Dynamical McKay correspondence conjecture via gradient-flow limits

Let $\Gamma$ be a finite subgroup of $SU(2)$, and let $X_\zeta$ be the space in the paper's theorem. Let $S_1,\ldots,S_r$ be the components of the $S^1$-fixed locus of $X_\zeta$ generating $H^2(X_\zeta)$. For each $j$, let

$$\rho_j(t)=P_j(t)RP_j(t)^{-1},\qquad 1\leq j\leq r,$$

be the one-parameter family of holonomy representations satisfying

$$\rho_j(t)=\rho_{(\alpha_j(t),\beta_j(t))},$$

with

$$\lim_{t\to-\infty}(\alpha_j(t),\beta_j(t))=(\alpha_j,\beta_j),$$

where $(\alpha_j,\beta_j)\in S_j$. Dynamical McKay correspondence conjecture. The limit

$$P_j^{\lim}=\lim_{t\to\infty}P_j(t)$$

is the projector onto the nontrivial irreducible representation $\rho_j$ of $\Gamma$, and every nontrivial irreducible representation of $\Gamma$ arises in this way. This conjecture proposes a dynamical identification of the generators of $H^2(X_\zeta)$ with the nontrivial irreducible representations of $\Gamma$; the supplied text gives no evidence that it has been proved or disproved.