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.