Why is the $\operatorname{GL}_n$ character variety "cohomologically" the product of the $\operatorname{PGL}_n$ character variety and a torus?

Question

$\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\GL{\operatorname{GL}}$This question is about an assertion in Mixed Hodge polynomials of character varieties, 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?

Answer 1

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. 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 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.

See also my paper with Ramras Covering spaces of character varieties 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.