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.

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And this caret-like symbol appears again on the next page (and a few times thereafter).

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Does anyone know what this symbol means?

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    $\begingroup$ $\mathfrak{S}$ - stunning. \mathfrak{S} - it's an S. $\endgroup$ – davidlowryduda May 1 '12 at 23:17

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.

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  • $\begingroup$ Oh - this seems to make a lot of sense. Writing math notation was likely far more constricted in 1973 than it is now. $\endgroup$ – davidlowryduda May 1 '12 at 22:12
  • $\begingroup$ Yes indeed - it makes complete sense. Thanks for that. $\endgroup$ – davidlowryduda May 1 '12 at 23:17

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