The equivariant cohomological realization conjecture for local Dahmen–Micchelli modules

Let $G$ be the compact abelian group, with Lie algebra $[?]$; let $X$ be the finite list or configuration used to define the local Dahmen–Micchelli module $D_{\mathcal T}(X)$, and let $M_X^{\mathcal T}$ be the corresponding submanifold associated with a semilocal set $\mathcal T\subseteq 2^X$. The cotangent bundle is taken in the equivariant sense, $T_G^*M_X^{\mathcal T}$, and $S[\mathfrak g^*]$ denotes the symmetric algebra on $\mathfrak g^*$, with its natural grading. The set $\mathcal T$ is semilocal if it is closed under taking supersets and there exists a big cell $\Omega$ such that

$$L(X)\subseteq \mathcal T\subseteq L_\Omega(X).$$

Equivariant cohomological realization conjecture. For every semilocal $\mathcal T$, there is an isomorphism of graded $S[\mathfrak g^*]$-modules

$$H_G^*(T_G^*M_X^{\mathcal T})\cong D_{\mathcal T}(X)$$

that is given by the infinitesimal index.

This is the cohomological analogue of the preceding theorem, which establishes the corresponding realization for the equivariant $K$-theory module $DM_{\mathcal T}(X)$. The authors indicate that analogous methods should prove the statement, but the conjecture itself is presented as an assertion whose proof is not supplied in the source.