The parabolic category-O Grothendieck group is the integral Fock module

The standard-class map is an isomorphism. In each finite block the standard classes form a basis of the Grothendieck group and characters are injective, while the residue functors act on the standard basis by the explicit single-entry moves; the Fock monomials satisfy the identical relations and have the same degrees. Hence $[M(A)]\mapsto M_A$ is a degree-preserving isomorphism in each block, and it extends to the direct sum over the blocks because the direct sum decomposition is by central character on both sides.

The transition matrix and duality at $1$. Write $D_{A,B}=d_{A,B}(1)=[M(B):L(A)]$ with simple rows and standard columns. Then $[M(B)]={\sum }_A{D}_{A,B}[L(A)]$, so it is the simple-to-standard expansion that uses $D^{-1}$. BGG reciprocity instead gives $[P(A):M(B)]=D_{A,B}$, hence $[P(A)]={\sum }_B{D}_{A,B}[M(B)]$; this matrix is the transposed standard/simple multiplicity array when rows are projective labels, not its inverse. Under the standard-class map these are exactly the defining $P_A$ and $L_A$ expansions. On the actual finite type-$A$ cuts, each nonconstant term of the quasi-$R$ root-factor expansion has a factor $s-s^{-1}$ and coefficients regular at $s=1$ (the quantum-factorial denominators have nonzero ordinary-factorial values there). Thus the tensor bar matrix specializes to the identity; passage to the primitive wedge factors preserves this specialization. The source monomial Gram matrices, which are its inverse, are consequently the identity at $1$. Multiplying the $D$ projective array by the $D^{-1}$ simple array yields $⟨P_A,L_B⟩={\delta }_{AB}$. No inverse BGG multiplicity matrix is asserted.

The general Hom pairing. For a projective $P$ and a finite-length $N$, the functor $\mathrm{Hom}(P,-)$ is exact, so its dimension on $N$ is additive on a composition series and equals the pairing of $[P]$ with $[N]$ under the $P\mathrm{/}L$ duality of step 2.1; in particular $⟨P_A,L_B⟩={\delta }_{AB}$. This is a projective/simple statement: the standard/simple matrix is the unitriangular mixed multiplicity matrix, not the identity, so no standard/simple orthogonality is claimed.

Conclusion. The standard-class map identifies the direct-sum $K_0$ with the integral Fock module compatibly with degrees and the residue Chevalley action; the projective/simple pairing is the Fock pairing at $s=1$, giving $\dim \mathrm{Hom}(P,N)=⟨[P],[N]⟩$ and $⟨P_A,L_B⟩={\delta }_{AB}$, while standard/simple orthogonality is false; the projective, simple and tilting bases are proved separately.