Dihedral long root A-packets of \(p\)-adic \(G_2\) via theta correspondence (Q6608018)

Let \(G\) stand for a split exceptional group of type \(G_2\) over a number field \(F\). Then the \(L\)-group of \(G\) can be replaced by \(G(\mathbb{C})\), and there are four different conjugacy classes of morphisms \(\mathrm{SL}_2(\mathbb{C}) \rightarrow G(\mathbb{C})\). It is also known that such classes give rise to families of different nontempered A-parameters \(\psi\) for \(G_2\).\N\NIn the paper under the review, the authors study the case when \(\psi|_{\mathrm{SL}_2(\mathbb{C})}\) gives the long root \(\mathrm{SL}_2\) in \(G_2\). They provide a construction of the local nonarchimedean A-packets associated to dihedral long root A-parameters, using a theta lift from \(\mathrm{PU}_3 \rtimes \mathbb{Z} / 2\mathbb{Z}\) to \(G_2\) arising from the exceptional theta correspondence for the dual pair \((\mathrm{PU}_3 \rtimes \mathbb{Z} / 2\mathbb{Z} ) \times G_2\).