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Let |$G$| be a connected complex semi-simple Lie group and |${\mathcal {B}}$| its flag variety. For every positive integer |$n$|⁠ , we introduce a Poisson groupoid over |${{\mathcal {B}}}^n$|⁠ , called the |$n$| th total configuration Poisson groupoid of flags of |$G$|⁠ , which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in |${\mathcal {B}}^n$|⁠. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to |$G$|⁠. [ABSTRACT FROM AUTHOR]

The goal of this paper is to construct a Frobenius splitting on |$G/U$| via the Poisson geometry of |$(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$|⁠ , 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 |$\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}}$| is the standard Poisson structure on |$G/U$|⁠. We first study the Poisson geometry of |$(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$|⁠. 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 subvarieties. Lastly, we apply our general theory to construct a Frobenius splitting on |$G/U$|⁠. [ABSTRACT FROM AUTHOR]