Why is that the theory of automorphic forms concentrates on the case of half-integral weight? I read in Borel's book "Automorphic forms on $SL_2$" (Section 18.5) that by considering the finite or universal coverings of $SL_2$, one can define modular forms with rational or even real weight. Borel mentions in passing that the case of half-integral weight is particularly important. Why?!
[Math] Why only half-integral weight automorphic forms
automorphic-formshalf-integral-weightmodular-formsnt.number-theory
Best Answer
Since the other MO answer linked-to in Alison Miller's comment did not mention this (although Alison M. did make a comment in this direction there): indeed, the Segal-Shale-Weil/oscillator repn produces representations and automorphic forms for two-fold covers of $Sp_{2n}$, but not for more general covers. This gives the classical theta correspondences, which accounted for nearly all the "early" examples of Langlands functoriality. The technical point seems to be that the two-fold cover of $Sp_{2n}$ admits a very small "minimal" repn, which then usefully decomposes (see also "Howe correspondence") over mutually commuting subgroups ("dual reductive pairs").
I may be mistaken, but it is my impression that other covers do not have such small "minimal" repns, so that the analogues of theta correspondences are either not present at all, or don't have the multiplicity-one properties.
Another key-phrase is "first occurrence" in work of Kudla-Rallis and collaborators, examining in precise detail the "theta correspondences"' repn-theoretic properties. I am not aware of any comparable successes for other covers. Nevertheless, there is work of Brubaker-Bump-Friedberg on "multiple Dirichlet series" that looks for applications of more general covers.