A note on the arithmetic of residual automorphic representations of reductive groups

We prove an arithmeticity result for a class of cohomological residual automorphic representations of a general reductive group $G$. More precisely, we show that this class of residual representations is stable under the action of $\mathrm{Aut}(\mathbb{C})$. This complements numerous results on the stability of cohomological cuspidal automorphic representations due by several people. We conclude by showing that the rationality field of such a cohomological residual automorphic representation is a number field.

2010 Mathematics Subject Classification

Primary 11F70, 11F75, 22E47. Secondary 11F67.

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Accepted 8 July 2014

Published 13 April 2015