# Exotic Chains for Group Cohomology of a Complex Lie Group

Related Question: Exotic Chains for Group Homology of a Complex Lie Group

Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free resolution of $\mathbb Z$ over $\mathbb Z[G]$ is given by: $$\cdots\to\mathbb Z[G^3]\to\mathbb Z[G^2]\to\mathbb Z[G]\xrightarrow\epsilon\mathbb Z\to 0$$ If we want to compute $H^\ast(G^\delta,\mathbb C)=\operatorname{Ext}_{\mathbb Z[G]}(\mathbb Z,\mathbb C)$, then we just apply $\operatorname{Hom}_{\mathbb Z[G]}(-,\mathbb C)$ to the above complex, and thus get: $$\cdots\leftarrow\operatorname{Hom}_{\mathbb Z[G]}(\mathbb Z[G^3],\mathbb C)\leftarrow\operatorname{Hom}_{\mathbb Z[G]}(\mathbb Z[G^2],\mathbb C)\leftarrow\operatorname{Hom}_{\mathbb Z[G]}(\mathbb Z[G],\mathbb C)$$

Now let's take a leap of faith and ask what happens if we consider the subcomplex consisting of algebraic functions on $G^\bullet$ which satisfy the requisite invariance property under left-multiplication by $G$. In other words, we use the complex: $$\cdots\leftarrow(\mathcal O(G)^{\otimes 3})^G\leftarrow(\mathcal O(G)^{\otimes 2})^G\leftarrow(\mathcal O(G))^G$$ where $^G$ means take $G$-invariants, and $G$ acts on each function space by left translation on all factors. To be explicit, the coboundary maps are $d:\mathcal O(G)^{\otimes (k+1)}\to\mathcal O(G)^{\otimes(k+2)}$ given by: $$d(f_0\otimes\cdots\otimes f_k)=\sum_{i=0}^{k+1}(-1)^i(f_0\otimes\cdots\otimes f_{i-1}\otimes 1\otimes f_i\otimes\cdots\otimes f_k)$$ Does anyone know what the resulting cohomology groups are, or if they have been studied before?

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