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Frobenius splitting and geometry of $G$-Schubert varieties

By Xuhua He and Jesper Funch Thomsen
Preprints
No. 03, April 2007
Abstract:
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 $\mathrm{diag}(G) \cdot V$, where $V$ is a $B \times B$-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 that compatibly splits all the $G$-Schubert varieties. Moreover, any $G$-Schubert variety admits stable Frobenius splittings along ample divisors in case X is projective. Although this indicates that $G$-Schubert varieties have nice singularities we give an example, in the wonderful compactification of a group of adjoint type, which is not normal. Finally we also extend the Frobenius splitting results to the more general class of $R$-Schubert varieties.
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