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Mathematical Research Letters
Volume 27 (2020)
Number 6
Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations
Pages: 1669 – 1696
DOI: https://dx.doi.org/10.4310/MRL.2020.v27.n6.a4
Authors
Abstract
In recent work, the authors proved a general result on lifting $G$-irreducible odd Galois representations $\operatorname{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_\ell)$, with $F$ a totally real number field and $G$ a reductive group, to geometric $\ell$‑adic representations. In this note we take $G$ to be a classical group and construct many examples of $G$-irreducible representations to which these new lifting methods apply, but to which the lifting methods currently provided by potential automorphy theorems do not.
We are grateful to Wushi Goldring for stimulating conversations. N.F. was supported by the DAE, Government of India, project no. RTI4001. C.K. would like to thank TIFR, Mumbai for its hospitality, in periods when some of the work was carried out. S.P. was supported by NSF grants DMS-1700759 and DMS-1752313. We also thank the referees for their comments and corrections which helped to improve the exposition.
Received 5 May 2020
Accepted 20 July 2020
Published 17 February 2021