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32 Dual topology of the motion groups SO(n) ⋉ R n
In this paper we consider the motion groups Mn := SO(n)⋉R n and we show
that in this case the topology of their unitary dual spaces ˆ Mn is determined
by the topology of the space of admissible co-adjoint orbits. For every admissible
linear functional ℓ of the Lie algebra mn of Mn, we can construct
an irreducible unitary representation πℓ by holomorphic induction and every
irreducible representation of Mn arises in this manner. We obtain in this fashion
a map from the set m ‡ n of the admissible linear functionals onto the
dual space ˆ Mn of Mn. Since πℓ is equivalent to πℓ ′ if and only if ℓ and ℓ′
are in the same Mn-orbit, we obtain finally a homeomorphism between the
space of admissible co-adjoint orbits m ‡ n/Mn and the dual space ˆ Mn of Mn
in Theorem 2.4.6.
The dual topology of the semi-direct products K ⋉ N, where N is an abelian
group and K is a compact group, is determined by Baggett in terms of the
Fell topology (see Theorem 6.2-A of [Ba]). Other results have already been
obtained on the topology of the dual space of Mn. For instance the cortex
for general motion groups K ⋉ R n has been determined in [Be-Ka] and it has
been shown in [Kan-Ta] that for all compact subsets L of Mn, the mapping
defined by
ψL(π) = inf
ξ∈H 1 π
(max�π(x)ξ
− ξ�)
x∈L
is continuous on ˆ Mn\ � SO(n), that is, on the set of infinite dimensional representations
of Mn, where H 1 π is the unit sphere in Hπ, the Hilbert space of π.
Here is a brief section-by-section description of the contents of the paper.
In paragraph 2, we describe the motion groups and we determine their dual
spaces ; the representations attached to an admissible linear functional are
obtained via Mackey’s little-group method and the dual space of Mn is given
�
by the parameter space Pn := {(r, ρ), r > 0, ρ ∈
SO(n − 1)} � � SO(n). In
section 3, referring to the paper [Ba] of Baggett, we shall link the convergence
of sequences of elements of ˆ Mn to the convergence in Pn. In the last section,
we use the convergence in the parameter space to show that the orbit space
m ‡ n/Mn and ˆ Mn are homeomorphic.
Let us remark that similar results are true for other kinds of motion groups,
for instance the groups SU(n) ⋉ C n . It suffices to adapt our proofs.