Let $X$ be an affine algebraic variety (over $\mathbb{C}$). Let $H$ be an algebraic group acting on $X$. Assume also that there exists an action of $\mathbb{C}^\times$ on $X$ commuting with the action of $H$ and contracting $X$ to one point. Let $\pi\colon X \rightarrow \operatorname{pt}$ be the projection.
Is it true that the pullback homomorphism $\pi^*\colon K^H(\operatorname{pt}) \rightarrow K^H(X)$ is an isomorphism? By $K^H(X)$ we mean the complexified Grothendieck ring of $H$-equivariant vector bundles on $X$.
One example I know when this is true is the following. Assume that $X=\mathcal{N}$ is a nilpotent cone of a simple Lie algebra $\mathfrak{g}$, let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. The $\mathbb{C}^\times$-action on $\mathcal{N}$ is given by $t.x = tx$. Then $K^G(\mathcal{N})$ has a basis consisting of $V_\lambda \otimes \mathcal{O}_{\mathcal{N}}$, where $V_\lambda$ are irreducible $G$-modules. This precisely means that the map $K^G(\operatorname{pt}) \rightarrow K^G(\mathcal{N})$ is an isomorphism in this case.
The reason why I am asking this question is the following. Assume that we are in the setting of this https://arxiv.org/abs/1610.08121 paper by McGerty and Nevins. It should be true that $K^{\mathbb{G}}(\operatorname{pt})$ is isomorphic (or at least maps surjectively) to $K^{\mathbb{G}}(\mu^{-1}(0))$? Is the same true after we replace $\mu^{-1}(0)$ by $\mu^{-1}(0) \setminus \mu^{-1}(0)^{s}$?
