Harmonic analysis on semisimple Hopf algebras

Let $H$ be a semisimple Hopf algebra. The relationship is studied between the character algebra of $H$ and that of a Hopf subalgebra. Hecke algebras are discussed, as well as their links with quantum spaces of double cosets. An explicit expression for spherical functions is given. Also, Gelfand pairs are studied, and a description of Fourier analysis on symmetric spaces via spherical functions is presented. It is shown that the pair $(D(H), H)$ is a Gelfand pair if and only if $H$ is almost cocommutative; here $D(H)$ is the Drinfeld double of $H$.