The space of commuting elements in a Lie group and maps between classifying spaces

Let $\pi$ be a discrete group, and let $G$ be a compact connected Lie group. Then there is a map $\mathrm{\Theta }\text{\hspace{0.17em}⁣}:\mathrm{H}\mathrm{o}\mathrm{m}(\pi ,G{)}_0\to {\mathrm{m}\mathrm{a}\mathrm{p}}_{\ast }(B\pi ,BG{)}_0$ between the null-components of the spaces of homomorphism and based maps, which sends a homomorphism to the induced map between classifying spaces. Atiyah and Bott studied this map for $\pi$ a surface group, and showed that it is surjective in rational cohomology. In this paper, we prove that the map $\mathrm{\Theta }$ is surjective in rational cohomology for $\pi ={\mathbb{Z}}^m$ and the classical group $G$ except for $SO(2n)$, and that it is not surjective for $\pi ={\mathbb{Z}}^m$ with $m\ge 3$ and $G=SO(2n)$ with $n\ge 4$. As an application, we consider the surjectivity of the map $\mathrm{\Theta }$ in rational cohomology for $\pi$ a finitely generated nilpotent group. We also consider the dimension of the cokernel of the map $\mathrm{\Theta }$ in rational homotopy groups for $\pi ={\mathbb{Z}}^m$ and the classical groups $G$ except for $SO(2n)$.