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

TL;DR

This paper computes the Aubert-Zelevinsky duality for the exceptional group G_2, linking it to Hecke algebra involutions and confirming parts of the Bernstein conjecture through explicit examples.

Contribution

It provides explicit computations of the Aubert-Zelevinsky duality for G_2's principal and mediate series, connecting these to Hecke algebra involutions and verifying aspects of the Bernstein conjecture.

Findings

Computed Aubert-Zelevinsky duality for G_2 series

Established correspondence with Hecke algebra involutions

Confirmed several instances of the Bernstein conjecture for G_2

Abstract

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.