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50 On the dual topology of the groups U(n) ⋉ Hn
with the pullback of the hull-kernel topology on the primitive ideal space of
C ∗ (G), the C ∗ -algebra of G. Besides the fondamental problem of determining
�G as a set, there is a genuine interest in a precise and neat description of the
topology on � G. For several classes of Lie groups, such as simply connected
nilpotent Lie groups or, more generally, exponential solvable Lie groups, the
Euclidean motion groups and also the extension groups U(n)⋉Hn considered
in this paper, there is a nice geometric object parametrizing � G, namely the
space of admissible coadjoint orbits in the dual g ∗ of the Lie algebra g of G.
In such a situation, the natural and important question arises of whether
the bijection between the orbit space, equipped with the quotient topology,
and � G is a homeomorphism. In [Lep-Lud], H. Leptin and J. Ludwig have
proved that for an exponential solvable Lie group G = expg, the dual space
�G is homeomorphic to the space of coadjoint orbits g ∗ /G through the Kirillov
mapping. On the other hand, we have recently shown in [El-Lu] that the dual
topology of the classical motion groups SO(n) ⋉ R n , n ≥ 2, can be linked to
the topology of the quotient space of admissible coadjoint orbits.
In this paper we consider the semi-direct product Gn = U(n) ⋉ Hn, n ≥ 1,
and we identify its dual space ˆ Gn with the lattice of admissible coadjoint
orbits. Lipsman showed in [Lip] that each irreducible unitary representation
of Gn can be constructed by holomorphic induction from an admissible linear
functional ℓ of the Lie algebra gn of Gn. Furthermore, two irreducible
representations in ˆ Gn are equivalent if and only if their respective linear functionals
are in the same Gn-orbit. We guess then that this identification is a
homeomorphism and we prove this conjecture for G1 = U(1) ⋉ H1.
This paper is structured in the following way. Section 2 contains preliminary
material and summarizes results from previous work concerning the
dual space of Gn which is identified with its admissible coadjoint orbit space.
The representations attached to an admissible linear functional are obtained
via Mackey’s little-group method and the dual space ˆ Gn is given by the parameter
space Pn = {α ∈ R∗ , r > 0, ρµ ∈ � U(n − 1), τλ ∈ � U(n)}. In section 3, we
shall link the convergence of sequences of admissible coadjoint orbits to the
convergence in Pn. Section 4 describes the dual topology of a second countable
locally compact group. In the last paragraph, we discuss the topology
of the dual space of our groups Gn.