Category-O translation intertwines residue induction and restriction

Restriction uses the dual natural module. For $N$ in the degree-$(d+1)$ summand and a basis $v_a$ of $V$ with dual basis $v^a$, tensor adjunction gives the natural map ${\rho }_N:{\mathrm{Hom}}_{\mathfrak{g}}(T_{d+1},N)\to {\mathrm{Hom}}_{\mathfrak{g}}(T_d,N\otimes V^{\ast })$, ${\rho }_N(h)(t)={\sum }_{a}h(t\otimes v_a)\otimes v^a$. Its inverse sends $g$ to $(t\otimes v)\mapsto (1\otimes {\mathrm{ev}}_v)g(t)$. These maps are basis-independent by coevaluation and evaluation and commute with the first $d$ tower generators, acting on the $T_d$ factors. Thus they identify restriction of ${\pi }_{d+1}N$ with ${\pi }_d(N\otimes V^{\ast })$ after the appropriate degree/block projection. The natural and dual natural modules are distinct ${\mathfrak{g}\mathfrak{l}}_n$-modules; no identification between them is used.

The induction map is an actual tensor map. Tensoring an endomorphism of $T_d$ with ${\mathrm{i}\mathrm{d}}_V$ gives the image-tower homomorphism $B_d\to B_{d+1}$; it sends the unit to the unit and the first $d$ generators to their tensor actions. For $M$ of degree $d$, put ${\tau }_M:B_{d+1}{\otimes }_{B_d}{\pi }_{d}M\to {\pi }_{d+1}(M\otimes V)$, $b\otimes f\mapsto (f\otimes {\mathrm{i}\mathrm{d}}_V)\circ R_b$, where $R_b$ is the right action on $T_{d+1}$. The first $d$ generators act on $T_d$, so this expression is balanced over $B_d$ and is $B_{d+1}$-linear for the left-module convention; naturality follows by postcomposition. To prove it invertible, on the finite-W side form ${\sigma }_X(b\otimes g)=(g⊛{\mathrm{i}\mathrm{d}}_V)\circ {\mu }_d^{-1}\circ R_b$. For $X=U_d$ this is the regular-module isomorphism $B_{d+1}{\otimes }_{B_d}{B}_d\cong B_{d+1}$. It remains an isomorphism on finite sums and summands. Every finite-W image object has a finite projective presentation by summands of finite sums of $U_d$, and both sides are right exact: induction is right exact, tensoring with $V$ is exact and Hom from $U_d,U_{d+1}$ is exact. Passing to cokernels proves ${\sigma }_X$ invertible for all $X$. Since the Morita functors and tensoring are exact, this also proves image induction exact; its residue summands are exact as well. Image restriction is exact directly. The coherent $\gamma ,\nu ,\mu$ square of [F3] sends $1\otimes f$ by either route to $(V_{\text{\hspace{0.17em}⁣}\mathrm{W}\mathrm{h}}(f)⊛{\mathrm{i}\mathrm{d}}_V){\mu }_d^{-1}$; $B_{d+1}$-linearity then makes the square commute on every $b\otimes f$. Consequently ${\tau }_M$ is invertible for every $M$. This presentation argument takes place in the finite-W image category, where $U_d$ is a full projective generator; it does not assert that $T_d$ is a full category-O generator.

The two residue projectors. The tower's last dot on $T_d\otimes V$ is the tensor Casimir, with the normalization of [F1]. Hence $\tau$ intertwines its generalized-$i$ eigenprojector with the projector defining $f_i^{\mathcal{O}}$. On the dual tensor factor one has the actual operator identity ${\mathrm{\Omega }}_{N,V^{\ast }}{\rho }_N(h)=-{\rho }_N(h{\mathrm{\Omega }}_{T_d,V})-n{\rho }_N(h)$. Indeed $E_{ba}{v}^c=-{\delta }_{bc}{v}^a$ and equivariance of $h$ give the first term $-{\sum }_{a,b}h(E_{ab}t\otimes v_b)\otimes v^a$ and the second term $-n{\sum }_{a}h(t\otimes v_a)\otimes v^a$, exactly that identity. Thus the last-dot generalized-$i$ projector corresponds to the dual-Casimir generalized-$(-n-i)$ projector, including its nilpotent part. This is the actual adjunction mate and agrees with the flag normalization in [F1]. Under $\rho$ the last-dot projector therefore corresponds to the generalized-$(-n-i)$ projector defining $e_i^{\mathcal{O}}$. Applying the source content projector $e_{\alpha }$ and target projector $e_{\alpha +{\alpha }_i}$ to steps 1.1–1.2 gives $u_{\alpha ,i}{\pi }_{\alpha +{\alpha }_i}N\cong {\pi }_{\alpha }{e}_i^{\mathcal{O}}N$ and $B_{\alpha +{\alpha }_i}{u}_{\alpha ,i}{\otimes }_{B_{\alpha }}{\pi }_{\alpha }M\cong {\pi }_{\alpha +{\alpha }_i}{f}_i^{\mathcal{O}}M$. Polynomial generalized-eigenspace projectors commute with these natural maps because the maps intertwine the operators, proving the asserted residue compatibility.

The natural duality comparison. The ordinary category-O duality $D$ uses matrix transpose and restricted weight-space duals. The product of the ground contravariant form and the natural-factor forms gives a nondegenerate detector form $(xtb,t^{\mathrm{\prime }})=(t,x^{\mathrm{t}\mathrm{r}}{t}^{\mathrm{\prime }}{b}^{\ast })$; the Casimir, crossings and content projectors are self-adjoint. Put $\delta M={\mathrm{Hom}}_{\mathfrak{g}}(M,T_d)$ with $(bf)(m)=f(m)b^{\ast }$ and $X^{\mathrm{\#}}={\mathrm{Hom}}_{B_d}(X,B_d)$ with the same twist. The detector form gives an actual isomorphism $\beta :T_d\cong D{T}_d$, $\beta (t)(t^{\mathrm{\prime }})=(t,t^{\mathrm{\prime }})$. Exact restricted contravariant duality, with its canonical bidual map, consequently gives $\delta M={\mathrm{Hom}}_{\mathfrak{g}}(M,T_d)\cong {\mathrm{Hom}}_{\mathfrak{g}}(M,D{T}_d)\cong {\mathrm{Hom}}_{\mathfrak{g}}(T_d,DM)={\pi }_{d}DM$. Its explicit map is $f\mapsto [t\mapsto (m\mapsto (t,f(m)))]$; its inverse sends $g:T_d\to DM$ to ${\beta }^{-1}\circ Dg\circ {\mathrm{bidual}}_M:M\to T_d$. These maps are inverse by the symmetry and nondegeneracy of the form and the bidual identities, and the form's $\ast$-invariance makes them Hecke-linear. This is a canonical natural Hom-duality isomorphism; it requires neither dimensions of simple images nor self-duality of individual projective summands. Tensor-Hom gives $\delta {\pi }_d^{\ast }\cong \mathrm{\#}$, while the symmetric trace of [F2] gives $\mathrm{\#}\cong D_H$ by $h\mapsto \mathrm{t}\mathrm{r}\circ h$. Define ${\theta }_M:M\to D{\pi }_d^{\ast }\delta M$ by ${\theta }_M(m)(t\otimes f)=(t,f(m))$. The displayed form identity makes it balanced over $B_d$ and $\mathfrak{g}$-linear. Applying ${\pi }_d$ gives a natural transformation ${\pi }_d\to {\pi }_{d}D{\pi }_d^{\ast }\delta \cong \mathrm{\#}{\pi }_{d}D$ between exact functors. It is an isomorphism on $T_d$ and hence on its finite sums and summands. Apply the copresentations in [F3] to prove it an isomorphism on every projective, by exactness and kernels; then apply projective presentations to prove it on every object, by cokernels. Dualizing yields the natural comparison ${\pi }_{d}D\cong D_H{\pi }_d$. The product forms, tensor evaluation and trace form are the same forms used for the dual adjunctions; consequently this comparison identifies their dual unit/counit squares. Since the residue projectors are self-adjoint, it also commutes with each $f_i,e_i$ separately. Thus $D{f}_i\cong f_{i}D$ and $D{e}_i\cong e_{i}D$. None of this identifies the usual tensor dual $V^{\ast }$ with $V$.

Grothendieck operators and quotient. Exactness of $\pi$ and of the residue functors makes the natural isomorphisms of step 2.1 induce $[f_i^H][\pi ]=[\pi ][f_i^{\mathcal{O}}]$ and $[e_i^H][\pi ]=[\pi ][e_i^{\mathcal{O}}]$. On $G_0$, the source simple-class basis shows that $\ker [\pi ]$ is exactly the span of the off-standard simple classes, and the induced map from this quotient onto $G_0(B)$ is an isomorphism. No isomorphism from the entire category-O Grothendieck group is asserted. On split projective groups only $\mathrm{add}(T)$ is equivalent to Hecke projectives; an ambient off-standard projective need not have projective image. These distinctions leave the displayed operator equalities intact. On the admitted Fock residue intervals, [F1] supplies the Chevalley relations, and the quotient operator equalities transport them to $G_0(B)$. When the selected contents are fully supported, $B$ is their full cyclotomic cut, so the same comparisons apply to that cut.

Adjunction maps. The first tensor unit is the coevaluation $m\mapsto {\sum }_a(m\otimes v_a)\otimes v^a$, and the tensor counit is evaluation. The induction/restriction unit is $z\mapsto 1\otimes z$, and the counit is $b\otimes z\mapsto bz$. Substituting these formulas into $\tau$ and $\rho$ gives the same composite maps, term by term: coevaluation is precisely the sum in $\rho$, and evaluation cancels that sum by $v^a(v_b)={\delta }_{ab}$. Restricting the commuting squares by the content and residue projectors proves compatibility for $f_i⊣e_i$. For the reverse adjunction use the biadjoint mate maps supplied by [F1]. By step 3.1 both image functors commute with duality, so dualizing their first adjunction gives the reverse one. Define its unit and counit by these dual mates, with the same normalization as the O-side maps. The first commuting squares then dualize to the reverse commuting squares. The two pairs are thus compared with their transported normalizations; this is stronger than asserting abstract uniqueness of adjoints across a quotient.

Filtrations and boundaries. Applying exact $\pi$ to the O-side residue standard flags preserves their successive quotients, and the intertwined generalized-eigenvalue projectors select the corresponding residue pieces. Admissible nonintegral origins retain the typed coset realization of [F4]; an integral highest-weight Fock interpretation is used only on its actual integral residue component. All comparisons here are ungraded at parameter one. Together with steps 2.1–3.2 this proves the statement.