The degenerate Ariki categorification theorem

Both identifications and their duality. By The degenerate cyclotomic Hecke $G_0$ is the dual highest weight module the map $[D(A)]\mapsto D_A$ identifies $G_0(\mathrm{R}\mathrm{e}\mathrm{p}(H^{\mathrm{\Lambda }}))$ with the full integral costandard lattice $V(\mathrm{\Lambda }{)}_{\mathbb{Z}}^{\ast }$ and is compatible with the standard classes $[S(B)]\mapsto S_B$; by The degenerate cyclotomic Hecke projective $K_0$ is the highest weight module the map $[Y(A)]\mapsto P_A$ identifies $K_0(\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}(H^{\mathrm{\Lambda }}))$ with the standard lattice $V(\mathrm{\Lambda }{)}_{\mathbb{Z}}$, the embedding of projectives induces the lattice inclusion, and the Cartan pairing is the normalized highest pairing $C_0(P_A,D_B)={\delta }_{AB}$ of The Fock pairing matches the projective-simple and standard-costandard pairings. Since the two lattice identifications are the actual source dictionary $[L(A)]\mapsto L_A$, $[P(A)]\mapsto P_A$ of Simples, projectives and tiltings map to the three distinguished Fock bases at parameter one, the identifications hold simultaneously and are mutually dual.

The actual surviving-head calculation. Fix $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ and let $M=f_i^{\mathcal{O}}L(A)$, or $M=e_i^{\mathcal{O}}L(A)$. The local residue theorem The residue translation functors give the integrable Fock-space action says $M=0$ exactly when the corresponding crystal edge is absent; otherwise its head is the simple $L(C)$ at that edge, and it is self-dual. The ground component ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ is closed under the two crystal operators by Standard tableau monomials form a basis of the highest weight Fock submodule, so $C$ is standard. The actual detector/corner Hom comparison of The Whittaker functor is exact and fully faithful on the required projectives applies to maps from a Std-head object to a Std-socle object. Applying it to $M$ and $L(B)$ for each standard $B$ gives ${\mathrm{Hom}}_H(\pi M,D(B))\cong {\mathrm{Hom}}_{\mathfrak{g}}(M,L(B))$, of dimension ${\delta }_{CB}$. Since the $D(B)$ are the complete distinct Hecke simples, this computes the head of $\pi M$ as the single simple $D(C)$ and proves $\pi M\mathrm{\ne }0$. The natural intertwining of Category-O translation intertwines residue induction and restriction identifies $\pi M$ with the appropriate Hecke translate of $D(A)$, and its natural duality comparison makes that translate self-dual. Its socle is therefore also $D(C)$. A direct sum of two nonzero finite-length modules has at least two nonzero simple head summands, so a module with simple head is indecomposable. If the edge is absent the translate is zero by the same natural isomorphism. This proves (iii) in both directions by a Hom calculation; no arbitrary head or socle preservation by an exact quotient is assumed.

The parameter-one normalization and label bijections. At parameter one the quasi-canonical family satisfies $P_A=T_{A^{\uparrow }}$ and the costandard family is $D_A={\pi }_{\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{k}}{L}_A$ with $A\mapsto A^{\uparrow }$ the inverse rectification from standard to reverse-standard tableaux and $\downarrow$ the rectification in the opposite direction, by The highest weight Fock submodule carries the quantum-group global bases and The dual-canonical, canonical and quasi-canonical Fock bases; this gives (iv).

Level zero. For level $0$ the tower has $H_0^{\mathrm{\Lambda }}=\mathbb{C}$ and $H_d^{\mathrm{\Lambda }}=0$ for $d>0$ by the level-zero convention of The degenerate cyclotomic Hecke tower and its residue functors, so only $\alpha =0$ and the ground class remain, consistently with the two identifications being the rank-one lattices $\mathbb{Z}{P}_{\mathrm{\varnothing }}=\mathbb{Z}{D}_{\mathrm{\varnothing }}$.

The decomposition formula (i). The formula $[S(B)]={\sum }_{A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}}{d}_{A,B}(1)[D(A)]$ for every $B\in {\mathrm{C}\mathrm{o}\mathrm{l}}^{\mathrm{\Lambda }}$ is the standard-class half of the identification of step 1.1; the coefficient dictionary $M_B={\sum }_A{d}_{A,B}(s)L_A$, $P_A={\sum }_B{d}_{A,B}(s)M_B$ is that of The dual-canonical, canonical and quasi-canonical Fock bases, and the full polynomial transition matrix, its unitriangularity and the restriction to standard simple labels are those of Parabolic Kazhdan-Lusztig polynomials give the dual-canonical transition matrix and Standard tableau monomials form a basis of the highest weight Fock submodule. This proves (i); no off-standard label is made a simple class.

The Chevalley operators (ii). The residue operators on $G_0$ are the Chevalley operators by The degenerate cyclotomic Hecke $G_0$ is the dual highest weight module, and the corresponding statement on $K_0$ for projective classes is The degenerate cyclotomic Hecke projective $K_0$ is the highest weight module; on the category-O side both statements are the integrable Fock action of The residue translation functors give the integrable Fock-space action transferred by the actual natural residue comparisons. This proves (ii).

Arbitrary consecutive $I$. Let $I$ be any nonempty consecutive set (possibly unbounded above or below), fix a content $\alpha$ and a finite list of residue operations, and choose a finite lower endpoint $k$ and upper endpoint $r$ so that the LETTER alphabet $[k,r+1]$ contains every charge and the ROOT interval $[k,r]$ contains the support of $\alpha$ and every shifted support occurring in the operations. The common-tail and double-cut compatibility of Standard tableau monomials form a basis of the highest weight Fock submodule and The highest weight Fock submodule carries the quantum-group global bases, together with the contractible-central-corner restriction of The higher-level Schur-Weyl projective generator and Schur functor and The higher-level Schur-Weyl double centralizer and Schur quotient, shows that added bottom occupancy is maximal in every column (so every new entry is forced ground) and added top occupancy is zero, and that the monomial, bar, global and projected basis equations are unchanged on the selected finite weight set. Hence the Hecke block is the same content component of the same $H_d(m)$, with the same intrinsic Specht modules, projective and simple labels, residue projectors and residue maps, and the blockwise statements of (i)-(iv) glue along these compatible finite pieces to give the arbitrary-$I$ statement. At the level of the block decomposition of the tower, the excluded blocks for $I\mathrm{\ne }\mathbb{Z}$ are real (the supported cut is proper by The degenerate cyclotomic Hecke tower and its residue functors), and no equality of $H^{\mathrm{\Lambda }}$ with ${⨁}_d{H}_d(m)$ is used.

Conclusion. Step 1.1 assembles the simultaneous full upper/lower integral lattice identifications and their perfect mutual pairing; step 1.2 proves the residue head/socle/indecomposability rule; step 1.3 proves the parameter-one normalization with its correctly directed label bijections; steps 2.1–2.2 prove the decomposition formula and Chevalley action; step 2.3 glues the actual same-block and full-basis maps for arbitrary consecutive $I$; and step 1.4 gives the level-zero ground line. These are all clauses of the statement.