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 $S{p}_{2n}$. 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 $S{p}_{2n-r+1}$. 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 $S{p}_4$, 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.

23 Pages

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• Distinguished theta representations for certain covering groups