For a given topological group $G$ there are natural transformations $$K^* \leftarrow K^*_G \overset a\to H^{**}(EG \times_G -;\mathbb Q)$$ from equivariant K-theory, the first forgetting the $G$-structure of a bundle and $a$ inducing from an equivariant bundle over $X$ a nonequivariant one over $EG \times_G X$ and taking the Chern character. These maps do respectably at detecting equivariant K-theory classes in nice cases, so I'm interested in how they can fail.

Let us simplify matters by rationalizing and taking $X$ and $G$ compact. Then the forgetful map factors as $$K^*_G(X;\mathbb Q) \overset a\to H^{**}(EG \times_G X;\mathbb Q) \to H^{**}(X;\mathbb Q) \overset\cong\to K^*(X;\mathbb Q),$$ so we really only want the kernel of $a$. As $a$ can be viewed as completion of the $R(G) \otimes \mathbb Q$-module $K_G^*(X;\mathbb{Q})$ at the augmentation ideal $I(G;\mathbb Q)$, its kernel consists of classes annihilated by some element of $1 + I(G;\mathbb Q)$. Such classes exist when $G$ is discrete (already in $R(G) \otimes \mathbb Q$), but I don't know any where $G$ is connected.

**Does anyone have an explicit example of this happening when $G$ is connected**?