If we fix a reductif algebraic group $G$ defined over a local field $F$, an we put $\mathbf{G}$ the group of rationnel point of $G$, we denote $\mathcal{R}(\mathbf{G})$ the category of smooth representations of $G$, $\mathcal{S}(\mathbf{G})$ the category of $\mathbf{G}$-simplicial complex of finite dimensional, that is the category which objects are the simplicial compexes $X$ with an action of G preserving their structures and such that the action is proper (ie, the stabilizer of any simplex is open compact subgroup) and morphisms are the $G$-equivariant simplicial map. We have a contravariant functor $$H_{*}:\mathcal{S}(\mathbf{G})\longrightarrow\mathcal{R}(\mathbf{G})$$ defined by $H_{*}(X)=H^{n}(X,\mathbb{C})$, where $n$ is the dimension of $X$. My questions are :
1) This functor is exact ? 2) The image of this functor is $\mathcal{R}(\mathbf{G})$ ? if not, what are the representations in the image of this functor ?
For example, we know that the Steinberg representation of $G$ is in the image of this functor.