Theta Functions on Covers of Symplectic Groups

Abstract

We study the automorphic theta representation $\Theta_{2n}^{(r)}$ on the $r$-fold cover of the symplectic group $Sp_{2n}$. This representation is obtained from the residues of Eisenstein series on this group. If $r$ is odd, $n\le r <2n$, then under a natural hypothesis on the theta representations, we show that $\Theta_{2n}^{(r)}$ may be used to construct a generic representation $\sigma_{2n-r+1}^{(2r)}$ on the $2r$-fold cover of $Sp_{2n-r+1}$. 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 $Sp_4$, unconditionally. We expect that in fact the representation constructed here, $\sigma_{2n-r+1}^{(2r)}$, is precisely $\Theta_{2n-r+1}^{(2r)}$; that is, we conjecture relations between theta representations on different covering groups.