Thank you! Maybe I need a more explicit construction. For example we know that $Ext^2_{\mathbb{P}^2}(\mathcal{O},\mathcal{O}(-3))\neq 0$ hence as you pointed out we have a complex $\mathcal{O}(-3)\rightarrow K \rightarrow L\rightarrow \mathcal{O}$. Now could we find an explicit expression of the $K$ and $L$?

@JasonStarr Thank you very much! Is there any illustrative description in easy cases?

@QiaochuYuan Sure, and, as pointed out by S. Carnahan, the map I wrote in the question is just the $0$-cells of the mapping simplicial set. After considering the whole mapping simplicial set we can get the nerve of the action groupoid of the adjoint action of $G$ on itself.

@S.Carnahan Yes you are right! After considering the mapping simplicial set I think I solved this problem. Thank you very much!