zbMATH — the first resource for mathematics

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 \(G_2\).

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
Full Text: DOI