Theta functions on covers of symplectic groups



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 globally 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.