Local Coefficient Matrices of Metaplectic Groups

The principal series representations of the n-fold metaplectic covers of the general linear group GLr(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 GL2(F), where 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 ζ-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 GL2(F) is intimately related to the irreducibility of its d-fold counterpart.