Translation identifies the regular integral blocks

Since $\lambda$ and $\mu$ are dominant regular integral, the shifted weights $a=\lambda +\rho$ and $b=\mu +\rho$ are strictly dominant and have trivial stabilizers in $W$; the dominant member $\nu$ of $W(\mu -\lambda )=W(b-a)$ satisfies $\mathrm{\mid }\nu \mathrm{\mid }=\mathrm{\mid }b-a\mathrm{\mid }$, and by [F5] and [F10] every weight $\gamma$ of $E=L(\nu )$ has $\mathrm{\mid }\gamma \mathrm{\mid }\le \mathrm{\mid }b-a\mathrm{\mid }$, with equality exactly on $W\nu$, where the multiplicity is one.

By [F2] the functors $F$ and $G$ are exact, and $G$ is both a left and a right adjoint of $F$ (with the roles of $E$ and $E^{\ast }$ exchanged for the reverse pair); exactness of the right adjoint on the other side makes $F$ preserve projectives and $G$ preserve projectives. Write $\eta :\mathrm{Id}\to GF$ and $\epsilon :FG\to \mathrm{Id}$ for the unit and counit of the adjunction $F⊣G$; both composites $GF$ and $FG$ are exact as compositions of exact functors.

Compute $F\mathrm{\Delta }(y\cdot \lambda )={\mathrm{pr}}_{{\chi }_{\mu }}(E\otimes \mathrm{\Delta }(y\cdot \lambda ))$ for $y\in W$. By [F3] the object $E\otimes \mathrm{\Delta }(y\cdot \lambda )$ has a Verma flag with factors $\mathrm{\Delta }(y\cdot \lambda +\gamma )$, $\gamma$ running over the weights of $E$ with multiplicity $\dim E_{\gamma }$, and by [F1] the exact projection ${\mathrm{pr}}_{{\chi }_{\mu }}$ maps this flag to a flag of the projected object whose factor at $\gamma$ is $\mathrm{\Delta }(y\cdot \lambda +\gamma )$ when $y\cdot \lambda +\gamma \in W\cdot \mu$ and $0$ otherwise. A retained factor has $y\cdot \lambda +\gamma =x\cdot \mu$ for some $x$, that is $ya+\gamma =xb$ with $a,b$ strictly dominant and regular. Then $\mathrm{\mid }\gamma \mathrm{\mid }=\mathrm{\mid }xb-ya\mathrm{\mid }=\mathrm{\mid }b-x^{-1}ya\mathrm{\mid }\ge \mathrm{\mid }b-a\mathrm{\mid }$ by [F4] applied to the dominant vectors $b$ and $a$; regularity of $b$ makes equality of the norm comparison force $x^{-1}ya=a$, and regularity of $a$ then forces $x=y$. Comparing with the bound $\mathrm{\mid }\gamma \mathrm{\mid }\le \mathrm{\mid }b-a\mathrm{\mid }$ of [F5] gives equality, so $\gamma \in W\nu$ and $x=y$; then $\gamma =y(b-a)=y(\mu -\lambda )$, the factor is $\mathrm{\Delta }(y\cdot \lambda +y(\mu -\lambda ))=\mathrm{\Delta }(y\cdot \mu )$, and by the multiplicity-one clause of [F5] it occurs exactly once. No other retained factor exists, since every retained factor was shown to equal this one. The projected object therefore has a one-step flag with quotient $\mathrm{\Delta }(y\cdot \mu )$, so $F\mathrm{\Delta }(y\cdot \lambda )\cong \mathrm{\Delta }(y\cdot \mu )$ for every $y$.

The dual computation gives $G\mathrm{\Delta }(y\cdot \mu )\cong \mathrm{\Delta }(y\cdot \lambda )$ for every $y$. By [F3] applied to the finite-dimensional module $E^{\ast }$ of [F10], the flag factors of $E^{\ast }\otimes \mathrm{\Delta }(y\cdot \mu )$ are $\mathrm{\Delta }(y\cdot \mu +{\gamma }^{\mathrm{\prime }})$ with multiplicities $\dim (E^{\ast }{)}_{{\gamma }^{\mathrm{\prime }}}$, and a factor retained by ${\mathrm{pr}}_{{\chi }_{\lambda }}$ satisfies $yb+{\gamma }^{\mathrm{\prime }}=xa$ for some $x\in W$. Then $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\mid }xa-yb\mathrm{\mid }=\mathrm{\mid }a-x^{-1}yb\mathrm{\mid }\ge \mathrm{\mid }b-a\mathrm{\mid }$ by [F4] applied to the dominant vectors $a$ and $b$, and regularity of $a$ forces $x^{-1}yb=b$ and then $x=y$; the weight bound [F5] applied to the weights of $E^{\ast }$ (whose extremal orbit is $W(\lambda -\mu )$ with multiplicity one, [F10]) gives equality $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\mid }b-a\mathrm{\mid }$, ${\gamma }^{\mathrm{\prime }}=y(a-b)=y(\lambda -\mu )$, and the retained factor $\mathrm{\Delta }(y\cdot \mu +y(\lambda -\mu ))=\mathrm{\Delta }(y\cdot \lambda )$ occurs once. Hence $G\mathrm{\Delta }(y\cdot \mu )\cong \mathrm{\Delta }(y\cdot \lambda )$.

For each $y\in W$, the unit ${\eta }_{\mathrm{\Delta }(y\cdot \lambda )}:\mathrm{\Delta }(y\cdot \lambda )\to GF\mathrm{\Delta }(y\cdot \lambda )$ corresponds under the adjunction isomorphism $\mathrm{Hom}(F\mathrm{\Delta }(y\cdot \lambda ),F\mathrm{\Delta }(y\cdot \lambda ))\cong \mathrm{Hom}(\mathrm{\Delta }(y\cdot \lambda ),GF\mathrm{\Delta }(y\cdot \lambda ))$ to the identity of $F\mathrm{\Delta }(y\cdot \lambda )$; since $F\mathrm{\Delta }(y\cdot \lambda )\cong \mathrm{\Delta }(y\cdot \mu )\mathrm{\ne }0$ by step 2.1, the unit is nonzero. Its target is $GF\mathrm{\Delta }(y\cdot \lambda )\cong G\mathrm{\Delta }(y\cdot \mu )\cong \mathrm{\Delta }(y\cdot \lambda )$ by steps 2.1 and 2.2, so ${\eta }_{\mathrm{\Delta }(y\cdot \lambda )}$ is a nonzero endomorphism of a Verma module after transport along this isomorphism, and by [F6] every nonzero endomorphism of a Verma module is an isomorphism. Hence $\eta$ is an isomorphism on every standard object of ${\mathcal{O}}_{\lambda }$.

The same argument for the counit: for each $y\in W$, ${\epsilon }_{\mathrm{\Delta }(y\cdot \mu )}:FG\mathrm{\Delta }(y\cdot \mu )\to \mathrm{\Delta }(y\cdot \mu )$ corresponds to the identity of $G\mathrm{\Delta }(y\cdot \mu )\mathrm{\ne }0$ under the adjunction, hence is nonzero, and its source is $FG\mathrm{\Delta }(y\cdot \mu )\cong F\mathrm{\Delta }(y\cdot \lambda )\cong \mathrm{\Delta }(y\cdot \mu )$ by steps 2.1 and 2.2; by [F6] it is an isomorphism on every standard object of ${\mathcal{O}}_{\mu }$.

The isomorphisms of steps 3.1 and 3.2 extend to every Verma-filtered object. If $0\to A\to B\to C\to 0$ is a short exact sequence in ${\mathcal{O}}_{\lambda }$ with $A$ and $C$ Verma-filtered and ${\eta }_A,{\eta }_C$ isomorphisms, then $GF$ is exact by step 1.2, and naturality of $\eta$ gives a map of short exact sequences from $0\to A\to B\to C\to 0$ to $0\to GFA\to GFB\to GFC\to 0$ with isomorphisms at the outer terms; the five lemma makes ${\eta }_B$ an isomorphism. Inducting along a finite Verma flag of $X$ (Finite Verma flags and their multiplicities, the flag decomposition $0=X_0\subseteq \cdots \subseteq X_n=X$ with Verma quotients) gives that ${\eta }_X$ is an isomorphism for every Verma-filtered $X$, and [F7] makes every projective of ${\mathcal{O}}_{\lambda }$ Verma-filtered; the identical induction with $FG$ exact and the counit gives that ${\epsilon }_Y$ is an isomorphism for every Verma-filtered $Y$ of ${\mathcal{O}}_{\mu }$, in particular for every projective.

The isomorphisms extend to all objects by finite projective presentations. By [F8] every object $X$ of ${\mathcal{O}}_{\lambda }$ has finite length and is a quotient of a finite direct sum of projective covers; choosing such a cover $P_0\twoheadrightarrow X$ and applying the same to the kernel of this map (an object of ${\mathcal{O}}_{\lambda }$ by [F9]) produces an exact sequence $P_1\to P_0\to X\to 0$ with $P_0,P_1$ finite direct sums of projective covers, hence projective and Verma-filtered by [F7]. Applying the exact functor $GF$ and using naturality of $\eta$ identifies ${\eta }_X$ with the map induced on the cokernel of ${\eta }_{P_1},{\eta }_{P_0}$, which are isomorphisms by step 4.1; therefore ${\eta }_X$ is an isomorphism. The same argument in ${\mathcal{O}}_{\mu }$ with a finite projective presentation and the exact functor $FG$ makes ${\epsilon }_Y$ an isomorphism for every $Y$ in ${\mathcal{O}}_{\mu }$. Thus the unit and counit of the adjunction $F⊣G$ are natural isomorphisms, so $F$ and $G$ are mutually inverse equivalences of categories.

Being inverse equivalences, $F$ is exact and reflects the categorical structures. For the labels: $F\mathrm{\Delta }(w\cdot \lambda )\cong \mathrm{\Delta }(w\cdot \mu )$ was proved in step 2.1. For each $w$, $FL(w\cdot \lambda )$ is a simple object, and the exact functor $F$ turns the epimorphism $\mathrm{\Delta }(w\cdot \lambda )\twoheadrightarrow L(w\cdot \lambda )$ onto its unique simple quotient into an epimorphism $\mathrm{\Delta }(w\cdot \mu )\cong F\mathrm{\Delta }(w\cdot \lambda )\twoheadrightarrow FL(w\cdot \lambda )$; since the simple quotient of $\mathrm{\Delta }(w\cdot \mu )$ is unique and equal to $L(w\cdot \mu )$, one has $FL(w\cdot \lambda )\cong L(w\cdot \mu )$. Finally $P(w\cdot \lambda )$ is a projective cover of $L(w\cdot \lambda )$ by [F8] and [F11], so $FP(w\cdot \lambda )$ is a projective object mapping onto $FL(w\cdot \lambda )\cong L(w\cdot \mu )$ with essential kernel: if a subobject $Z^{\mathrm{\prime }}\subseteq FP(w\cdot \lambda )$ mapped onto $FL(w\cdot \lambda )$, then applying the inverse equivalence would exhibit a subobject of $P(w\cdot \lambda )$ mapping onto $L(w\cdot \lambda )$, so essentiality on the source forces $Z^{\mathrm{\prime }}=FP(w\cdot \lambda )$. By uniqueness of projective covers in [F8], $FP(w\cdot \lambda )\cong P(w\cdot \mu )$. The dual statements $G\mathrm{\Delta }(w\cdot \mu )\cong \mathrm{\Delta }(w\cdot \lambda )$, $GL(w\cdot \mu )\cong L(w\cdot \lambda )$ and $GP(w\cdot \mu )\cong P(w\cdot \lambda )$ follow from step 2.2 and the identical argument with the roles exchanged.

The multiplicities are transported termwise. Since $F$ is an equivalence, it is full and faithful, so applying $F$ to a finite Verma flag of $P$ and applying $G$ to a finite Verma flag of $FP$ are inverse bijections between the flag multiplicities of $P$ and of $FP$; by steps 6.1 and 2.2 the labels match, giving $(FP(w\cdot \lambda ):F\mathrm{\Delta }(y\cdot \lambda ))=(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))$ and, with step 6.1, $(P(w\cdot \mu ):\mathrm{\Delta }(y\cdot \mu ))=(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))=P_{y,w}(1)$ by [F11]; BGG reciprocity [F11] likewise transports the composition multiplicities and their vanishing. Transporting the identity of [F12] along the induced isomorphism of Grothendieck groups gives $[L(w\cdot \mu )]={\sum }_x{q}_{w,x}^{\mathrm{\prime }}(1)[M(x\cdot \mu )]$ with the same weighted-inverse coefficients and the same support $x\ge w$, because $F$ carries $L(w\cdot \lambda )$ to $L(w\cdot \mu )$ and $M(x\cdot \lambda )=\mathrm{\Delta }(x\cdot \lambda )$ to $M(x\cdot \mu )$. The conventions are preserved because the label $w$ is unchanged: dominant and antidominant labels, Bruhat comparisons, left/right labels and inverse Soergel labels carry over; the formal characters are read in the respective weight series $x\cdot \lambda$ and $x\cdot \mu$ and need not coincide. Nothing above uses or produces a grading on $\mathcal{O}$, an $\mathrm{E}\mathrm{x}\mathrm{t}$-algebra or a Koszul structure.