Non-generic unramified representations in metaplectic covering groups

Abstract

Let G^(r) denote the metaplectic covering group of the linear algebraic group G. In this paper we study conditions on unramified representations of the group G^(r) not to have a nonzero Whittaker function. We state a general Conjecture about the possible unramified characters χ such that the unramified subrepresentation of $Ind_{{B^{\left( r \right)}}}^{{G^{\left( r \right)}}}{X^{\delta _B^{1/2}}}$ will have no nonzero Whittaker function. We prove this Conjecture for the groups GL ^{( r)} _n with r ≥ n − 1, and for the exceptional groups G ^{( r)}₂ when r ≠ 2.