In this article we define and study the Shahidi local coefficients matrix associated with a genuine principal series representation I (σ) of an n-fold cover of p-adic SL 2 (F) and an additive character ψ. The conjugacy class of this matrix is an invariant of the inducing representation σ and ψ and its entries are linear combinations of Tate or Tate type γ-factors. We relate these entries to functional equations associated with linear maps defined on the dual of the space of Schwartz functions. As an application we give new formulas for the Plancherel measures and use these to relate principal series representations of different coverings of SL 2 (F). While we do not assume that the residual characteristic of F is relatively prime to n we do assume that n is not divisible by 4.
|Number of pages||43|
|State||Published - 5 May 2019|
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Acknowledgements. We would like to thank Fan Gao for his remarks on earlier versions of the manuscript. We have essentially an equal role in writing Sect. 4.1. We would also like to thank Alexander Burstein, Francois Ramaroson, Sankar Sitaraman and Valentin Buciumas for useful discussions on the subject matter. Finally we would like to thank the referee for numerous suggestions which significantly improved the style and clarity of this paper. At the time this manuscript was prepared, the author was partially supported by a Simons Foundation Collaboration Grant 426446.