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.
buddenma@mail.armstrong.edu
[Abstract-pdf]
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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