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On triviality: suppose there is a central element $z$ in $G$ which acts by multiplication by a scalar different from $1$. In that case $c=\{z\}$ is a conjugacy class, $G_c=G$, $\lambda(g,z)=\det g$ for all $g$, and the component of $c(V)$ in $\Ext^1_{\ZZ G}(\ZZ c,\CC^\times)=\hom(G,\CC^\times)$ is the determinant map. If the representation does not land in $SL(V)$, 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 $-1$ —see this question and its answers for the list of which types work— or abelian groups acting with some element not having $1$ in its spectrum.

On triviality: suppose there is a central element $z$ in $G$ which acts by multiplication by a scalar different from $1$. In that case $c=\{z\}$ is a conjugacy class, $G_c=G$, $\lambda(g,z)=\det g$ for all $g$, and the component of $c(V)$ in $\Ext^1_{\ZZ G}(\ZZ c,\CC^\times)=\hom(G,\CC^\times)$ is the determinant map. If the representation does not land in $SL(V)$, 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 $-1$ —see this question and its answers for the list of which types work— or abelian groups acting with some element not having $1$ in its spectrum.

On triviality: suppose there is a central element $z$ in $G$ which acts by multiplication by a scalar different from $1$. In that case $c=\{z\}$ is a conjugacy class, $G_c=G$, $\lambda(g,z)=\det g$ for all $g$, and the component of $c(V)$ in $\Ext^1_{\ZZ G}(\ZZ c,\CC^\times)=\hom(G,\CC^\times)$ is the determinant map. If the representation does not land in $SL(V)$, 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 $-1$; see this question and its answers for the list of which types work

On triviality: suppose there is a central element $z$ in $G$ which acts by multiplication by a scalar different from $1$. In that case $c=\{z\}$ is a conjugacy class, $G_c=G$, $\lambda(g,z)=\det g$ for all $g$, and the component of $c(V)$ in $\Ext^1_{\ZZ G}(\ZZ c,\CC^\times)=\hom(G,\CC^\times)$ is the determinant map. If the representation does not land in $SL(V)$, 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 $-1$ —see this question and its answers for the list of which types work— or abelian groups acting with some element not having $1$ in its spectrum.

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?). Since $\lambda(g,1_G)=\det g$, the determinant of the linear map representing $g$, the component in the trivial conjugacy class is just that. In particular, the class is in general non-zero: the component at the identity is zero iff the representation land in $SL(V)$.

This class shows up when one computes the Hochschild cohomology of the cross product algebra $S(V)\rtimes G$, but explaining how would make this post too much longer.

Note. If $e$ is the exponent of $G$ and $\Omega$ is any set of $e$th roots of unity, we can consider the subspace $V_g^\Omega$ spanned by eigenvectors of $g$ corresponding to eigenvalues in $\Omega$, and proceeding as above we get a class $c_\Omega(V)$ in the same direct sum. The class above corresponds to taking $\Omega$ the set of all $e$th roots of unity different from $1$. The assigment $V\mapsto\bigoplus_{g\in G}V_g^\Omega$, for $c$ a conjugacy class of $G$, is a nice functor into Yetter-Drinfeld modules, by the way.

Note. If the representation respects a symplectic form $\omega$ on $V$, then all the $d(g)$ are even and we may take $\omega_g$ to be the restriction of the $\tfrac12d(g)$th power $\omega^{d(g)/2}$ to $V_g$. In this case, $\lambda(h,g)=1$ identically, so $c(V)=0$.

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 $S(V)\rtimes G$, 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 $z$ in $G$ which acts by multiplication by a scalar different from $1$. In that case $c=\{z\}$ is a conjugacy class, $G_c=G$, $\lambda(g,z)=\det g$ for all $g$, and the component of $c(V)$ in $\Ext^1_{\ZZ G}(\ZZ c,\CC^\times)=\hom(G,\CC^\times)$ is the determinant map. If the representation does not land in $SL(V)$, 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 $-1$; see this question and its answers for the list of which types work

In the other direction, if the representation respects a symplectic form $\omega$ on $V$, then all the $d(g)$ are even and we may take $\omega_g$ to be the restriction of the $\tfrac12d(g)$th power $\omega^{d(g)/2}$ to $V_g$. In this case, $\lambda(h,g)=1$ identically, so $c(V)=0$.

Note. If $e$ is the exponent of $G$ and $\Omega$ is any set of $e$th roots of unity, we can consider the subspace $V_g^\Omega$ spanned by eigenvectors of $g$ corresponding to eigenvalues in $\Omega$, and proceeding as above we get a class $c_\Omega(V)$ in the same direct sum. The class above corresponds to taking $\Omega$ the set of all $e$th roots of unity different from $1$. The assigment $V\mapsto\bigoplus_{g\in G}V_g^\Omega$, for $c$ a conjugacy class of $G$, is a nice functor into Yetter-Drinfeld modules, by the way.