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Endoscopic lifts from $$\mathrm{PGL}_3$$ to $$G_2$$. (English) Zbl 1071.22007
The authors almost completely determine the theta correspondence arising from the dual pair $$\mathrm{PGL}_3\times G_2\hookrightarrow F_6$$. Their first result determines the lift of any non-supercuspidal representation $$\pi$$ of $$\mathrm{PGL}_3$$. This lifting respects Langlands functoriality, i.e., the Gross-Savin conjecture is verified for each such $$\pi$$. Their second result shows that the theta lift $$\theta(\pi)$$ of a non-self-dual supercuspidal representation $$\pi$$ of $$\mathrm{PGL}_3$$ is an irreducible generic supercuspidal representation of $$G_2$$.

##### MSC:
 22E35 Analysis on $$p$$-adic Lie groups 22E50 Representations of Lie and linear algebraic groups over local fields 11F70 Representation-theoretic methods; automorphic representations over local and global fields
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