Suppose that is a finite group and that it acts on a finite dimensional complex vector space .

For each we may consider the subspace of spanned by the eigenvectors of corresponding to eigenvalues different from , and put . We pick any nonzero linear map .

If then the map makes sense, and we write it . Since it is not zero, there is a nonzero scalar such that . Associativity of the group action implies that for all , , . This means that the function is a -cocycle in the complex which computes , when we use a bar resolution of the -module (which is the permutation -module constructed from the conjugation action of on itself) We may therefore take its class in .

This class depends only on the representation and not on the choice of the 's, so we write it .

The long exact sequence corresponding to the exponential sequence allows us to identify that with . If is the set of conjugacy classes and for each we let be the permutation module corresponding to the conjugation action on , we have . If denotes the centralizer of an element of , some form of Shapiro's lemma allows us to identify with and then our class is an element of This can be written a bit more canonically as with now the centralizer of and acting on the direct sum in the only sensible way. If is abelian, this can be identified further to .

Question: What is this cohomology class ?

This reminds me of Burghelea's description of the cyclic homology of the group algebra, so it might be some form of Chern class (if some vector bundle constructed from V on the free loop space of BG?).

This class shows up when one computes the Hochschild cohomology of the cross product algebra , but explaining how would make this post too much longer. It appears as an obstruction to making this nice :-/

On triviality: suppose there is a central element in which acts by multiplication by a scalar different from . In that case is a conjugacy class, , for all , and the component of in is the determinant map. If the representation does not land in , then this is not zero. Examples of representations like this are the geometric representation of Weyl groups which have nontrivial center, in which case the longest element is central and acts as —see this question and its answers for the list of which types work— or abelian groups acting with some element not having in its spectrum.

In the other direction, if the representation respects a symplectic form on , then all the are even and we may take to be the restriction of the th power to . In this case, identically, so .

Note. If is the exponent of and is any set of th roots of unity, we can consider the subspace spanned by eigenvectors of corresponding to eigenvalues in , and proceeding as above we get a class in the same direct sum. The class above corresponds to taking the set of all th roots of unity different from . The assigment , for a conjugacy class of , is a nice functor into Yetter-Drinfeld modules, by the way.