Dunkl localization gives flat connections on the regular locus

By [F3], the assignments $x\mapsto m_x$, $w\mapsto w$, and $y\mapsto T_y$ respect every defining relation, so they give an algebra homomorphism into $\mathcal{D}({\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}})⋊W$. Filter $H_{1,c}$ by degree in $\mathfrak{h}$ with ${\mathfrak{h}}^{\ast }$ and $W$ in degree zero, and filter the target by differential order. The principal symbol of $T_y$ is ${\mathrm{\partial }}_y$, so the associated-graded map is the localization $(\mathbb{C}[{\mathfrak{h}}^{\ast }]\otimes \mathbb{C}[\mathfrak{h}])⋊W⟶\mathbb{C}[{\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}}\times {\mathfrak{h}}^{\ast }]⋊W$, with the diagonal $W$-action on both sides. It is injective: the crossed products are direct sums over $W$, and each polynomial coefficient embeds after localization by the nonzero $\delta$. Therefore the original homomorphism is injective: a nonzero element with its finite filtration degree would have a nonzero leading symbol, contradicting injectivity of the associated-graded map.

After inverting $\delta$, every ${\alpha }_s$ is a unit by [F1]. Hence ${\mathrm{\partial }}_y=T_y+{\sum }_s{c}_s{\alpha }_s(y){\alpha }_s^{-1}(1-s)$ belongs to the localized image for every $y\in \mathfrak{h}$. The localized image contains all regular functions, $W$, and all constant coordinate derivations, which generate $\mathcal{D}({\mathfrak{h}}_{\mathrm{r}\mathrm{e}\mathrm{g}})⋊W$. It is the whole target; injectivity is preserved by exact localization over $P$.

Localization is exact because it is localization over the commutative ring $P$. By [F4], $M_{\mathrm{r}\mathrm{e}\mathrm{g}}$ is finite over $P[{\delta }^{-1}]$. Step 2.1 makes it a module over the differential operators on that ring. The coordinate derivations satisfy the Leibniz rule and commute, so they define an integrable connection. The crossed-product relations make this connection $W$-equivariant.

By step 3.1, the finite localized module has the required Leibniz operators over the polynomial principal open. [F5] proves local freeness at each complex point and its extension to an algebraic neighborhood, so this is a finite-rank algebraic vector bundle. Its connection is integrable and equivariant by step 3.1.

Localization sends each short exact sequence in ${\mathcal{O}}_c$ to a short exact sequence of vector bundles with connection by step 3.1 and local freeness from step 4.1. A short exact sequence of vector bundles is locally split, so its fiber sequence is exact. Parallel transport commutes with morphisms of connections; hence the fiber monodromy representations form a short exact sequence.