Finite-reductive dual pairs in \(G_2\) (Q5956244)

Barchini and Sepanski show that, up to conjugation, all dual pairs \((F,G')\) in complex \(G_2\), where \(F\) is finite and \(G'\) is infinite complex reductive, are only the following NEWLINE\[NEWLINE ({\mathbb{Z}}_2,SL(2,{\mathbb{C}})\times_{{\mathbb{Z}}_2} SL(2,{\mathbb{C}})),\quad (S_3,SO(3,{\mathbb{C}})),\quad ({\mathbb{Z}}_3,SL(3,{\mathbb{C}})), NEWLINE\]NEWLINE where \(S_3\) is the symmetric group on three letters. They realize \(G_2\) by means of the Cayley algebra and, for each nontrivial nonabelian semisimple subgroup \(G_0\), there are six, of \(G_2\), they calculate \(N_{G_2}(G_0)\) straightforwardly. For the remaining exceptional cases, they announce an approach which reduces the regular case to manageable calculations.