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Some irreducibility theorems of parabolic induction on the metaplectic group via the Langlands-Shahidi method

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Abstract

Let \(\overline {S{p_{2n}}({\rm{<Emphasis FontCategory="NonProportional">F</Emphasis>}})} \) be the metaplectic double cover of F where F is a local field of characteristic 0. We use the Uniqueness of Whittaker model to define a metaplectic analog to Shahidi local coefficients and we use these coefficients to define gamma factors. We show that these gamma factors are multiplicative and satisfy the crude global functional equation. Then, we compute these factors in various cases and obtain explicit formulas for Plancherel measures. These computations are then used to prove some irreducibility theorems for parabolic induction on the metaplectic group over p-adic fields. In particular, we show that all principal series representations induced from unitary characters are irreducible. We also prove that parabolic induction from unitary supercuspidal representation of the Siegel parabolic sub group is irreducible if and only if a certain parabolic induction on F is irreducible.

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  1. The Center for Advanced Studies in Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva, 84105, Israel

    Dani Szpruch

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Dani Szpruch

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Szpruch, D. Some irreducibility theorems of parabolic induction on the metaplectic group via the Langlands-Shahidi method. Isr. J. Math. 195, 897–971 (2013). https://doi.org/10.1007/s11856-012-0140-y

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