The degenerate cyclotomic Hecke $G_0$ is the dual highest weight module

The Schur quotient and its simple images. Assume AC ([F3]) as the imported suppliers do. In the admissible stable realization, the exact Schur quotient $\pi$ of Schur functor images of standard, simple, projective and tilting objects is the Serre quotient killing exactly the simples $L(A)$ with $A\mathrm{\notin }{\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$; it satisfies $\pi (N(A))=S(A)$ for the intrinsic Specht modules and $\pi (L(A))=D(A)$ for $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, these $D(A)$ being pairwise nonisomorphic and exhausting the simples of $\mathrm{R}\mathrm{e}\mathrm{p}(H^{\mathrm{\Lambda }})$, while $\pi (L(A))=0$ for $A\mathrm{\notin }{\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$. By [F1] the classes $[D(A)]$, $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, form a $\mathbb{Z}$-basis of $G_0(\mathrm{R}\mathrm{e}\mathrm{p}(H^{\mathrm{\Lambda }}))$. The ingredients are the Schur-functor image theorem, exactness of $\pi$ and the double-centralizer/quotient description of The higher-level Schur-Weyl double centralizer and Schur quotient.

The Fock side of the dictionary. On the Fock side the normalized highest projection ${\pi }_{\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{k}}$ is the linear map with ${\pi }_{\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{k}}{L}_A=D_A$ for $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ and ${\pi }_{\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{k}}{L}_A=0$ for $A\mathrm{\notin }{\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, and ${\pi }_{\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{k}}{M}_B=S_B$ with $M_B={\sum }_A{d}_{A,B}(s)L_A$ and $S_B={\sum }_{A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}}{d}_{A,B}(1)D_A$ at source parameter $1$; the vectors $\{D_A\}$ span the full integral costandard lattice $V(\mathrm{\Lambda }{)}_{\mathbb{Z}}^{\ast }$ and the vectors $\{S_B:B\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}\}$ form the standard monomial basis of that lattice, a second integral basis. These are the definitions and the standard-monomial basis theorem of The dual-canonical, canonical and quasi-canonical Fock bases and Standard tableau monomials form a basis of the highest weight Fock submodule, together with the O-side dictionary $[L(A)]\mapsto L_A$, $[P(A)]\mapsto P_A$, $[T(A)]\mapsto T_A$ of Simples, projectives and tiltings map to the three distinguished Fock bases.

The O-side Grothendieck group is the integral Fock module. At parameter one the parabolic category-O Grothendieck group is identified with the integral Fock module by $[M(A)]\mapsto M_A$, with $\dim \mathrm{Hom}(P,N)=⟨[P],[N]⟩$ and $⟨P_A,L_B⟩={\delta }_{AB}$, by The parabolic category-O Grothendieck group is the integral Fock module; here $M(A)$, $L(A)$, $P(A)$ are the generalized standard, simple and projective objects of the genuine block ${\mathcal{O}}_{\alpha }^{\mathrm{\Lambda }}$ of The integral parabolic category-O blocks for higher-level Schur-Weyl duality.

The identification of $G_0$ with the costandard lattice. Define $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:G_0(\mathrm{R}\mathrm{e}\mathrm{p}(H^{\mathrm{\Lambda }}))\to V(\mathrm{\Lambda }{)}_{\mathbb{Z}}^{\ast }$ on the basis of step 1.1 by $\mathrm{\Phi }([D(A)])=D_A$. This is a well-defined group isomorphism because both sides are free $\mathbb{Z}$-modules with the same basis set ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$: on the module side by [F1] and step 1.1, on the Fock side by step 1.2, which exhibits $\{D_A\}$ as a $\mathbb{Z}$-basis of the FULL integral costandard lattice and makes no integral ground-generation claim. This proves (i) for the simple classes and, together with step 1.2, identifies $\mathrm{\Phi }([S(B)])$ with $S_B$.

The Specht classes. For the intrinsic Specht module $S(B)=\pi (N(B))$ of step 1.1, apply the exact functor $\pi$ to a Jordan–Hölder composition series of $N(B)$; exactness and [F2] give $[S(B)]={\sum }_A[N(B):L(A)][D(A)]$ in $G_0$, the sum running over the simples, i.e. over $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ by step 1.1. The multiplicity $[N(B):L(A)]$ equals the Fock coefficient $d_{A,B}(1)$ of step 1.2 by the O-side multiplicity dictionary of step 1.3 and the Schur-image theorem of step 1.1. Hence $\mathrm{\Phi }([S(B)])={\sum }_{A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}}{d}_{A,B}(1)D_A=S_B$, and the square submatrix on standard columns is unitriangular, so $\{S_B:B\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}\}$ is an integral basis; this completes (i).

Divided-power integral forms. The lower library global vectors $G_{\mathrm{l}\mathrm{o}\mathrm{w}}(A)=T_{A^{\uparrow }}=s^{-a(\alpha )}{P}_A$ and the form-dual upper vectors $G_{\mathrm{u}\mathrm{p}}(A)=s^{a(\alpha )}{D}_A$ with $a(\alpha )=(2\mathrm{\Lambda }-\alpha ,\alpha )\mathrm{/}2$ span integral forms on which the divided powers of the Chevalley generators act integrally, by The highest weight Fock submodule carries the quantum-group global bases; at parameter one the standard lattice is therefore generated over $\mathbb{Z}$ by its ground vector $P_{A_{\mathrm{\Lambda }}}$ under the divided-power operators $f_i^{(n)}$, while the costandard lattice is its full weightwise $C_0$-dual: at $s=1$, $C_0(P_A,D_B)={\delta }_{AB}$ identifies it with ${⨁}_{\alpha }{\mathrm{Hom}}_{\mathbb{Z}}(V(\mathrm{\Lambda }{)}_{\mathbb{Z},\alpha },\mathbb{Z})$. In particular it is stable under divided powers, since the $e_i^{(n)}$ and $f_i^{(n)}$ are adjoint and preserve the lower lattice. Nondegeneracy identifies their rational scalar extensions, but does not identify the two integral lattices. The ground-generation use is the actual definition $V_{\mathcal{A}}(\mathrm{\Lambda })=U_{\mathcal{A}}^{-}{v}_{\mathrm{\Lambda }}$ of The lower global basis of $U_q^{-}$ and of a highest weight module, specialized at one; it concerns the standard lattice only. This gives the divided-power clause of (ii) and separates the two lattices.

Residue operators. The residue functors $e_i^H,f_i^H$ are exact, so they induce endomorphisms of $G_0$ by [F2]. The natural intertwining isomorphisms $f_i^H\pi \cong \pi f_i^O$ and $e_i^H\pi \cong \pi e_i^O$ of Category-O translation intertwines residue induction and restriction, together with the category-O Chevalley action of The residue translation functors give the integrable Fock-space action, show that $[\pi ]$ is a module map. Steps 1.1–2.2 identify its kernel with the span of the off-standard simple classes, precisely the kernel of the Fock projection. Thus the induced quotient map $\mathrm{\Phi }$ intertwines $e_i,f_i$, proving the ordinary Chevalley clause of (ii). No head, socle or indecomposability assertion is inferred merely from exactness of a quotient.

Level zero. If the level is $0$, then $\mathrm{\Lambda }=0$, the only simple is the ground class in degree $0$, and the tower is $H_0^{\mathrm{\Lambda }}=\mathbb{C}$ by the level-$0$ convention of The degenerate cyclotomic Hecke tower and its residue functors; then $G_0=\mathbb{Z}[D(\mathrm{\varnothing })]\cong \mathbb{Z}{D}_{\mathrm{\varnothing }}$ is the ground line alone, with both lattices equal to the ground line and every residue operator zero. This is the level-$0$ clause of (v).

Crystal reachability and composition factors. The source upper ground component is exactly ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ by Standard tableau monomials form a basis of the highest weight Fock submodule. The normalized highest form is nondegenerate on each finite weight space, so over $\mathbb{Q}$ it identifies the full costandard lattice's scalar extension with the irreducible highest module. Its highest vector therefore generates that module under lowering divided powers. Consequently every basis vector $D_A$ is a rational linear combination of lowering words applied to the ground vector. By step 3.1 each ordinary lowering word is the class of the corresponding exact residue-word module, with nonnegative simple-class coefficients by [F1]. If $D(A)$ occurred in none of these modules, every such word and every rational linear combination would have zero $D_A$ coefficient, contrary to the coefficient one in $D_A$. Thus some residue word contains $D(A)$ as a composition factor. This proves (iv) without requiring integral costandard generation or transport of head and socle through the quotient.

The maps for arbitrary consecutive $I$. Fix a finite set of contents and residue words. Choose $J=[k,r]$ so that $[k,r+1]$ contains every charge and $J$ contains every intermediate root support; decrease $k$ further to reach the needed stable range. The Hecke-side map is literally the identity on $H_d(m)e_{\alpha }$: its spectral idempotent is the same sum over words of content $\alpha$, independently of whether the ambient supported interval is $I$ or $J$. It therefore fixes the intrinsic residue corner $u_{\alpha ,i}$ and the bimodules defining $e_i,f_i$. On tableaux the lower transition adjoins the prescribed ground entries to every column, and the upper transition regards the same entries in a larger alphabet with no new top entries. Removing that common tail is the inverse on the selected weight. Column ground differences, hence intrinsic Specht partitions, are unchanged. The actual finite-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 gives $D_{\mathrm{ȷ}A}=\mathrm{ȷ}{D}_A$, $S_{\mathrm{ȷ}B}=\mathrm{ȷ}{S}_B$, $P_{\mathrm{ȷ}A}=\mathrm{ȷ}{P}_A$ and commuting normalized highest projections. Thus both upper and lower lattices are identified by their full bases, not just their rational spans. On overlapping cuts these maps compose by adjoining/removing the same ground entries and by the intrinsic Hecke identity; the normalized projections agree by highest-vector uniqueness. Every content, vector and finite residue word fits such a cut. Applying the bounded-below identifications and the rational-cyclicity argument of steps 1.1–4.1 there, and taking the compatible weightwise direct sum, proves all clauses for arbitrary admitted $I$. No full degree algebra is identified with a proper supported cut.

Rational ground cyclicity. The normalized $E\mathrm{/}F$-contravariant highest form is nondegenerate on each finite weight space. Thus its weightwise duality identifies $V(\mathrm{\Lambda }{)}_{\mathbb{Q}}^{\ast }$, interpreted as the direct sum of the costandard weight spaces, with the irreducible rational highest module. The ground vector maps to the normalized highest vector; its orbit under lowering divided powers therefore spans every weight space, as used in step 4.1. This proves the rational clause of (iii). The full unrestricted algebraic dual of an infinite-dimensional module is not used.

The rank-one index-two witness. Take $I=\{0\}$ and $\mathrm{\Lambda }=2{\mathrm{\Lambda }}_0$. The supported degree-one block is $B_0=\mathbb{C}[x]\mathrm{/}{x}^2$ and the supported degree-two block is $M_2(\mathbb{C})$, with its simple represented by $x_1=N=E_{12}$, $s_1=-(E_{12}+E_{21})$ and $x_2=-N$. For completeness the whole degree-two presentation gives an isomorphism $H_2(0,0)\cong M_2(\mathbb{C})\oplus B_{+}\oplus B_{-}$ with $B_{\pm }=\mathbb{C}[ϵ]\mathrm{/}{ϵ}^2$: send $(x_1,x_2,s_1)$ to $(N,-N,-J)$ in the matrix factor and to $(ϵ,\pm 1+ϵ,\pm 1)$ in the other factors. The eight PBW spanning words are $1,x_1,x_2,x_1{x}_2,s_1,x_1{s}_1,x_2{s}_1,x_1{x}_2{s}_1$. The central sum $z=x_1+x_2$ has spectra $0,1,-1$ on the three factors; its Hermite polynomials $(1-z^2{)}^2$ and $\{1-(1-z^2{)}^2\pm (3z-z^3)\mathrm{/}2\}\mathrm{/}2$ project onto the three factors. Each factor is generated by the images, making the map onto an eight-dimensional product surjective and hence an isomorphism by the spanning bound. The all-zero-residue cut is exactly the matrix factor. In degree three an all-zero-residue module would have $x_2=-x_1$, $x_1^2=0$ from the matrix factor; $s_2$ commutes with $x_1$ and therefore with $x_2$, so $x_3=s_2{x}_2{s}_2+s_2=x_2+s_2$ has eigenvalues $\pm 1$, contradiction. The same argument excludes all higher degrees. The first dot is not zero on the matrix simple. Induction from degree zero gives the regular $B_0$-module, with two simple factors. The matrix algebra is right-free of rank two over $B_0$ under $x\mapsto N$, so induction of its one-dimensional simple has dimension two and is the matrix simple; restricting that matrix simple gives the regular $B_0$-module. Restricting the degree-one simple to degree zero has dimension one. Thus the residue Ind/Res computations give $f{D}_{00}=2D_{01}$, $f{D}_{01}=D_{11}$ and $e{D}_{01}=D_{00}$, $e{D}_{11}=2D_{01}$, with the end actions zero. Their divided squares give $f^{(2)}{D}_{00}=D_{11}$ and $e^{(2)}{D}_{11}=D_{00}$. Therefore $\mathbb{Z}{D}_{00}\oplus 2\mathbb{Z}{D}_{01}\oplus \mathbb{Z}{D}_{11}$ is stable under $e,f$ and all their divided powers, contains the ground vector and excludes $D_{01}$. This is a proper integral costandard submodule, whereas over $\mathbb{Q}$ division by two recovers $D_{01}$. Degree one is not called a corner sitting inside degree two.

Conclusion. Step 2.1 identifies $G_0$ with the full integral costandard lattice and step 2.2 computes the Specht classes, proving (i); step 2.3 gives the divided-power integral forms and step 3.1 identifies the residue operators with the Chevalley operators, proving (ii); step 5.1 proves rational ground cyclicity, step 2.3 proves integral ground generation of the standard lattice and step 6.1 exhibits the index-two failure of integral costandard cyclicity, proving (iii) and its sharpness; step 4.1 proves the crystal reachability statement (iv); and step 3.2 (level zero) together with steps 6.1 and 4.2 separates the rational-cyclicity and categorical reachability conclusions from integral costandard cyclicity, as asserted in (v); step 4.2 supplies the arbitrary-$I$ maps and full integral extension. No step identifies $\{D_A\}$ with the standard divided-power lattice, and no integral costandard generation is used.