Dressing orbits and a quantum Heisenberg group algebra

Abstract

In this paper, as a generalization of Kirillov's orbit theory, we explore the relationship between the dressing orbits and irreducible ${}^*$-representations of the Hopf $C^*$-algebras $(A,\Delta)$ and $(\tilde{A},\tilde{\Delta})$ we constructed earlier. We discuss the one-to-one correspondence between them, including their topological aspects. On each dressing orbit (which are symplectic leaves of the underlying Poisson structure), one can define a Moyal-type deformed product at the function level. The deformation is more or less modeled by the irreducible representation corresponding to the orbit. We point out that the problem of finding a direct integral decomposition of the regular representation into irreducibles (Plancherel theorem) has an interesting interpretation in terms of these deformed products.