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

TL;DR

This paper explicitly parametrizes and constructs representatives for rational nilpotent orbits in orthogonal groups over large residual characteristic p-adic fields, linking DeBacker's parametrization with Bruhat-Tits building theory.

Contribution

It provides an explicit algorithmic construction of nilpotent orbit representatives and a new characterization of DeBacker's building set in classical groups.

Findings

Explicit parametrization of rational nilpotent orbits in orthogonal groups.

Algorithmic construction of orbit representatives.

Proof that the construction aligns with DeBacker's parametrization.

Abstract

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 $\mathrm{GL}_n(D)$, $\mathrm{SL}_n(D)$ (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.