On embeddings of certain spherical homogeneous spaces in prime   characteristic
  Let $\mc G$ be a reductive group over an algebraically closed field of characteristic $p>0$. We study homogeneous $\mc G$-spaces that are induced from the $G\times G$-space $G$, $G$ a suitable reductive group, along a parabolic subgroup of $\mc G$. We show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen-Macaulay.   Examples are the $G\times G$-orbits in normal reductive monoids with unit group $G$. Our class of homogeneous spaces also includes the open orbits of the well-known determinantal varieties and the varieties of (circular) complexes.   We also show that all $G$-orbit closures in a spherical variety which is canonically Frobenius split are normal.   Finally we study the Gorenstein property for the varieties of circular complexes and for a related reductive monoid. 
Rudolf Tange
math.AG math.GR