On the standard Poisson structure and a Frobenius splitting of the basic affine space
arXiv:1804.10815 · doi:10.1093/imrn/rnz179
The paper constructs a Frobenius splitting of the basic affine space G/U by exploiting the standard Poisson structure on G/U, develops a general theory of Frobenius splittings for torus‑equivariant Poisson varieties, and shows that compatibly split subvarieties must be torus‑Poisson subvarieties.
Abstract
The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,Ï_{G/U})$, 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 $Ï_{G/U}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,Ï_{G/U})$. 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 $G/U$.