exact functor and representations of p-adic groups 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.
 A: I do not think that your procedure gives the whole ${\mathcal R}({\mathbf G})$. In particular you'll have difficulties to get irreducible representations different from the Steinberg representation. However I do not know how to prove this fact!
To see examples you may read my papers:
Representations of ${\rm PGL}(2)$ of a local field and harmonic cochains on graphs. Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), no. 3, 495–513.
Simplicial complexes lying equivariantly over the affine building of ${\rm GL}(N)$. Math. Ann. 329 (2004), no. 3, 495–511.
Un revêtement de l'arbre de ${\rm GL}_2$ d'un corps local. (French) [A covering of the ${\rm GL}_2$ tree of a local field] Compositio Math. 135 (2003), no. 1, 37–47.
I construct certain simplicial complexes and study their cohomology.
On the other hand, if you fix $X$ as being the Bruhat-Tits building and if you allow general equivariant coefficients (i.e. coefficient systems in the language of Schneider and Stuhler), then taking the (co)homology gives all smooth representations (maybe you have to add finiteness conditions here). This fact is proved in
P. Schneider, U. Stuhler, * representation theory and sheaves on the Bruhat-Tits buildings*, Publ. Math. IHES.
