On the standard Poisson structure and a Frobenius splitting of the basic affine space

TL;DR

This paper constructs a Frobenius splitting on the basic affine space $G/U$ using Poisson geometry, developing a general theory for Frobenius splittings on $ ext{T}$-Poisson varieties and applying it to $G/U$.

Contribution

It introduces a novel method to obtain Frobenius splittings via Poisson structures and develops a general framework for $ ext{T}$-Poisson varieties, with applications to affine spaces.

Findings

Established a Frobenius splitting on $G/U$ using Poisson geometry.

Proved that compatibly split subvarieties are $ ext{T}$-Poisson sub-varieties.

Developed a general theory for Frobenius splittings on $ ext{T}$-Poisson varieties.

Abstract

The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,\pi_{G/U})$, where $G$ is a semi-simple algebraic group of classical type defined over an algebraically closed field of characteristic $p > 3$, $U$ is the uniradical of a Borel subgroup of $G$ and $\pi_{G/U}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,\pi_{G/U})$. Then, we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvarieties of Frobenius splittings constructed in this way must be $\mathbb{T}$-Poisson sub-varieties. Lastly, we apply our general theory to construct a Frobenius splitting on $G/U$.