Duality for finite Gelfand pairs

Let $\mathrm{G}$ be a split reductive group, $K$ be a non-Archimedean local field, and $O$ be its ring of integers. Satake isomorphism identifies the algebra of compactly supported invariants $\mathbb{C}_c[\mathrm{G}(K)/\mathrm{G}(O))]^{\mathrm{G}(O)}$ with a complexification of the algebra of characters of finite-dimensional representations $\mathcal{O}(\mathrm{G}^L(\mathbb{C}))^{\mathrm{G}^L(\mathbb{C})}$ of the Langlands dual group. In this note we report on the results of the study of analogues of such an isomorphism for finite groups. In our setup we replaced Gelfand pair $\mathrm{G}(O)\subset \mathrm{G}(K)$ by a finite pair $H\subset G$. It is convenient to rewrite the character side of the isomorphism as $\mathcal{O}(\mathrm{G}^L(\mathbb{C}))^{\mathrm{G}^L(\mathbb{C})}=\mathcal{O}((\mathrm{G}^L(\mathbb{C})\times \mathrm{G}^L(\mathbb{C}))/\mathrm{G}^L(\mathbb{C}))^{\mathrm{G}^L(\mathbb{C})}$. We replace diagonal Gelfand pair $\mathrm{G}^L(\mathbb{C})\subset \mathrm{G}^L(\mathbb{C})\times \mathrm{G}^L(\mathbb{C})$ by a dual finite pair $\check{H}\subset \check{G}$ and use Satake isomorphism as a defining property of the duality. In this text we make a preliminary study of such duality and compute a number of nontrivial examples of dual pairs $(H,G)$ and $(\check{H}, \check{G})$. We discuss a possible relation of our constructions to String Topology.