This is an excellent and carefully constructed question. The cohomology class you've defined is a well-known object in the context of equivariant K-theory and index theory, often appearing as a component of an equivariant Chern character. Let's break down its identity.

Summary of the Identification

The cohomology class can be identified as follows: for each conjugacy class of an element , its component in is the first Chern class of a specific complex line bundle over the classifying space . This line bundle is associated with the 1-dimensional representation of the centralizer group given by the determinant of its action on the subspace .

More precisely, the component of corresponding to the conjugacy class of is: where is the dual representation of for the group , and is the top exterior power, which is a 1-dimensional representation.

Detailed Explanation

Let's trace the construction step-by-step to see why this is the case.

1. The Component in

   As you've noted, the direct sum decomposition and Shapiro's Lemma give an isomorphism: where is the centralizer of any chosen representative of the conjugacy class . Let's fix a representative for a given conjugacy class, and let be its centralizer.

   The component of the class in is given by a character . Let's determine this character.

   For any , we have . The defining equation for your cocycle is . Since , this simplifies to: By definition, for any , the left-hand side is . The action of preserves the space , since if , then .

   So, for , the map is a linear operator on . Its induced action on the top exterior power is multiplication by . Therefore: Comparing this with , we find that the character is given by: This is the character of the 1-dimensional representation of on the line , which we can denote as .

2. From to

   You correctly identified that the long exact sequence in group cohomology associated to the exponential sequence of coefficients gives a connecting homomorphism: Since is a finite group, for (as is a divisible group), so this map is an isomorphism.

   Topologically, we can interpret this isomorphism as follows:

   • classifies isomorphism classes of 1-dimensional complex representations of . A character corresponds to a representation space on which acts as multiplication by .
   • Such a representation defines a complex line bundle over the classifying space .
   • is the group where characteristic classes of such bundles live.
   • The isomorphism maps the class of the representation to the first Chern class of the associated line bundle.

   Therefore, the component of your class in is precisely the first Chern class of the line bundle associated with the character . This is .

Connection to Equivariant K-theory and Chern Characters

Your intuition about Chern classes and Burghelea's work is spot on. This construction is a concrete manifestation of ideas from equivariant topology.

Let be a representation of . This corresponds to an element , the representation ring of . There is an equivariant Chern character map: for any -space . For , this is a map from to the algebra of class functions.

The construction of is related to the process of localization in equivariant cohomology. The Atiyah-Segal-Singer localization theorem relates the equivariant cohomology of a space to the cohomology of its fixed-point sets. For the "free loop space of BG", which is the homotopy quotient , the "fixed points" under the action of a given correspond to the component .

The class can be interpreted as the degree-2 component of the Chern character of a certain virtual bundle, localized at each conjugacy class. Specifically, for each , we have the decomposition of into eigenspaces for the action of : , where is the subspace with eigenvalue 1. As representations of , we have a virtual representation . The character you constructed is . The corresponding class in is: This is the restriction of from to . Your class captures the "non-trivial eigen-part" of the first equivariant Chern class of at each conjugacy class.

Verifying Your Examples

• Symplectic Representation: If has a -invariant symplectic form , then for any , . The centralizer also preserves . The restriction of to is non-degenerate (since and are orthogonal with respect to any invariant Hermitian form, and a similar property holds for symplectic forms). Thus, the representation of on is also symplectic, which implies for all . The character is trivial, and so is the class . This matches your observation.

• Central Element : If acts by a scalar , then and . The corresponding character is given by . This is the inverse of the determinant character of the representation. If the representation is not contained in , this character is non-trivial, and the class has a non-trivial component in , hence in . This also matches your observation.

In summary, your class elegantly packages the first Chern classes of the "non-trivial eigenspace bundles" associated with the representation over the classifying spaces of the centralizers of all elements of . It is a natural and important invariant.