The explicit Zelevinsky-Aubert duality

TL;DR

This paper presents an explicit algorithm for computing the Zelevinsky-Aubert dual of irreducible representations of p-adic symplectic and odd special orthogonal groups, with formulas and criteria for irreducibility.

Contribution

It introduces a new explicit, computable algorithm and formulas for derivatives and socles, advancing understanding of duality in p-adic representation theory.

Findings

Provided explicit formulas for derivatives and socles.

Developed a combinatorial criterion for irreducibility.

Established an explicit algorithm for the Zelevinsky-Aubert dual.

Abstract

In this paper, we give an explicit computable algorithm for the Zelevinsky-Aubert dual of irreducible representations of $p$-adic symplectic and odd special orthogonal groups. To do this, we establish explicit formulas for certain derivatives and socles. We also give a combinatorial criterion for the irreducibility of certain parabolically induced representations.