The intersection-sheaf characterization of the canonical basis

Let ${\mathcal{P}}$ be the category of semisimple perverse sheaves on ${\mathfrak{T}(\mathbf{d})}'$ constructible with respect to the stratification by the ${A_{\mathbf{w},\mathbf{r},\mathbf{n}}}'$, and let $\theta:{\mathcal{P}}\to\mathcal{T}(\mathbf{d})$ be defined by

$$(\theta(\mathbf{B}^{\bullet}))(x)=\sum_i(-1)^i q^i(\mathbf{B}_x^i).$$

For each irreducible component $\overline{Z_{\mathbf{w}}'}$, let $IC_{\mathbf{w}}^{\bullet}$ be the associated intersection sheaf complex, and let $M=M(\mathbf{d},\mathbf{w})$. The intersection-sheaf characterization.

$$\theta\left(IC_{\mathbf{w}}^{\bullet}\right)=k_{\mathbf{w},\mathbf{r}^M,\mathbf{n}^M}^{-1}g^{\mathbf{d}}_{\mathbf{w}}.$$

The factor records the normalization on the dense stratum $A_{\mathbf{w},\mathbf{r}^M,\mathbf{n}^M}'$. Proving this characterization is expected to involve identifying Verdier duality with the involution $\Psi^{(k)}$ on $\mathcal{T}(\mathbf{d})$.