Nilpotent Orbits of Orthogonal Groups over p-adic Fields, and the DeBacker Parametrization

  • Tobias Bernstein
  • Jia-Jun Ma
  • Monica NevinsEmail author
  • Jit Wu Yap


For local non-archimedean fields k of sufficiently large residual characteristic, we explicitly parametrize and count the rational nilpotent adjoint orbits in each algebraic orbit of orthogonal and special orthogonal groups. We separately give an explicit algorithmic construction for representatives of each orbit. We then, in the general setting of groups GLn(D), SLn(D) (where D is a central division algebra over k) or classical groups, give a new characterisation of the “building set” (defined by DeBacker) of an \(\mathfrak {sl}_{2}(k)\)-triple in terms of the building of its centralizer. Using this, we prove our construction realizes DeBacker’s parametrization of rational nilpotent orbits via elements of the Bruhat-Tits building.


p-adic groups Nilpotent orbits DeBacker classification Quadratic forms Bruhat-Tits buildings 

Mathematics Subject Classification (2010)

20G25 (17B08, 17B45) 


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.School of Mathematical SciencesShanghai Jiao Tong UniversityShanghaiChina
  3. 3.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  4. 4.Department of MathematicsNational University of SingaporeSingaporeSingapore

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