Triality and adjoint lifting for GL(3)

Using the stable twisted trace formula for the triality automorphism, we show the adjoint lifting (to GL(8)) of cuspidal representations of GL(3) with a discrete series local component. We also describe the possible isobaric decompositions of the resulting automorphic representations on GL(8) and give an application towards Ramanujan bounds for GL(3).

💡 Research Summary

The paper “Triality and adjoint lifting for GL(3)” investigates a new instance of Langlands functoriality that arises from the triality outer automorphism of the split simply‑connected group Spin 8 (equivalently, its adjoint form PGSO₈). The authors combine the stable twisted trace formula of Moeglin–Waldspurger with explicit descriptions of the triality automorphism to construct the adjoint lift of a cuspidal automorphic representation π of GL 3 to an automorphic representation Ad(π) of GL 8, under the modest hypothesis that π has a discrete‑series component at some finite place.

The paper begins with a detailed algebraic description of triality. Using the theory of symmetric composition algebras, two split 8‑dimensional algebras are presented: the para‑octonion algebra and a 3×3 trace‑zero matrix algebra M equipped with a twisted product. The automorphism group of M is PGL₃, and the induced map \