## Frobenius splitting and geometry of G-Schubert varieties

Publikation: Working paperForskning

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.