Theta functions on covers of symplectic groups

Abstract. We study the automorphic theta representation Θ(r) 2n on the r-fold cover of the symplectic group Sp2n. This representation is obtained 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 Θ(r) 2n may be used to construct a globally generic representation σ (2r) 2n−r+1 on the 2r-fold cover of Sp2n−r+1. Moreover, 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 Θ(2r) 2n−r+1; that is, we conjecture relations between theta representations on different covering groups.