The B-orbits on a Hermitian symmetric variety in characteristic 2

TL;DR

This paper studies the classification and properties of Borel group orbits on Hermitian symmetric varieties over fields of characteristic 2, providing combinatorial descriptions and formulas for orbit dimensions.

Contribution

It offers a parametrization of Borel orbits on Hermitian symmetric varieties in characteristic 2 and characterizes the Bruhat order and orbit dimensions combinatorially.

Findings

Parametrization of Borel orbits on $G/L$ in characteristic 2.

Combinatorial characterization of Bruhat order for simply laced root systems.

Formula for computing orbit dimensions from combinatorial data.

Abstract

Let $G$ be a reductive linear algebraic group over an algebraically closed field $\mathbb{K}$ of characteristic $2$. Fix a parabolic subgroup $P$ such that the corresponding parabolic subgroup over $\mathbb{C}$ has abelian unipotent radical and fix a Levi subgroup $L\subseteq P$. We parametrize the orbits of a Borel $B\subseteq P$ over the Hermitian symmetric variety $G/L$ supposing the root system $\Phi$ is irreducible. For $\Phi$ simply laced we prove a combinatorial characterization of the Bruhat order over these orbits. We also prove a formula to compute the dimension of the orbits from combinatorial characteristics of their representatives.