MathSciDoc: An Archive for Mathematician ∫

Let X be an equivariant embedding of a connected reductive group X over an algebraically closed field X of positive characteristic. Let X denote a Borel subgroup of X . A X -Schubert variety in X is a subvariety of the form $\diag (G)\cdot V $, where X is a X -orbit closure in X . In the case where X is the wonderful compactification of a group of adjoint type, the X -Schubert varieties are the closures of Lusztig's X -stable pieces. We prove that X admits a Frobenius splitting which is compatible with all X -Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any X -Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that X -Schubert varieties have nice singularities we present an example of a non-normal X -Schubert variety in the wonderful compactification of a group of type X . Finally we also extend the Frobenius splitting results to the more general class of X -Schubert varieties.

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