Residues in group completions and the Cech cohomology of BG

Let $G$ be a connected affine algebraic group over $\mathbb{C}$, $G \to X$ be an open immersion of $G$-varieties, $Z = X-G$ and $i: Z \to X$ be the inclusion. Let $α\in H^*(G,\mathbb{C})$ be primitive. We give a method to compute the image of $α$ in $H^*(Z, i^!\mathbb{C}_X)$, using a lift of $α$ along the first edge map of the Čech spectral sequence for $H^*(BG, \mathbb{C})$. We apply it to the wonderful compactification of a centerless semisimple group $G$.