q-Analogues of representation theoretic measures

I will discuss joint work with Grigori Olshanski on remarkable families of probability measures, which arise from the theory of q-hypergeometric orthogonal polynomials. These measures can be seen as sophisticated q-analogues of (relatives of) the well-known Plancherel measure on partitions. I will review the definition of the Plancherel measure and compare it to our q-version.