Why unitary characters for the dual group in Pontryagin duality if $G$ is not compact?

In harmonic analysis, for any locally compact abelian group, one constructs the dual group as the group of homomorphisms into the unit circle with the compact open topology. In other words, unitary linear characters of the group. We know from representation theory that we should expect irreducible reps of an abelian group to be 1 dimensional, so it makes sense to use linear characters. We also know that if the group is compact, we can always take the (finite dim) rep to be unitary, hence for compact groups, our characters should be unitary.

So what about when $G$ is not compact? Are we losing something by skipping nonunitary characters (homomorphisms into $\mathbb{C}^\times$)? Obviously not, since Pontryagin duality works, but is there a nice way to understand that? If there is no invariant inner product, how do we even define unitarity? More generally, what's so special about $\text{U}(1)$ that it should be the "universal dualizer"?

Solution 1:

The more general maps KCd mentions were called ‘generalized characters’ by George Mackey in his 1948 paper, The Laplace Transform for Locally Compact Abelian Groups, http://www.pnas.org/content/34/4/156.full.pdf.