It seems pretty clear a priori that Fourier analysis on a discrete cocompact group of motions of $\mathbb{E}^n$ should be quite tractable (since, by Bieberbach, such groups are almost $\mathbb{Z}^n,$ but I have not seen a lucid exposition of such, so any pointers or words of wisdom would be welcome.
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$\begingroup$ I don't too much about this area, but I think the jump is not quite as small as one might first hope. That said, one place to start with some aspects of their Fourier analysis might be with a description of their $C^*$-algebras, for which this paper of K. F. Taylor mathstat.dal.ca/~kft/ResearchRelated/C_star_crystal.pdf could be worth a look. $\endgroup$– Yemon ChoiJul 2, 2012 at 4:40
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1$\begingroup$ The finite dimensional reps are given here: encyclopediaofmath.org/index.php/Crystallographic_group $\endgroup$– Marc PalmJul 2, 2012 at 12:11
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$\begingroup$ I will check these out, thanks to both @Yemon and @Mrc... $\endgroup$– Igor RivinJul 2, 2012 at 21:36
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