Rational singularities of $G$-saturation

Let $G$ be a semisimple algebraic group defined over an algebraically closed field of characteristic 0 and $P$ be a parabolic subgroup of $G$. Let $M$ be a $P$-module and $V$ be 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 $R^i\text{ind}_P^G(-)$ satisfies certain vanishing condition then the variety $G\cdot V$ has rational singularities. This generalizes the main result of Kempf in [Invent. Math., 37 (1976), no. 3]. As an application, we prove the property of having rational singularities for nilpotent commuting varieties over $3\times 3$ matrices.

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