The Zelevinsky-Aubert duality for classical groups (Automorphic forms, Automorphic representations, Galois representations, and its related topics)

Abstract

In 1980, Zelevinsky [14] studied the representation theory of p-adic general linear groups. He introduced an involution on the Grothendieck group of smooth representations of finite length, which exchanges the trivial representation with the Steinberg representation. In fact, he conjectured that it preserves the irreducibility. Aubert [5] extended this involution to p-adic reductive groups, which is now called the Zelevinsky-Aubert duality. It is expected that this duality preserves the unitarity. In this article, based on the joint work with Alberto Mfnguez [3], we give an algorithm to compute the Zelevinsky-Aubert duality for odd special orthogonal groups or symplectic groups.