Nilpotent orbits of orthogonal groups over p-adic fields, and the\n DeBacker parametrization

2018/08/10 by Tobias Bernstein, Jiajun Ma, Bernstein, Tobias +5
Computer Science · Mathematics · #17B45) #20G25 (17B08 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1808.03593

openalex publication_date 2018/08/10 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

Abstract

For local non-archimedean fields k of sufficiently large residual\ncharacteristic, we explicitly parametrize and count the rational nilpotent\nadjoint orbits in each algebraic orbit of orthogonal and special orthogonal\ngroups. We separately give an explicit algorithmic construction for\nrepresentatives of each orbit. We then, in the general setting of groups\n\GL_n(D), \SL_n(D) (where D is a central division algebra\nover k) or classical groups, give a new characterisation of the "building\nset" (defined by DeBacker) of an mathfraksl₂(k)-triple in terms of the\nbuilding of its centralizer. Using this, we prove our construction realizes\nDeBacker's parametrization of rational nilpotent orbits via elements of the\nBruhat-Tits building.\n

Related

• On Rational Nilpotent Orbits of SL_n and Sp_2n over a Local Non-Archimedean Field
• Parametrizing nilpotent orbits via Bruhat-Tits theory
• Computing with nilpotent orbits in simple Lie algebras of exceptional type
• Parametrizing nilpotent orbits in p-adic symmetric spaces using Bruhat-Tits theory
• Nilpotent orbits: finiteness, separability and Howe's conjecture