Algorithms on the Pyasetskii involution on local Langlands parameters of classical groups

TL;DR

This paper presents an algorithm to compute the Pyasetskii involution for classical groups, combining existing algorithms and providing a geometric interpretation for certain cases.

Contribution

It introduces a new algorithm for the Pyasetskii involution on classical groups and offers a geometric perspective on the bad parity case.

Findings

Algorithm successfully computes the Pyasetskii involution for specified groups.

Combines Moeglin-Waldspurger and Lanard-Mínguez algorithms.

Provides geometric interpretation for bad parity representations.

Abstract

We give an algorithm to compute the Pyasetskii involution for $\mathrm{Sp}_{2n}$, $\mathrm{SO}_{2n+1}$ and $\mathrm{O}_{2n}$. The algorithm is a combination of Moeglin-Waldspurger's algorithm for the Pyasetskii involution for $\mathrm{GL}_n$ ([MW86]) and Lanard-M${\'i}$nguez's algorithm for the Aubert-Zelevinsky involution of bad parity representations for classical groups ([LM25]). In particular, we give a geometric interpretation of the bad parity case of Lanard-M${\'i}$nguez's algorithm.