Abstract
Let F be a non-Archimedean local field of residual characteristic two and let d be an odd positive integer. Let D be a central F-division algebra of dimension d 2. Let π be one of: an irreducible smooth representation of D × , an irreducible cuspidal representation of GL d (F), an irreducible smooth representation of the Weil group of F of dimension d. We show that, in all these cases, if π is self-contragredient then it is defined over \({\mathbb Q}\) and is orthogonal. We also show that such representations exist.
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Bushnell, C.J., Henniart, G. Self-dual representations of some dyadic groups. Math. Ann. 351, 67–80 (2011). https://doi.org/10.1007/s00208-010-0592-5
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DOI: https://doi.org/10.1007/s00208-010-0592-5