The degenerate cyclotomic Hecke projective $K_0$ is the highest weight module

The split $K_0$ basis of covers. Every simple finite-dimensional $H^{\mathrm{\Lambda }}$-module has a projective cover, unique up to isomorphism, and the classes $[Y(A)]$, $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, of the covers of the simples $D(A)$ of The degenerate cyclotomic Hecke $G_0$ is the dual highest weight module form a $\mathbb{Z}$-basis of the split Grothendieck group of finite-dimensional projective modules; this is [F1] applied block by block and summed over $\alpha \in Q^{+}$.

The left adjoint on projectives. The Schur quotient $\pi$ has a left adjoint ${\pi }^{\ast }$ which is right exact on all modules and sends $H$-projectives to category-O projectives; for $A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$ one has ${\pi }^{\ast }(Y(A))=P(A)$ and $\pi {\pi }^{\ast }(Y(A))=Y(A)$, so ${\pi }^{\ast }$ recovers the standard-labelled prinjectives, and its exactness on arbitrary modules is not needed. These are the statements of The higher-level Schur-Weyl projective generator and Schur functor and Schur functor images of standard, simple, projective and tilting objects.

The standard lattice and its ground generation. The standard lattice $V(\mathrm{\Lambda }{)}_{\mathbb{Z}}$ has $\mathbb{Z}$-basis $\{P_A:A\in {\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}\}$, the source vector $P_A$ lies in the highest constituent with ${\pi }_{\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{k}}{P}_A=P_A$, and at parameter one $P_A=T_{A^{\uparrow }}=G_{\mathrm{l}\mathrm{o}\mathrm{w}}(A)$, so $V(\mathrm{\Lambda }{)}_{\mathbb{Z}}$ is integrally generated by its ground vector $P_{A_{\mathrm{\Lambda }}}$ under the divided-power operators: the actual lower lattice is $V_{\mathcal{A}}(\mathrm{\Lambda })=U_{\mathcal{A}}^{-}{v}_{\mathrm{\Lambda }}$ by The lower global basis of $U_q^{-}$ and of a highest weight module. This is the parameter-one specialization of The highest weight Fock submodule carries the quantum-group global bases, together with the basis statements of The dual-canonical, canonical and quasi-canonical Fock bases and Simples, projectives and tiltings map to the three distinguished Fock bases.

The identification $K_0\cong V(\mathrm{\Lambda }{)}_{\mathbb{Z}}$. Define $\mathrm{\Psi }$ on the basis of step 1.1 by $\mathrm{\Psi }([Y(A)])=P_A$; by step 1.3 both sides are free $\mathbb{Z}$-modules on the same index set ${\mathrm{S}\mathrm{t}\mathrm{d}}^{\mathrm{\Lambda }}$, so $\mathrm{\Psi }$ is a group isomorphism onto $V(\mathrm{\Lambda }{)}_{\mathbb{Z}}$, giving (i). The identification is realized functorially by ${\pi }^{\ast }$: step 1.2 shows that on projective classes the composite of ${\pi }^{\ast }$ with the category-O identification $[P(A)]\mapsto P_A$ at parameter one (the projective half of the tableau $K_0$/Fock identification of The parabolic category-O Grothendieck group is the integral Fock module, proved in Simples, projectives and tiltings map to the three distinguished Fock bases) is $\mathrm{\Psi }$, so no exactness of ${\pi }^{\ast }$ on non-projectives is used.

The inclusion is the lattice inclusion. The embedding of projectives into all finite-dimensional modules induces a map on Grothendieck groups; in the category of finite-dimensional $H^{\mathrm{\Lambda }}$-modules a projective class $[Y(A)]$ maps to the class of the same module, which under the identification $\mathrm{\Phi }$ of The degenerate cyclotomic Hecke $G_0$ is the dual highest weight module is the costandard coordinate vector of the module $Y(A)$. By step 1.2, $\pi {\pi }^{\ast }(Y(A))=Y(A)$, so this vector equals the image of $[P(A)]$ under the exact quotient, namely $P_A$ by step 1.3. Hence the composite $\mathrm{\Phi }\circ (\text{inclusion})\circ {\mathrm{\Psi }}^{-1}$ is the identity on the basis $\{P_A\}$, i.e. the natural inclusion $V(\mathrm{\Lambda }{)}_{\mathbb{Z}}\hookrightarrow V(\mathrm{\Lambda }{)}_{\mathbb{Z}}^{\ast }$. It is injective because the vectors $P_A$ are linearly independent in the rational highest module, and hence also as elements of its full costandard lattice. The normalized highest form is nondegenerate on each finite weight space; no claim that the embedded standard and costandard lattices coincide is used. This proves (ii).

The Cartan pairing is the normalized highest pairing. By [F2] the Cartan pairing descends to $K_0\times G_0$ and pairs the two bases dually; under the identifications of step 2.1 and of The degenerate cyclotomic Hecke $G_0$ is the dual highest weight module the value on basis vectors is computed on the Fock side, where the pairing between the standard and costandard families at parameter one is the normalized highest pairing $C_0$ with $C_0(P_A,D_B)={\delta }_{AB}$ by The Fock pairing matches the projective-simple and standard-costandard pairings. This proves (iii); in particular the costandard vectors entering the pairing are the projected simples ${\pi }_{\mathrm{F}\mathrm{o}\mathrm{c}\mathrm{k}}{L}_B=D_B$, not unprojected monomials.

Chevalley operators on $K_0$. Use the actual exact residue biadjunction from the tower supplier. For projective $P$ over the source, $\mathrm{Hom}(f_{i}P,-)=\mathrm{Hom}(P,e_i-)$ is exact, so $f_{i}P$ is projective; for projective $Q$ over the target, $\mathrm{Hom}(e_{i}Q,-)=\mathrm{Hom}(Q,f_i-)$ is exact, so $e_{i}Q$ is projective. Applying an idempotent by itself would not prove projectivity over a different algebra. These functors induce the adjoint operators of [F3] under the Hom pairing. Their $G_0$ counterparts are the already identified Chevalley operators, and the normalized highest pairing makes $e_i,f_i$ adjoint, so the same identification holds on $K_0$.

The full integral extension. The arbitrary-$I$ construction of The degenerate cyclotomic Hecke $G_0$ is the dual highest weight module fixes each selected finite-dimensional algebra $H_{\alpha }^{\mathrm{\Lambda }}$, its simple labels and residue bimodules. An algebra identity fixes projective covers of the identified simples, uniquely up to isomorphism, so it fixes their split-$K_0$ basis. The lower lattice map is $P_A\mapsto P_{\mathrm{ȷ}A}$ under the same common-ground-tail label map; the actual lower/global cut compatibility makes this an integral basis identification commuting with divided powers and with the pairing against the full upper basis. It therefore transports $\mathrm{\Psi }$, the Cartan map and its pairing on each content; overlaps agree by the intrinsic algebra identity and the composable tail maps. Their weightwise direct sum proves (i)–(iv) for every admitted consecutive $I$, without claiming integral ground cyclicity for the costandard lattice.

Conclusion. Step 2.1 proves (i), step 3.1 proves the lattice-inclusion statement (ii), step 3.2 proves the pairing statement (iii) with its projected-simple qualification, step 4.1 proves (iv), step 5.1 gives their full integral arbitrary-$I$ extension, and the properties of the right-exact left adjoint stated in step 1.2 are exactly the assertion (v); at no point is exactness of ${\pi }^{\ast }$ on arbitrary modules, or a recovery of all category-O projectives, assumed.