On the Standard Poisson Structure and a Frobenius Splitting of the Basic Affine Space

Abstract

The goal of this paper is to construct a Frobenius splitting on G/U via the Poisson geometry of (G/U, π G/U ), where G is a simply-connected semi-simple algebraic group defined over an algebraically closed field of characteristic p > 3, U is the uniradical of a Borel subgroup of G and π G/U is the standard Poisson structure on G/U . We first study the Poisson geometry of (G/U, π G/U ). Then, we develop a general theory for Frobenius splittings on T-Poisson varieties, where T is an algebraic torus. In particular, we prove that compatibly split subvarieties of Frobenius splittings constructed in this way must be T-Poisson subvarieties. Lastly, we apply our general theory to construct a Frobenius splitting on G/U . 0