[Math] Topology on the Unitary Dual

Question

Suppose I have a locally compact topological group G. The unitary dual of G is the set of equivalence classes of irreducible unitary representations of G. Now, it seems to me that the sensible way of putting a topology on this space is as follows:

1. Fix a hilbert space Hn of cardinality n.

2. Consider the set R(G,Hn), the set of unitary representations of G on Hn. We can give it the topology of uniform convergence on compact sets. Specifically, reps pn approach p if for any compact K in G and v in Hn, pn(g)v -> p(g)v uniformly on K.

3. Now take the subspace I(G,Hn) of irreducible representations, with the subspace topology. Then quotient by unitary equivalence, and give the resulting space the quotient topology.

4. Finally, take a disjoint union over all (countable) n.

I am not sure, however, if this is commonly done. The popular topology on the unitary dual seems to be the Fell topology. Is what I described equivalent? If not, what advantages does the Fell topology have? Also, there is the perspective that the unitary dual is more importantly a measure space than a topological space- is a topological structure significant or important?

Thanks.

Answer 1

Let me partly comment on "what advantages does the Fell topology have"?

One of "advantages" is that it is compatible with Kirillov's orbit method. Let me quote from Boyrchenko&K paper THE ORBIT METHOD FOR PROFINITE GROUPS AND A p-ADIC ANALOGUE OF BROWN’S THEOREM

An important feature of all four situations mentioned above is that both $\hat G$ and $g^∗/G$ are equipped with a natural topology. The topology on the former is the so-called Fell topology (see §3.2). The topology on the latter is the quotient of the standard (compact-open) topology on $g^*$ Moreover, in all four cases the orbit method bijection turns out to be a homeomorphism. This is a nontrivial result which has useful applications. For an interesting application in the p-adic setting we refer the reader to [GK92]. In the setting of real Lie groups this statement was originally conjectured by Kirillov, who also proved that the bijection $g^ ∗/G$ −→ $\hat G$ is continuous. The proof that this bijection is also open is substantially more difficult, and was given by Ian Brown about 10 years later in [Br73].