The residue translation functors on the sum of parabolic blocks

Definition of the functors. Tensoring with $V$ or $V^{\ast }$ preserves $\mathcal{O}$ and parabolic local finiteness: tensor a finite parabolic span of a vector with the finite tensor factor. By the fixed-centre equivalence in [F4], the exact semisimple central-character projections in [F1] extend to ${\mathfrak{g}\mathfrak{l}}_n$: first project to an $I_n$ eigenvalue and then to the corresponding ${\mathfrak{s}\mathfrak{l}}_n$ central character. Tensoring with $V$ shifts the centre scalar by $1$, and tensoring with $V^{\ast }$ shifts it by $-1$. Thus these projections define the stated functors. Replacing $i$ by $i+1$ changes the Fock weight by $-{\alpha }_i$, so a source weight $\mathrm{\Lambda }-\alpha$ becomes $\mathrm{\Lambda }-(\alpha +{\alpha }_i)$; the reverse move gives $\alpha -{\alpha }_i$. A target with no admitted tableau is the zero category and its projection is zero.

Exactness and biadjunction. Tensoring with a finite-dimensional module is exact, and the fixed-centre block projections of step 1.1 are exact, hence so are $f_i,e_i$. For $W=V$ or $V^{\ast }$, evaluation and coevaluation give both adjunctions between $-\otimes W$ and $-\otimes W^{\ast }$; $W^{\ast \ast }=W$ supplies the second direction. These maps are ${\mathfrak{g}\mathfrak{l}}_n$-linear, preserve the Cartan-semisimple and locally parabolic-finite categories, and respect the scalar-centre shifts. For a source block $\alpha$ and target block $\beta =\alpha +{\alpha }_i$, no maps connect distinct central-character components. Therefore projection restricts both adjunctions to $\mathrm{Hom}(f_{i}M,N)\cong \mathrm{Hom}(M,e_{i}N)$ and $\mathrm{Hom}(N,f_{i}M)\cong \mathrm{Hom}(e_{i}N,M)$, with the evaluation/coevaluation units and counits. This proves biadjunction without a regular-wall hypothesis, including zero target blocks.

The Casimir description. The well-definedness supplier [F6] computes the tensor-Casimir scalars on generalized-standard flag sections: a valid $\lambda \mapsto \lambda +{\epsilon }_j$ section has scalar $a_j$, and a valid $\lambda \mapsto \lambda -{\epsilon }_j$ section has scalar $-a_j-n+1$. In particular the reverse move from entry $i+1$ has scalar $-n-i$. Its extension from these flag sections to arbitrary finite-length objects proves that the generalized $i$-eigencomponent is $f_i$ and the generalized $-n-i$-eigencomponent is $e_i$, with the block shifts of step 1.1.

Commutation with duality. Extend the semisimple restricted duality [F3] through the fixed-centre equivalence [F4]; transpose fixes $I_n$, so its scalar is retained. For a finite-dimensional weight module $W$, the evaluation map $DM\otimes DW\to D(M\otimes W)$, $(\phi \otimes \eta )(m\otimes w)=\phi (m)\eta (w)$, is a natural isomorphism: at each total weight only finitely many weights of $W$ occur, and all the participating weight spaces are finite dimensional. The diagonal tensor action and the transpose anti-involution make this map ${\mathfrak{g}\mathfrak{l}}_n$-linear. In the standard coordinate bases, the dot forms on $V$ and $V^{\ast }$ identify $DV\cong V$ and $D(V^{\ast })\cong V^{\ast }$: the matrices are $x$ on $V$ and $-x^{\mathsf{T}}$ on $V^{\ast }$, and in each case the transpose action is the adjoint for the dot form. Thus $D(M\otimes V)\cong DM\otimes V$, and likewise for $V^{\ast }$. Exactness and preservation of the simple composition factors show that $D$ preserves every finite-length central-character block, so $D{\mathrm{pr}}_{\beta }\cong {\mathrm{pr}}_{\beta }D$. Combining the two natural isomorphisms gives $D{f}_i\cong f_{i}D$ and $D{e}_i\cong e_{i}D$.

Projectives and injectives. The biadjunction of step 2.1 holds inside the parabolic blocks. For a projective $P$, the identity $\mathrm{Hom}(f_{i}P,-)\cong \mathrm{Hom}(P,e_i-)$ is exact because $e_i$ is exact, so $f_{i}P$ is projective; interchange the functors for the other direction. For an injective $J$, $\mathrm{Hom}(-,f_{i}J)\cong \mathrm{Hom}(e_i-,J)$ is exact, and again interchange the functors. Thus both preserve projectives and injectives in the stated categories.

Conclusion. The residue translation functors $f_i,e_i$ on the direct sum of finite classical blocks are exact, biadjoint, preserve projectives and injectives, commute with restricted duality, and agree with the stated Casimir eigencomponents; they are special residue translations with the Fock root label shifts, not ordinary regular-wall translations and not classical highest-weight shifts.