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Apologies for not having looked hard enough, but what I ask essentially follows from repeated application of the projective bundle formula in de Rham cohomology, at least for $G=GL_n$$G=\operatorname{GL}_n$.

Apologies for not having looked hard enough, but what I ask essentially follows from repeated application of the projective bundle formula in de Rham cohomology, at least for $G=GL_n$.

Apologies for not having looked hard enough, but what I ask essentially follows from repeated application of the projective bundle formula in de Rham cohomology, at least for $G=\operatorname{GL}_n$.

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Apologies for not having looked hard enough, but what I ask essentially follows from repeated application of the projective bundle formula in de Rham cohomology, at least for $G=GL_n$.