  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.
