The Aubert-Zelevinsky involution for $G_2$ and its associated Hecke algebras

Motivated by the recent work of Aubert-Xu and the techniques in G. Muic's article, we provide examples of computations of the Aubert-Zelevinsky duality functor for the principal and mediate series of the exceptional group $G_2$, and deduce corresponding results regarding the involution on the Hecke algebra side. These computations also allow us to confirm several instances of the Bernstein conjecture for $G_2$. This article is developed from part of the author's PhD thesis.