Questions about the Koszul differential in this case The following statement is from the article by Kumar and Vergne on equivariant cohomology page 133:

Let us now consider the lie group G with lie algebra $\mathfrak{g}$. Recall the definition of te Koszul differential $c_L$ on the space $T^*= L \otimes \wedge^*\mathfrak{g}^*$ calculating the cohomology of a $\mathfrak{g}$-module L: for $ \alpha \in L \otimes \wedge^p\mathfrak{g}^*$, $c_L \alpha \in L \otimes \wedge^{p+1}\mathfrak{g}^*$ is defined by
$$ (c_L \alpha)(X_1,...,X_{p+1}) = \sum_i (-1)^{i+1}X_i.\alpha(X_1,..., \hat{X_i},...,X_{p+1}) + \sum_{i<j}(-1)^{i+1}\alpha([X_i,X_j]X_1,..., \hat{X_i},...\hat{X_j},...,X_{p+1})$$ where $X_1,...,X_{p+1}$ are elements of $\mathfrak{g}$

What I don't understand in this paragraph is why are the entries of $c_L \alpha$ elements of $\mathfrak{g}$. I mean $c_L \alpha$ is defined on the tensor product of L and $\wedge^* \mathfrak{g}^*$, why isn't  kind of this  $ c_L \alpha(l_1 \otimes X_1 ,....,l_k \otimes X_k)$ , where $l_i \otimes X_i \in L \otimes \wedge^*  \mathfrak{g}^*$ ?
I understand from that statement that the Koszul differential on the space $ L \otimes \wedge^*\mathfrak{g}^*$ helps to  calculate the cohomology of a  L, am I right ? Please could you explain to me why it is useful to work with Koszul differential in this setting .
Thanks!
 A: If $\mathfrak{g}$ is a Lie algebra, its graded differential algebra is $\Lambda^* \mathfrak{g}^*$, and its elements are alternate -or differential- forms: they take value in the field of scalar $\mathbb{K}$ of $\mathfrak{g}$.
But sometimes, you may want to consider the same sort of structures but with value in another space than the field $\mathbb{K}$. For example, if $V$ is a $\mathbb{K}$ vector space, you may want to study "alternate forms" with values in $V$. A way to do that is considering the tensorial product $\left(\Lambda^*\mathfrak{g}^*\right)\otimes_{\mathbb{K}}V$: its elements are of the form $\sum \omega^i \otimes v_i$ where $\omega^i$ is a differential form on $\mathfrak{g}$ and $v_i$ is an element of $V$. For instance, in the simplest case, if $\omega$ is a linear form on $\mathfrak{g}$ (i.e $\omega \in \mathfrak{g}^*$) and if $v \in V$, then one can think of $\omega\otimes v$ as a linear function on $\mathfrak{g}$ with value in $V$: indeed, if $X \in \mathfrak{g}$, then $\left(\omega\otimes v \right)(X) = \omega(X)\cdot v \in V$.
One can extend this construction to many algebraic objects. In the case you are talking about, consider $L$ a $\mathfrak{g}$-module. Then we can think of $\mathfrak{g}^*\otimes_{\mathfrak{g}}L$ as the set of linear functions from $\mathfrak{g}$ to $L$, and more generally, if $\omega \in \Lambda^p \mathfrak{g}^*$, and $l\in L$, $\omega\otimes l$ can be thought as a $p$-alternate form with value in $L$, etc. I just wrote things on simple tensors $\omega \otimes l$, but of course, the linearity is the key to define everything.
There is a canonical isomorphism between $A \otimes B$ and $B\otimes A$, thus the order does not really matter. I would rather consider the alternate form on the left to keep thinking about it as a coefficient, but it seems the authors of the cited article prefer writing it to the right.
It is not surprising that this can help computing some cohomology classes: it is a generalisation of the de Rham complex and its exterior differential.
