aligned}$$

and, by assumption, the right hand side is an $s_{\lambda }$-monomial of degree at most

$$\begin{aligned} \alpha _1+\cdots +\alpha _{i_{N+2}}-1+\cdots +\alpha _d=N+1-1=N. \end{aligned}$$

Hence, applying $(D_{i_1}-\lambda _{i_1}) (D_{i_2}-\lambda _{i_2}) \cdots (D_{i_{N+1}}-\lambda _{i_{N+1}})$ on both sides, the statement follows. $\square$

It turns out that all $s_{\lambda }$-monomials of the form $\partial ^{\alpha }s$ with $|\alpha |\le N$ span the space of K-monomials of degree at most N, as the following theorem shows.

Theorem 3.7

Every $s_{\lambda }$-monomial of degree at most N is a linear combination of the functions $\partial ^{\alpha }s_{\lambda }$ with $|\alpha |\le N$.

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 $f\ne 0$ be an $s_{\lambda }$-monomial of degree at most $N+1$; it follows that the functions $(D_i-\lambda _i)f$ are $s_{\lambda }$-monomials of degree at most N, hence, by our assumption, we have the representations

$$\begin{aligned} (D_i-\lambda _i)f=\sum _{k=0}^N\, \sum _{|\alpha |=k} a_{i,\alpha } \,\partial ^{\alpha }s \end{aligned}$$

with some complex numbers $a_{i,\alpha }$ for $i=1,2,\ldots ,r$. We define the polynomials

$$\begin{aligned} Q_i(z)=\sum _{k=0}^N\, \sum _{|\alpha |=k} a_{i,\alpha } \,z^{\alpha } \end{aligned}$$

for z in ${\mathbb {C}}^r$ and $i=1,2,\ldots ,r$. Then

$$\begin{aligned} (D_i-\lambda _i)f=Q_i(\partial )s_{\lambda }, \end{aligned}$$

and, by Lemma 3.2

$$\begin{aligned} (D_j-\lambda _j)(D_i-\lambda _i)f= (\partial _jQ_i)(\partial )s_{\lambda }. \end{aligned}$$

Similarly, we have

$$\begin{aligned} (D_i-\lambda _i)(D_j-\lambda _j )f=(\partial _iQ_j)(\partial )s_{\lambda }. \end{aligned}$$

Consequently, we have

$$\begin{aligned} (\partial _jQ_i)(\partial )s_{\lambda }=(\partial _iQ_j)(\partial )s_{\lambda }. \end{aligned}$$

By Lemma 3.5, we have $\partial _iQ_j=\partial _jQ_i$ for each $i,j=1,2,\ldots ,r$. We infer that there exists a complex polynomial P in r variables such that $\partial _iP=Q_i$ for $i=1,2,\ldots ,r$. Clearly, the degree of P is at most $N+1$. We define

$$\begin{aligned} \varphi =P(\partial )s_{\lambda }. \end{aligned}$$

Then we have

$$\begin{aligned} (D_i-\lambda _i)f=Q_i(\partial )s_{\lambda }, \end{aligned}$$

and

$$\begin{aligned} (D_i-\lambda _i )\varphi =(D_i-\lambda _i )P(\partial )s_{\lambda }=(\partial _iP)(\partial )s_{\lambda }= Q_i(\partial )s_{\lambda } \end{aligned}$$

for $i=1,2,\ldots ,r$. It follows that

$$\begin{aligned} (D_i-\lambda _i )(f-\varphi )=0\quad \text {for } i=1,2,\ldots ,r. \end{aligned}$$

We conclude that $f-\varphi$ is a joint eigenfunction of the generators of ${\mathcal {D}}_K({\mathbb {R}}^n)$ with the same eigenvalues as $s_{\lambda }$, hence, by the uniqueness of the K-spherical function $s_{\lambda }$, it is a constant multiple of $s_{\lambda }$: $f-\varphi =c s_{\lambda }$ with some complex number c. As $\varphi$ is a linear combination of the partial derivatives $\partial ^{\alpha } s_{\lambda }$ with $|\alpha |\le N+1$, our theorem is proved. $\square$

Corollary 3.8

Let $K\subseteq GL({\mathbb {R}}^d)$ be a compact subgroup. Then the functions $\partial ^{\alpha } s_{\lambda }$ with $|\alpha |\le N$ form a basis of the linear space of $s_{\lambda }$-monomials of degree at most N.

Corollary 3.9

Let $K\subseteq GL({\mathbb {R}}^d)$ be a compact subgroup. Then every K-monomial is analytic.

The functions $\partial ^{\alpha } s_{\lambda }$ are called elementary $s_{\lambda }$-monomials. Since $s_{\lambda }(0)=1$ holds for each $\lambda$ in ${\mathbb {C}}^d$, we have $\partial ^{\alpha }s_{\lambda }(0)=0$ for $|\alpha |\ge 1$.

Observe that in the case $d=1$ the group $GL({\mathbb {R}})$ is identified with the multiplicative group ${\mathbb {R}}_{o}$ of nonzero real numbers. The only compact subgroups are the trivial one, and $\{-1,1\}$. If $K=\{1\}$, then K-translation is the ordinary translation, hence K-varieties are exactly the translation invariant closed linear spaces, simply called varieties. Every function and measure is K-invariant, and the K-invariant differential operators are the differential operators with constant coefficients, that is the algebra ${\mathcal {D}}_K({\mathbb {R}})$ is generated by $\frac{d}{dx}$. The K-spherical functions are exactly the exponential functions: $e_{\lambda }(x)=e^{\lambda x}$ with arbitrary complex numbers $\lambda$. The elementary $e_{\lambda }$-monomials are the functions

$$\begin{aligned} \frac{d^j\exp \lambda x}{d\lambda ^j}=x^je_{\lambda }(x),\quad j=0,1,\ldots \end{aligned}$$

(3.2)

These functions for $j=0,1,\ldots ,m_j-1$ are exactly the basis of the solution space of the differential equation

$$\begin{aligned} \bigl (\frac{d}{dx}-\lambda _j\bigr )^{m_j}f=0. \end{aligned}$$

In fact, the solution space $V_j$ of this differential equation is an indecomposable variety corresponding to the exponential $e_{\lambda _j}$, $j=0,1,\ldots ,m_j-1$. If V is a nonzero finite dimensional variety, then we have $V=V_1+V_2+\cdots +V_p$, which consists of exponential polynomials, that is, linear combinations of the functions in (3.2). We can apply a similar argument in the case $K=\{-1,1\}$ to obtain the following theorem:

Theorem 3.10

Let $K\subseteq GL({\mathbb {R}})$ be a compact subgroup and $V\ne \{0\}$ a finite dimensional K-variety. Then there exist positive integers p, different complex numbers $\lambda _j$, positive numbers $m_j$ and K-invariant differential operators $D_j$ $(j=1,2,\ldots ,p)$ such that V is the solution space of the differential equation

$$\begin{aligned} (D_1-\lambda _1)^{m_1} (D_2-\lambda _2)^{m_2}\cdots (D_p-\lambda _p)^{m_p}f=0. \end{aligned}$$

(3.3)

In the case $K=\{1\}$ we have $D_j=\frac{d}{dx}$, and in the case $K=\{-1,1\}$ we have $D_j=\frac{d^2}{dx^2}$ for $j=1,2,\ldots ,p$. In both cases the dimension of V is $\sum _{j=1}^p m_j$.

Proof

The first statement about the $D_j$’s is obvious. Suppose that $K=\{-1,1\}$. For each f in ${\mathcal {C}}^{\infty }({\mathbb {R}})$ we have

$$\begin{aligned} \Bigl [\bigl (\frac{d^2}{dx^2}f\bigl )\circ (-id)\Bigr ](x)=f''(-x), \end{aligned}$$

and

$$\begin{aligned} \Bigl [\frac{d^2}{dx^2}\bigl (f\circ (-id)\bigr )\Bigr ](x)=f''(-x) \end{aligned}$$

for each x in ${\mathbb {R}}$, hence $\frac{d^2}{dx^2}$ is a K-invariant differential operator. As the rank of ${\mathcal {D}}_K({\mathbb {R}})$ is obviously one, we have that ${\mathcal {D}}_K({\mathbb {R}})$ is the polynomial algebra consisting of the polynomials $P(\frac{d^2}{dx^2})$ where P is an arbitrary complex polynomial.

$\square$

It follows that in the case $K=\{1\}$ every finite dimensional nonzero K-variety is the linear span of solutions of the differential equation

$$\begin{aligned} \left( \frac{d}{dx}-\lambda _1\right) ^{m_1}\left( \frac{d}{dx}-\lambda _2\right) ^{m_2}\ldots \left( \frac{d}{dx}-\lambda _p\right) ^{m_p}f=0, \end{aligned}$$

where the $\lambda$’s are different complex numbers and the m’s are positive integers, and in the case $K=\{-1,1\}$ every finite dimensional nonzero K-variety is the linear span of the even solutions of the differential equation

$$\begin{aligned} \left( \frac{d^2}{dx^2}-\lambda _1\right) ^{m_1} \left( \frac{d^2}{dx^2}-\lambda _2\right) ^{m_2}\ldots \left( \frac{d^2}{dx^2}-\lambda _p\right) ^{m_p}f=0, \end{aligned}$$

where the $\lambda$’s are different complex numbers and the m’s are positive integers. In both cases $m_1+m_2+\cdots +m_p=\dim V$.

Further examples

In this section we give more examples of finite dimensional varieties.

• (1) Our first example is the polynomial hypergroup X associated with the sequence of polynomials $(P_n)_{n\in {\mathbb {N}}}$. Let $K=\{e\}$, the trivial subgroup, then the double coset space X//K is identical to X, every function and measure is K-invariant. K-varieties are the solution spaces of linear homogeneous difference equation systems with constant coefficients. It follows easily that each proper K-variety is finite dimensional (see [5]).

• (2) Let K be a compact, and D a discrete hypergroup. We denote by X the hypergroup join $K\vee D$. The normalized Haar measure on K is denoted by $\omega _K$. We recall that we identify the identity elements of K and D, and it will be the identity e of X. We write $K_e=K\backslash \{e\}$ and $D_e=D\backslash \{e\}$. The involution on X is defined as the extension of the involutions on K and D. The convolution on X is defined as follows: if x, y are in K, then $\delta _x*\delta _y= \delta _x{*|_K}\,\delta _y$, where $*|_K$ is the convolution in K. If x is in K and y is in $D_e$, then we have $\delta _x*\delta _y=\delta _y*\delta _x=\delta _x$. Finally, if x, y are in D and $x\ne {\check{y}}$, then we have $\delta _x*\delta _y= \delta _x{*|_D}\,\delta _y$, where $*|_D$ is the convolution in D. Finally, if x is in $D_e$, then we have

  $$\begin{aligned} \delta _x*|_D\,\delta _{{\check{x}}}=\sum _{w\in D_e} c(w) \delta _w+c(e) \delta _e, \end{aligned}$$

  where $c:D\rightarrow {\mathbb {C}}$ is a finitely supported function with $c(e)\ne 0$. In this case the convolution of $\delta _x$ and $\delta _{{\check{x}}}$ in X is defined as

  $$\begin{aligned} \delta _x*\delta _{{\check{x}}}=\sum _{w\in D_e} c(w) \delta _w+c(e)\omega _K. \end{aligned}$$

  It is known that with these definitions X is a hypergroup, and K is a compact subhypergroup of X. It is known that the double coset hypergroup X//K is topologically isomorphic to D (see [1]). As (X, K) is a Gelfand pair if and only if the double coset hypergroup X//K is commutative, in our case this is equivalent to the commutativity of the hypergroup D. It follows that K-spherical functions and K-monomials can be identified with the exponentials, resp. exponential monomials on D.

• (3) If $K=SO({\mathbb {R}}^d)$, then ${\mathcal {C}}_K({\mathbb {R}}^d)$ can be identified with the space of continuous radial functions, and K-spherical functions have the form

  $$\begin{aligned} s_{\lambda }(x)=J_{\lambda }(\Vert x\Vert ) \end{aligned}$$

  for each x in ${\mathbb {R}}^d$, where $J_{\lambda }$ is the Bessel function defined by

  $$\begin{aligned} J_{\lambda }(r)=\Gamma \Bigl (\frac{d}{2}\Bigr ) \sum _{k=0}^{\infty } \frac{\lambda ^k}{k!\Gamma (k+\frac{d}{2})} \Bigl (\frac{r}{2}\Bigr )^{2k} \end{aligned}$$

  for r in ${\mathbb {R}}$. Here $s_{\lambda }$ is the unique K-spherical function corresponding to the eigenvalue $\lambda$ of the Laplacian in radial form:

  $$\begin{aligned} \frac{d^2s_{\lambda }}{dr^2}+\frac{d-1}{r}\frac{ds_{\lambda }}{dr}=\lambda s_{\lambda }. \end{aligned}$$

  (see [7]). For every nonzero finite dimensional K-variety V there exists a positive integer r, different complex numbers $\lambda _1,\lambda _2,\ldots , \lambda _r$ and nonnegative integers $k_1,k_2,\ldots ,k_r$ such that V is spanned by the K-monomials $\frac{d^js_{\lambda _i}}{d\lambda ^j}$ for $i=1,2,\ldots ,r$ and $j=0,1,\ldots ,k_r-1$, which can be obtained by termwise differentiation of the series of $J_{\lambda }$ with respect to $\lambda$. In [7], it was proved that K-spectral synthesis holds for each K-variety, i.e. every K-variety is the topological sum of its finite dimensional K-varieties.