Reducibility points for representations induced from cuspidal ones
type: Item
instance of: scholarly article
title: Reducibility points for representations induced from cuspidal ones
zbMATH Open document ID: 1028.22016
DOI: 10.1016/S0021-8693(03)00172-8
published in: Journal of Algebra
publication date: 2003-10-22T00:00:00Z
review text: Let \(G(n)\) be a symplectic or orthogonal group of rank \(n\) defined over a \(p\)-adic field \(F\). Let \(\pi\) be an irreducible cuspidal representation of \(G(n,F)\). The subject of the paper is the reducibility of the representations \(\rho\times\pi\), where \(\rho\) is an irreducible cuspidal representation of some \(\text{GL}(d,F)\) and \(\rho\times\pi\) denotes the representation of \(G(n+d,F)\) parabolically induced from the representation \(\rho\otimes\pi\) of \(\text{GL}(d,F)\times G(n,F)\) (\(G(n+d)\) is obtained from \(G(n)\) by adding \(d\) hyperbolic planes to the vector space on which \(G(n)\) acts). There are results concerning the reducibility points of \(\rho||^x \times\pi\) (\(\rho\) unitary, \(x\in {\mathbb R}\)). The connection with Jordan blocks, defined in \textit{C. Moeglin} [J. Eur. Math. Soc. 4, 143-200 (2002; Zbl 1002.22009)], is discussed. A particular case is treated, in which reducibility for some \(x > 0, x \neq {1\over 2}\), occurs only if \(\rho\) is a quadratic character. Then \(\pi\) is a ``quadratic unipotent representation''. An important tool in the proofs is the wave front of representations.
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 1995423
zbMATH Keywords: p-adic groups
zbMATH Keywords: Jordan blocks
zbMATH Keywords: reducibility
zbMATH Keywords: irreducible cuspidal representation
author: Colette Moeglin
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1995423