Let $V=\mathbb{R}^d$ be a $d$-dimensional (Euclidian) vector space over real numbers. Let $G=SO(V)$ be a compact Lie group of linear orthogonal transformations of $V$.

Let $Conf_n(V)$ be the space of $n$-tuples $\{(\overrightarrow{x_1},\ldots,\overrightarrow{x_n}), \overrightarrow{x_i}\in V\}$ of pairwise distinct points in $V$ ($\overrightarrow{x_i} \neq \overrightarrow{x_j}$). I.e. $Conf_n(V)$ is an open space that is a complement in $V\times V \times \ldots \times V = \mathbb{R}^{n d}$ to the union of arrangements of codimension $d$: $\cup_{i,j} \{\overrightarrow{x_i} = \overrightarrow{x_j}\}$ There is a natural action of $G$ on $Conf_{n}(V)$ (namely componentwise).

Question 1): Compute the equivariant (co)homology of $Conf_{n}(V)$ with respect to this action of $SO(V)$: $$H^{\bullet}_{SO(V)} (Conf_n(V);\mathbb{R})$$

Question 2): The same question for the complex situation. I.e. $V = \mathbb{C}^d$ and the group is $G = SU(V)$. and we are interested in the description of $$H^{\bullet}_{SU(V)} (Conf_n(V);\mathbb{C})$$

Note, that in the case $V= \mathbb{C} = \mathbb{R}^2$ the answer is known to coincide with the cohomology of the open moduli spaces of curves with zero genus and $n+1$ marked points. Unfortunately for the case of $d>2$ the action of $G$ is no more the free action and the total answer should be infinite at least for $d>n\geq 2$, but I will be happy with any reasonable description, even if it will be in terms of cohomology of some finetely generated differential graded algebra.