The integral regular localization equivalence restricts to Category O

Let $V\in {\mathcal{O}}_{{\chi }_{\kappa }}^{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}$. By [F1] it has a finite composition series with factors $L(\lambda )$. Since the central ideal $\ker {\chi }_{\kappa }$ annihilates $V$, it annihilates every subquotient; thus each factor has central character ${\chi }_{\kappa }$. By [F2], this means $\lambda +\rho$ and $\kappa$ have the same values on all Weyl-invariant polynomials. These invariants separate the finite $W$-orbits: for two disjoint finite orbits, polynomial interpolation separates their point sets, and averaging the separating polynomial over $W$ makes it invariant. Therefore $\lambda +\rho =w\kappa$ for some $w\in W$. Since $\kappa$ is integral, $\lambda$ is integral, and every weight of $L(\lambda )$ differs from $\lambda$ by an element of the root lattice. All weights of $V$ are therefore integral.

For a weight vector $v\in V$, the subspace $U({\mathfrak{n}}^{+})v$ is finite-dimensional by [F3] and is stable under $\mathfrak{h}$, since $[\mathfrak{h},{\mathfrak{n}}^{+}]\subseteq {\mathfrak{n}}^{+}$. Its weights are integral by step 1.1. Define the rational $T$-action on each such finite-dimensional subspace by its integral weight characters. Each positive-root operator is nilpotent there because it strictly raises weights in a finite weight set; more generally, a real linear functional positive on all positive roots gives a finite weight filtration strictly raised by every element of ${\mathfrak{n}}^{+}$, so $\pi (X)$ is nilpotent for each $X\in {\mathfrak{n}}^{+}$. The Lie representation of the nilpotent algebra ${\mathfrak{n}}^{+}$ integrates by $u=\exp (X)\mapsto \exp (\pi (X))$: the finite BCH polynomial is respected by the Lie representation, and the exponential coordinates ${\mathfrak{n}}^{+}\to U$ and their multiplication law are those of [F5]. For $t_0\in T$, the weight action satisfies ${\rho }_T(t_0)\exp (\pi (X)){\rho }_T(t_0{)}^{-1}=\exp (\pi (\mathrm{Ad}(t_0)X))$, so [F5] combines the $T$ and $U$ actions into a rational $B=T⋉U$-action differentiating to the given $\mathfrak{b}$-action. Any finite sum of these subspaces is again finite-dimensional and $\mathfrak{b}$-stable; uniqueness of the exponential $U$-action and the prescribed weight action makes the constructions compatible under inclusions. The subspaces form a directed cover of $V$, so their compatible actions define a rational $B$-action on $V$.

The TDO ${\mathcal{D}}_{\kappa }$ is $G$-equivariant by its action-algebroid construction. The diagonal $B$-action therefore makes ${\mathrm{Loc}}_{\kappa }(V)$ strongly $B$-equivariant; it is coherent by [F4]. Conversely, if $M$ is strongly $B$-equivariant and coherent, its global sections inherit a rational $B$-action. The torus action is a direct sum of integral weight spaces, and the unipotent action makes each $U({\mathfrak{n}}^{+})$-orbit finite-dimensional. By [F4], $\mathrm{\Gamma }M$ is finitely generated over $A_{\kappa }$, hence over $U(\mathfrak{g})$, and its central ideal acts by zero because $A_{\kappa }$ is the central quotient in Flag-variety twisted differential operators and their modules. Thus $\mathrm{\Gamma }M$ lies in ${\mathcal{O}}_{{\chi }_{\kappa }}^{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}$.

The unit and counit of the equivalence in [F4] are natural and $B$-equivariant, because all functors and maps in their construction commute with the diagonal $B$-action. Their restrictions are still isomorphisms, so the two restricted functors are inverse equivalences.