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Fourier Coefficients for Theta Representations on Covers of General Linear Groups

We show that the theta representations on certain covers of general linear groups support certain types of unique functionals. The proof involves two types of Fourier coefficients. The first are semi-Whittaker coefficients, which generalize coefficients introduced by Bump and Ginzburg for the double cover. The covers for which these coefficients vanish identically (resp. do not vanish for some choice of data) are determined in full. The second are the Fourier coefficients associated with general unipotent orbits. In particular, we determine the unipotent orbit attached, in the sense of Ginzburg, to the theta representations.