$\SL_3$ dual pairs

Source abstract

We study the exceptional theta correspondence for dual pairs of the form $\SL_3\times G$ inside $p$-adic groups of type $E_n$, $n=6,7,8$. We establish Howe duality for lifts of all Whittaker-generic unitarizable representations of $\SL_3$. Our dual pair is in a see-saw relationship with a dual pair of the form $G_2 \times H$ such that $G/H$ is a rank-two symmetric space. Using the theta correspondence for $\SL_3 \times G$, we prove a multiplicity-one result for $H$-invariant functionals on representations of $G$.