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.

中文翻译：

---

超群上的有限维变体

假设X是一个超群，K是它的紧凑子超群，并假设（X，  K）是一个Gelfand对。讨论了有限维变种和X上的K多项式之间的联系。结果表明，一个ķ上-variety X是有限维当且仅当它是由有限个跨越ķ -monomials。接下来，讨论\（{\ mathbb {R}} ^ d \）上的仿射组的有限维变体，其中d为正整数。给出了使用偏微分方程对这些品种的完整描述。