Type-A parabolic category-O Ringel duality via the opposite parabolic

The equivalence on flags and its variance. By [F1] the functor $F=D\mathrm{\Psi }$ is exact and covariant on the finite generalized-standard flag categories, sends $\bar{M}(A)$ to $\mathrm{\nabla }(A)$, preserves the central blocks and maps the reverse-column labels to the original ones by the explicit offset computation; it is an equivalence onto the costandard-filtered objects.

Actual projective-to-tilting transport. Put $X(A)=\mathrm{\Psi }(\bar{P}(A))$. In the source highest-weight block the labelled projective has an exact sequence $0\to \bar{K}(A)\to \bar{P}(A)\to \bar{\mathrm{\Delta }}(A)\to 0$, with $\bar{K}(A)$ standard-filtered by labels strictly above $A$. The exact contravariant equivalence $\mathrm{\Psi }$ turns it into $0\to \mathrm{\Delta }(A)\to X(A)\to \mathrm{\Psi }\bar{K}(A)\to 0$; the reversed label order makes the quotient standard-filtered by labels strictly below $A$. If $0\to X(A)\to Z\to \mathrm{\Delta }(B)\to 0$ is any extension in the target block, then $Z$ is standard-filtered. Applying the inverse of $\mathrm{\Psi }$ gives a surjection onto the source projective $\bar{P}(A)$, so the transformed extension splits, and therefore the original one splits. Thus ${\mathrm{Ext}}^1(\mathrm{\Delta }(B),X(A))=0$ for every $B$. The homological characterization in [F2] makes $X(A)$ a genuine tilting object. Its endomorphism ring is the opposite of the local ring $\mathrm{End}(\bar{P}(A))$, so it is indecomposable. Its actual bottom-standard sequence and uniqueness in [F2] give $X(A)\cong T(A)$. Finally $F(\bar{P}(A))=DX(A)\cong DT(A)\cong T(A)$.

Reciprocity. The exact covariant functor $F$ takes the source standard flag to a target costandard flag, so $(T(A):\mathrm{\nabla }(B))=(\bar{P}(A):\bar{\mathrm{\Delta }}(B))=[\bar{\mathrm{\Delta }}(B):\bar{L}(A)]$ by source BGG reciprocity. Since $DT(A)\cong T(A)$ and $D\mathrm{\Delta }(B)=\mathrm{\nabla }(B)$, its standard and costandard multiplicities agree. This proves $(T(A):M(B))=[\bar{M}(B):\bar{L}(A)]$.

Full blocks and proper ideals. Since both filtered equivalences are fully faithful, the endomorphism ring of the opposite projective generator equals that of the characteristic tilting generator, so the full-block Ringel dual is the opposite-parabolic block. For a proper original lower ideal $\mathrm{\Gamma }$, every $T(A)$ with $A\in \mathrm{\Gamma }$ has support $\le A$ and remains unchanged in the Serre truncation; its end algebra equals that of the corresponding sum of opposite full projectives. The functor $\mathrm{Hom}$ from that sum is exact, kills precisely the complementary simples, and is dense by projective presentations, realizing the Serre quotient as the corner $e_{\mathrm{\Gamma }}\bar{A}{e}_{\mathrm{\Gamma }}$; this is the opposite highest-weight quotient by the complementary labels, an upper ideal in the opposite order, not the same down-set subcategory.

Conclusion. The covariant semi-regular equivalence identifies the opposite standard flags with target costandard flags, sends opposite projectives to tiltings, preserves labels and central blocks, gives the reciprocity $(T(A):M(B))=[\bar{M}(B):\bar{L}(A)]$, and realizes the Ringel dual of the full block as the opposite-parabolic block and of a proper lower ideal as the opposite highest-weight Serre quotient; no unchanged full-longest Borel twisting or arbitrary-object equivalence is asserted.