Quasi-admissible, raisable nilpotent orbits and covering Barbasch-Vogan duality

TL;DR

This paper determines the minimal cover degrees for nilpotent orbits in type E groups over p-adic fields and establishes their quasi-admissibility via covering Barbasch-Vogan duality.

Contribution

It extends the understanding of nilpotent orbit covers and quasi-admissibility for simply-connected Lie groups of type E over p-adic fields, linking geometry and duality.

Findings

Identifies the degree of cover needed for quasi-admissibility of nilpotent orbits.

Proves that orbits in the image of the Barbasch-Vogan duality map are always quasi-admissible.

Combines new results with previous data for other types to generalize the criteria.

Abstract

For simply-connected Lie groups of type E over \( p \)-adic local field \( F \), we determine the degree of the cover required for a given \( F \)-split nilpotent orbit to be quasi-admissible or raisable, respectively. Combining this result with the previously computed data for other types by Gao-Liu-Tsai, we prove that all \( F \)-split nilpotent orbits whose geometry type contained in the image of the covering Barbasch-Vogan duality map \( d_{\mathrm{BV},G}^{(n)} \) of almost-simple Lie groups \( G \) in each Cartan type are always\( \overline{G}^{(n)} \)-quasi-admissible.