A note on the Global Langlands Conjecture

The theory of base change is used to give some new examples of the Global Langlands Conjecture. The Galois representations involved have solvable image and are not monomial, although some multiple of them in the Grothendieck group is monomial. Thus, it gives nothing new about Artin's Conjecture itself. An application is given to a question which arises in studying multiplicities of cuspidal representations of $SL_n$. We explain how the (conjectural) adjoint lifting can prove GLC for a family of representations containing the tetrahedral $2$-dimensional ones.

1991 Mathematics Subject Classification: Primary 11R39;Secondary 11F70, 22E55.

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