Abstract:Let $F$ be a locally compact non-Archimedean field of characteristic $0$, and let $G$ be either the split special orthogonal group $\mathrm{SO}_{2n+1}(F)$ or the symplectic group $\mathrm{Sp}_{2n}(F)$. The goal of this paper is to give an explicit description of the Aubert-Zelevinsky duality for $G$ in terms of Langlands parameters. We present a new algorithm, inspired by the Moeglin-Waldspurger algorithm for $\mathrm{GL}_n(F)$, which computes the dual Langlands data in a recursive and combinatorial way. Our method is simple enough to be carried out by hand and provides a practical tool for explicit computations. Interestingly, the algorithm was discovered with the help of machine learning tools, guiding us toward patterns that led to its formulation.
| Comments: | 86 pages. Added a determinant = 1 condition in the definition of symmetrical multisegment |
| Subjects: | Representation Theory (math.RT); Number Theory (math.NT) |
| Cite as: | arXiv:2509.13231 [math.RT] |
| (or arXiv:2509.13231v2 [math.RT] for this version) | |
| https://doi.org/10.48550/arXiv.2509.13231 arXiv-issued DOI via DataCite |
From: Thomas Lanard [view email]
[v1]
Tue, 16 Sep 2025 16:38:35 UTC (80 KB)
[v2]
Tue, 26 May 2026 14:04:15 UTC (81 KB)
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