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

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