I'm currently reading an article on Automorphic Forms, and I'm a bit confused about some of the notation used. The article is "On Some Results of Atkin and Lehner" by William Casselman, published in Mathematiche Annalen, 1973. Here is a link, but I've attached what I think are the relevant passages below.
In the image below, taken from the second page of the article, in the line that has "Thus we may describe $\varrho(w)$ by specifying for each $f \in \mathcal{G}(k^\times)$ and character $v$ what $(\varrho(w)f)$ caret-like symbol $(v,t)$ is", I don't know what the caret-like symbol means. The crazy G, which I denote by $\mathcal{G}$, refers to a space of Schwarz-Bruhat functions.
And this caret-like symbol appears again on the next page (and a few times thereafter).
Does anyone know what this symbol means?
I highly suspect it is a familiar notation for the Fourier transform $\mathcal{F}\{f\}=\hat{f}$, just put off to the side as if it were an exponent for some reason. (Perhaps using
\widehat, $\widehat{\varrho(w)f}$, wouldn't have looked so nice, or wasn't available etc.) Of course the usual notion of Fourier transform is generalized when we're talking about locally compact groups in general.