Frobenius splitting and geometry of $G$-Schubert varieties

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    Let X be an equivariant embedding of a connected reductive group G over an algebraically closed field k of positive characteristic. Let B denote a Borel subgroup of G. A G-Schubert variety in X is a subvariety of the form diag(G)dot operatorV, where V is a B×B-orbit closure in X. In the case where X is the wonderful compactification of a group of adjoint type, the G-Schubert varieties are the closures of Lusztig's G-stable pieces. We prove that X admits a Frobenius splitting which is compatible with all G-Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any G-Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that G-Schubert varieties have nice singularities we present an example of a nonnormal G-Schubert variety in the wonderful compactification of a group of type G2. Finally we also extend the Frobenius splitting results to the more general class of View the MathML source-Schubert varieties.
    Original languageEnglish
    JournalAdvances in Mathematics
    Pages (from-to)1469-1512
    Number of pages44
    Publication statusPublished - 2008


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