The motivic McKay correspondence for linear actions II

Let $\mathfrak{X}$ and $\mathfrak{V}$ be the centered log structures associated with $X=V/G$ and the linear $G$-variety $V$. For a $G$-cover $E\to D$ with connected component $F$ and stabilizer $H$, let $\mathfrak{V}^{|F|}$ be the associated centered log structure and let $C_G(H)$ act on it. Let $\tau$ be the tautological motivic measure on $G\text{-}\mathrm{Cov}(D)$.

The motivic McKay correspondence for linear actions II. We have

$$M_{\mathrm{st}}(\mathfrak{X})=\int_{G\text{-}\mathrm{Cov}(D)}M_{\mathrm{st},C_G(H)}(\mathfrak{V}^{|F|})\,d\tau.$$

This is the decomposition of the orbifold stringy motif into contributions from $G$-covers and is a reformulation of the first linear motivic McKay conjecture. Its validity depends on the preceding conjectural change-of-variables statements, and the source gives no resolution.