Theta functions on covers of symplectic groups

We study the automorphic theta representation ⊖^(r) _2n on the r-fold cover of the symplectic group Sp_2nThis representation is ob- tained from the residues of Eisenstein series on this group. If r is odd, n ≤ r < 2n, then under a natural hypothesis on the theta representations, we show that ⊖^(r) _2n may be used to construct a globally generic representation σ^(2r) _2n-r+1 on the 2r-fold cover of Sp_2n-r+1Moreover, when r = n the Whittaker functions of this representation attached to factorizable data are factorizable, and the unramified local factors may be computed in terms of n-th order Gauss sums. If n = 3 we prove these results, which in that case pertain to the six-fold cover of Sp₄, unconditionally. We expect that in fact the representation constructed here, σ^(2r) _2n-r+1, is precisely ⊖^(2r) _2n-r+1; that is, we conjecture relations between theta representations on different covering groups.