Has anyone ever studied the homology of the following situation?
a group $G$ acts on a space $X$; $\chi$ is an integer-valued character on the group; we take the subcomplex of the usual singular chain complex with integer (or rational) coefficients consisting of chains on which the group acts via its character $\chi$:
$C_k^{\chi}(X)= \{c \in C_k(X)\,|\, g_\#(c)=\chi(g)c\qquad \text{for all }g\in G\}$
Then define $H_k^{\chi}(X)$ as the k'th homology of this complex.
This must be a well-known homology theory. Has it a name?
