Consider $G=\left(\mathbb{R}^{+},\times\right)$, the group of positive real numbers with multiplication as the group operation. Let $\hat{G}$ denote the Pontryagin dual of $G$.
By the “duality bracket”, I mean the mathematical expression—which I denote by $\left\langle x,\xi\right\rangle$—with the property that, for each $\xi\in\hat{G}$, the map $x\in G\mapsto\left\langle x,\xi\right\rangle \in\partial\mathbb{D}$
defines a character on $G$, and, for each $x\in G$, the map $\xi\in\hat{G}\mapsto\left\langle x,\xi\right\rangle \in\partial\mathbb{D}$
defines a character on $\hat{G}$.
As an example, for $H=\left(\mathbb{R},+\right)$
, we have that $\hat{H}=\left(\mathbb{R},+\right)$
and that: $$\left\langle x,\xi\right\rangle =e^{2\pi ix\xi}$$
What is the corresponding formula for $\left\langle x,\xi\right\rangle $
for $G$
(and, in this context, what subset of $\mathbb{C}$
is $\hat{G}$
identified with so that the formula for $\left\langle x,\xi\right\rangle$
makes sense)?
Note: I don't want an explanation, I just want the formula. I can't find a straight answer on the internet, so I thought I might as well ask.
Additional Note: If anyone knows of a handy link to a chart or list of duality brackets for commonly-used groups, it would be a great help to have such a resource on hand for future reference.
Best Answer
The map $\log:G\to H$ is an isomorphism of topological groups. So, we can identify $\hat{G}$ with $\hat{H}=(\mathbb{R},+)$, with pairing $$\langle x,\xi \rangle= e^{2\pi i \log(x) \xi}.$$