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Irreducibility and cuspidality. (English) Zbl 1302.11034

Kobayashi, Toshiyuki (ed.) et al., Representation theory and automorphic forms. Based on the symposium, Seoul, Korea, February 14–17, 2005. Basel: Birkhäuser (ISBN 978-0-8176-4505-2/hbk). Progress in Mathematics 255, 1-27 (2008).
Summary: Suppose \(\rho\) is an \(n\)-dimensional representation of the absolute Galois group of \(\mathbb Q\) which is associated, via an identity of \(L\)-functions, with an automorphic representation \(\pi\) of \(\mathrm{GL}(n)\) of the adèle ring of \(\mathbb Q\). It is expected that \(\pi\) is cuspidal if and only if \(\rho\) is irreducible, though nothing much is known in either direction in dimensions \(> 2\). The objective of this article is to show for \(n < 6\) that the cuspidality of a regular algebraic \(\pi\) is implied by the irreducibility of \(\rho\). For \(n < 5\), it suffices to assume that \(\pi\) is semi-regular.
For the entire collection see [Zbl 1124.11004].

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F80 Galois representations
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