Let \(A\) be a compact quantum group and let \(B\) be its quantum subgroup. The authors define and study the sets of invariant elements of \(A\) with respect to \(B\). Let \(B \backslash A/B\) be the set of left- and right-invariant elements of \(A\) with respect to \(B\). In general, \(B \backslash A/B\) is not a coalgebra with respect to the comultiplication of \(A\). The authors introduce a new comultiplication \(\delta\) into \(B \backslash A/B\). Then \(B \backslash A/B\) is a coalgebra with respect to this comultiplication and the counit of \(A\). If the comultiplication \(\delta\) is cocommutative, then the pair \((A,B)\) is called a quantum Gelfand pair. If in addition the coalgebra \(B \backslash A/B\) is commutative, it is called a strict Gelfand pair. With every strict Gelfand pair, two mutually dual structures are associated. The first one is a commutative hypercomplex system with compact basis \(\text{Spec} (H^{inv})\) and a discrete commutative hypergroup with basis \(Z_+\). Characters of these hypercomplex systems are defined. As example, it is shown that the pair \((SU_q(n), U_q(n-1))\) is a strict Gelfand pair. Examples of characters, expressed in terms of Macdonald’s symmetric polynomials, are considered.