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 $G/U$.
Categories: math.AG