Mathematical Research Letters

Volume 27 (2020)

Number 6

Lifting $G$-irreducible but $\mathrm{GL}_n$-reducible Galois representations

Pages: 1669 – 1696



Najmuddin Fakhruddin (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India)

Chandrashekhar Khare (Department of Mathematics, University of California, Los Angeles, Calif., U.S.A.)

Stefan Patrikis (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.; and Department of Mathematics, Ohio State University, Columbus, Oh., U.S.A.)


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