Rational singularities of $G$-saturation

Abstract

Let $G$ be a semisimple algebraic group defined over an algebraically closed field of characteristic 0 and $P$ a parabolic subgroup of $G$. Let $M$ be a $P$-module and $V$ a $P$-stable closed subvariety of $M$. We show in this paper that, if the varieties $V$ and $G\cdot M$ have rational singularities, and the induction functor $R^i ind _P^G(-)$ satisfies certain vanishing conditions, then the variety $G\cdot V$ has rational singularities. This generalizes a result of Kempf on the collapsing of homogeneous bundles. As an application, we prove the property of having rational singularities for nilpotent commuting varieties over $3\times 3$ matrices.