Aubert duals of strongly positive representations for metaplectic groups

TL;DR

This work provides an explicit description of the Aubert duals for strongly positive representations of the metaplectic group $ ilde{Sp}(n)$ over a non-Archimedean field by leveraging Matić’s classification and a detailed analysis of Jacquet modules. The duals are described as Langlands subrepresentations of induced representations built from precise $ ext{GL}$-segment data (the $ ext{delta}$-segments) and a cuspidal component, with clear formulas for both the special and general cases. The authors also extend the same method to odd general spin groups $GSpin(2n+1)$, demonstrating a parallel structure and reinforcing the role of Aubert duality as a tool in understanding unitary representations for non-linear covering groups. This work thus bridges classical group results to metaplectic settings and lays groundwork for future unitary dual constructions in covering groups.

Abstract

We determine the Aubert duals of strongly positive representations of the metaplectic group \(\widetilde{Sp}(n)\) over a non-Archimedean local field $F$ of characteristic different from two. Using the classification of Matić and an explicit analysis of Jacquet modules, we describe these duals in terms of precise inducing data. Our results extend known descriptions for classical groups to the metaplectic groups case and clarify the role of Aubert duality for non-linear covering groups, providing a foundation for future applications to the study of unitary representations for those cases. Furthermore, We are able to show that the same method applies to odd general spin groups $GSpin(2n+1)$, yielding an explicit description of Aubert duals in that setting as well.

Theorems & Definitions (21)

• Theorem 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 4.1
• proof
• Lemma 4.2: M13