Geometry of $B \times B$-orbit closures in equivariant embeddings
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Let $X$ denote an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$. Let $B$ denote a Borel subgroup of $G$ and let $Z$ denote a $B \times B$-orbit closure in $X$. When the characteristic of $k$ is positive and $X$ is projective we prove that $Z$ is globally $F$-regular. As a consequence, $Z$ is normal and Cohen-Macaulay for arbitrary $X$ and arbitrary characteristics. Moreover, in characteristic zero it follows that $Z$ has rational singularities. This extends earlier results by the second author and M. Brion.
23 pages, revised version. Minor problem with definition of $\mathcal I$ in Section 5.3 resolved
23 pages, revised version. Minor problem with definition of $\mathcal I$ in Section 5.3 resolved