Dressing orbits and a quantum Heisenberg group algebra

摘要

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