## 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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