$\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\GL{\operatorname{GL}}$This question is about an assertion in [Mixed Hodge polynomials of character varieties](https://arxiv.org/abs/math/0612668), by Hausel and Rodriguez-Villegas. Fix positive integers $g$ and $n$ and let $\zeta$ be a primitive $n$-th root of unity.


Let
$$H = \left\{ (A\_1, A\_2, \ldots, A\_g, B\_1, B\_2, \ldots, B\_g) \in \GL\_n^{2g}(\CC) : \prod\_{i=1}^g (A\_i B\_i A\_i^{-1} B\_i^{-1}) = \zeta \cdot\mathrm{Id}\_n \right\}.$$
Let $M = H/{\GL\_n(\CC)}$, acting by conjugation, and let $\tilde{M} = (\CC^{\ast})^{2g} \backslash M$ acting by rescaling the $A\_i$ and $B\_i$.


At the top of page 4, Hausel and Rodriguez-Villegas say that $M$ is "cohomologically a product" of $(\CC^{\ast})^{2g}$ and $\tilde{M}$ which I interpret to mean that, although the $(\CC^{\ast})^{2g}$-bundle $M \to \tilde{M}$ may not be trivial, we nonetheless have
$$H^{\ast}(M, \ZZ) \cong H^{\ast}(\tilde{M}, \ZZ) \otimes\_{\ZZ} H^{\ast}((\CC^{\ast})^{2g}, \ZZ).$$
Conveniently, $H^{\ast}((\CC^{\ast})^{2g}, \ZZ)$ is torsion free, so there are no $\operatorname{Tor}\_1$ terms in Künneth.


This is not obvious to me. Why is it true? For what more general versions of character varieties is it true?



Why is the $\operatorname{GL}_n$ character variety "cohomologically" the product of the $\operatorname{PGL}_n$ character variety and a torus?
You need to keep reading to the proof of Theorem 2.2.12.


The main point is that the $PGL\_n$-character variety $\tilde{\mathcal{M}}\_n/\mathbb{C}$ is both a quotient by a torus $\mathcal{M}\_1/\mathbb{C}\cong (\mathbb{C}^\*)^{2g}$ of the $GL\_n$-character $\mathcal{M}\_n/\mathbb{C}$ and also a quotient of a finite group $\mu\_n^{2g}$ of the $SL\_n$-character variety $\mathcal{M}^\prime\_n/\mathbb{C}$ and the action of a finite group gives the invariant part in cohomology in one factor while trivial in the other (since it is a torus).


Thus, quoting the proof of Theorem 2.2.12, we have: \begin{eqnarray\*}H^∗(\mathcal{M}\_n/\mathbb{C}) &=&(H^∗(\mathcal{M}^\prime\_n/\mathbb{C}\times \mathcal{M}\_1/\mathbb{C}))^{\mu\_n^{2g}}\\ &=& H^∗(\mathcal{M}^{\prime}\_n
/\mathbb{C})^{\mu\_n^{2g}}\otimes H^∗(\mathcal{M}\_1/\mathbb{C})\\ &=& H^∗(\tilde{\mathcal{M}}\_n/\mathbb{C})\otimes H^∗(\mathcal{M}\_1/\mathbb{C}).\end{eqnarray\*}


This phenomenon occurs for other character varieties too.


For example, for free groups, see Theorem 2.4 in my paper with Florentino *[Singularities of free group character varieties](https://arxiv.org/abs/0907.4720)*. Note that the theorem is stated for $GL\_n$ and $SL\_n$ but it generalizes to connected reductive $G$ and its derived subgroup $DG$.


More generally, for $G\cong DG\times\_F T$ (central isogeny theorem for connected complex reductive $G$), we have:
$$
\begin{eqnarray\*}
\mathrm{Hom}^0(\Gamma,G)&\cong&\mathrm{Hom}^0(\Gamma,DG \times T)/\mathrm{Hom}(\Gamma,F)\\
 &\cong&(\mathrm{Hom}^0(\Gamma,DG) \times \mathrm{Hom}^0(\Gamma,T))/\mathrm{Hom}'(\Gamma,F), 
\end{eqnarray\*}$$
by Prop. 5(1) in *[G-Character varieties for G=SO(n,C) and other not simply connected groups](https://arxiv.org/abs/1303.7181)* by Sikora. Note that the superscript $0$ denotes the connected component of the trivial representation and $\mathrm{Hom}'(\Gamma,F)$ denotes the subgroup of $\mathrm{Hom}(\Gamma,F)$ mapping $\mathrm{Hom}^0(\Gamma,DG \times T)$ to itself. This then gives a similar decomposition for corresponding components of the character varieties by quotienting by $PG$ (since $F$ is central and so commutes with the conjugation action).


The above paragraph is quoted from my paper with Sikora *[Varieties of Characters](https://arxiv.org/abs/1604.02164)*.


See also my paper with Ramras *[Covering spaces of character varieties](https://arxiv.org/abs/1402.0781)* for related theorems. In particular, for "exponent canceling" groups we can remove the superscript $0$ above (for some $G$). Examples of exponent-canceling groups are free groups, surface groups, free abelian groups, and RAAGs.


