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Gamma Factors, Root Numbers, and Distinction
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- Journal:
- Canadian Journal of Mathematics / Volume 70 / Issue 3 / 01 June 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 683-701
- Print publication:
- 01 June 2018
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- Article
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We study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of
$p$-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.
On Induced Representations Distinguished by Orthogonal Groups
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- Journal:
- Canadian Mathematical Bulletin / Volume 56 / Issue 3 / 01 September 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 647-658
- Print publication:
- 01 September 2013
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Let
$F$ be a local non-archimedean field of characteristic zero. We prove that a representation of 
$GL\left( n,\,F \right)$ obtained from irreducible parabolic induction of supercuspidal representations is distinguished by an orthogonal group only if the inducing data is distinguished by appropriate orthogonal groups. As a corollary, we get that an irreducible representation induced from supercuspidals that is distinguished by an orthogonal group is metic.
