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Is there any literature about mixed Hodge structure on $H^*(GL(n,\mathbb{C}))$ ? I think it is a vary basic problem in mathematics, but there are no literature about it?

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  • $\begingroup$ is there some specific question with a specific answer you have about this Hodge structure? $\endgroup$
    – Nguyen
    Commented Nov 11, 2020 at 6:12

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Yes, there is literature about it, it is Théorème 9.1.5 of Deligne's Hodge III. He gives a precise description more generally for a connected algebraic group $G$. In brief: from general Hopf algebra theory one knows that $H^\ast(G,\mathbf Q)$ is an exterior algebra on a finite set of odd degree generators. Deligne shows that this is also true for the mixed Hodge structure: as a mixed Hodge structure the cohomology is an exterior algebra, and the generators are purely of Tate type, with the generators in degree $2i-1$ of Hodge type $(i,i)$.

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