## Abstract

We study the automorphic theta representation ⊖^{(r)} _{2n} on the r-fold cover of the symplectic group Sp_{2n}This representation is ob- tained 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 ⊖^{(r)} _{2n} may be used to construct a globally generic representation σ^{(2r)} _{2n-r+1} 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, σ^{(2r)} _{2n-r+1}, is precisely ⊖^{(2r)} _{2n-r+1}; that is, we conjecture relations between theta representations on different covering groups.

Original language | English |
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Pages (from-to) | 89-116 |

Number of pages | 28 |

Journal | BULLETIN OF THE IRANIAN MATHEMATICAL SOCIETY |

Volume | 43 |

Issue number | 4 Special Issue |

State | Published - Aug 2017 |

## Keywords

- Descent integral
- Generic representation
- Metaplectic cover
- Symplectic group
- Theta representation
- Unipotent orbit
- Whittaker function

## All Science Journal Classification (ASJC) codes

- General Mathematics