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Journal of Lie Theory 16 (2006), No. 2, 239--249
Copyright Heldermann Verlag 2006

Local Coefficient Matrices of Metaplectic Groups

Mark Budden
Dept. of Mathematics, Armstrong Atlantic State University, 11935 Abercorn Street, Savannah, GA 31419, U.S.A.


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.

Keywords: Principal series, automorphic forms, Shimura's correspondence.

MSC: 22D30, 11F32; 11F70, 11F85

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