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We define a notion of Hodge modules with rational singularities. A varietyhas rational singularities in the usual sense, if it is normal and the Hodgemodule related to intersection cohomology has rational singularities in thepresent sense. Our main result is a generalization of Boutot's theorem that ifa reductive group acts on an affine variety with a stable point, and $H$ is anequivariant Hodge module with rational singularities, then the induced moduleon the GIT quotient also has rational singularities.

Equivariant Hodge modules and rational singularities