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

We give an algorithm to compute the Pyasetskii involution for ${\mathrm{S}\mathrm{p}}_{2n}$, ${\mathrm{S}\mathrm{O}}_{2n+1}$ and ${\mathrm{O}}_{2n}$. The algorithm is a combination of Moeglin-Waldspurger’s algorithm for the Pyasetskii involution for ${\mathrm{G}\mathrm{L}}_n$ ([MW86]) and Lanard-M$\acute{\mathrm{ı}}$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$\acute{\mathrm{ı}}$nguez’s algorithm.