Journal of Lie Theory

The principal series representations of the $n$-fold metaplectic covers of the general linear group $\rm{GL}_r (\Bbb F)$ were described in the foundational paper ``Metaplectic Forms,'' by Kazhdan and Patterson (1984). In this paper, we study the local coefficient matrices for a certain class of principal series representations over $\rm{GL}_{2} (\Bbb F)$, where $\Bbb F$ is a nonarchimedean local field. The local coefficient matrices can be described in terms of the intertwining operators and Whittaker functionals associated to such representations in a standard way. We characterize the nonsingularity of local coefficient matrices in terms of the nonvanishing of certain local $\zeta$-functions by computing the determinant of the local coefficient matrices explicitly. Using these results, it can be shown that for any divisor $d$ of $n$, the irreducibility of the given principal series representation on the $n$-fold metaplectic cover of $\rm{GL}_2 (\Bbb F)$ is intimately related to the irreducibility of its $d$-fold counterpart.