  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.
