Theta Functions on Covers of Symplectic Groups

TL;DR

This paper investigates automorphic theta representations on covers of symplectic groups, establishing their properties and connections, including explicit computations of local factors and conjectured relations between different theta representations.

Contribution

It introduces new constructions of generic representations from theta representations on covering symplectic groups and computes their local factors explicitly.

Findings

Factorizable Whittaker functions for certain representations

Explicit unramified local factors in terms of Gauss sums

Conditional results for odd r with n ≤ r < 2n

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.