On an algorithm to compute derivatives

TL;DR

This paper extends Jantzen's algorithm to compute highest derivatives of irreducible representations of certain p-adic groups, aiding in understanding their Langlands data and duals.

Contribution

We complete Jantzen's algorithm for computing derivatives of irreducible representations of p-adic orthogonal and symplectic groups, providing new tools for representation analysis.

Findings

Extended Jantzen's algorithm successfully computes derivatives.

Examples of Langlands data for Aubert duals are provided.

Applications in integral reducibility cases are demonstrated.

Abstract

In this paper, we complete Jantzen's algorithm to compute the highest derivatives of irreducible representations of $p$-adic odd special orthogonal groups or symplectic groups. As an application, we give some examples of the Langlands data of the Aubert duals of irreducible representations, which are in the integral reducibility case.