The integral parabolic category-O blocks for higher-level Schur-Weyl duality

Reduction to the semisimple ambient algebra. For $n\ge 2$ write $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_n\oplus \mathbb{C}z$, $z=I_n$, and $\mathfrak{p}={\mathfrak{p}}^{\mathrm{\prime }}\oplus \mathbb{C}z$ with ${\mathfrak{p}}^{\mathrm{\prime }}=\mathfrak{p}\cap {\mathfrak{s}\mathfrak{l}}_n$. The universal enveloping algebra is $U({\mathfrak{s}\mathfrak{l}}_n)\otimes \mathbb{C}[z]$: the two factors commute, and the inverse maps follow from their defining universal properties. On the category where $z$ acts by a fixed scalar $c$, restriction to ${\mathfrak{s}\mathfrak{l}}_n$ and extension by $z=c$ are inverse exact functors. Finite generation, Cartan weight decomposition and local parabolic finiteness agree under these functors; induction of a finite Levi module agrees with the semisimple generalized Verma construction. Thus simples, projective covers, flags, duality, reciprocity and character bases in [F1]–[F2] transport to this fixed-centre category. Transpose fixes $z$, so restricted duality retains $c$. A finitely generated Cartan-semisimple ${\mathfrak{g}\mathfrak{l}}_n$ object has only finitely many $z$ eigenvalues: decompose its finite generating set into weight vectors, and centrality keeps every generated summand in its eigenvalue. Consequently these fixed-centre categories form an exact finite-support direct sum. Moreover $Z(U(\mathfrak{g}))=Z(U({\mathfrak{s}\mathfrak{l}}_n))\otimes \mathbb{C}[z]$, since commuting a polynomial in $z$ with ${\mathfrak{s}\mathfrak{l}}_n$ forces each coefficient to be central. Hence a highest-weight central character is determined by $c$ and the semisimple central character of the restricted weight. For $n=1$, the same construction reduces to finite-dimensional vector spaces with $z=c$; its one-simple block and all the stated structural assertions are immediate.

Levi dominance and the shifted multiset. Read each strict column from top to bottom, as stipulated for the ordinary highest-weight tuple. Then $a_j>a_{j+1}$ within each Levi block and the integral difference is at least one. Consequently ${\lambda }_{A,j}-{\lambda }_{A,j+1}=a_j-a_{j+1}-1\ge 0$, so ${\lambda }_A$ is weakly dominant and integral for the Levi. Consecutive ground entries may give equality of these highest-weight coordinates, which is allowed. This ordinary tuple follows the same normally ordered column reading. Therefore $L(A)$ and $M(A)$ are genuine objects of ${\mathcal{O}}^{\mathfrak{p}}$ by [F1] transported through step 1.1. Because ${\rho }_j=(n+1-2j)\mathrm{/}2$ has total sum $0$, the $j$-th shifted coordinate is ${\lambda }_{A,j}+{\rho }_j=a_j+(n-1)\mathrm{/}2$, so the shifted-coordinate multiset differs from the entry multiset by the common scalar $(n-1)\mathrm{/}2$; the centre scalar is $c_A={\sum }_j{\lambda }_{A,j}={\sum }_j{a}_j+n(n-1)\mathrm{/}2$. Restriction subtracts $c_A\mathrm{/}n$ from every shifted coordinate. By [F2] on ${\mathfrak{s}\mathfrak{l}}_n$ and step 1.1, equality of central characters is therefore exactly equality of the full shifted-coordinate multisets (their sums already recover $c_A$), with no undeclared full-staircase shift.

The block and its labels. Fix an admitted entry multiset $\alpha$; its shifted-coordinate multiset gives one central character, and the simple modules $L(A)$ of that central character form exactly the labels ${\mathrm{C}\mathrm{o}\mathrm{l}}_{\alpha }^{\mathrm{\Lambda }}$ with $\mathrm{w}\mathrm{t}(A)=\mathrm{\Lambda }-\alpha$. The induced modules $M(A)$ are the generalized standards of the block, the projectives $P(A)$ and costandards $\mathrm{\nabla }(A)$ exist by [F1], and BGG reciprocity and restricted duality hold blockwise because they are inherited from the ambient parabolic category.

Filtered and Grothendieck structure. By [F1] every object of the block has finite length with simple factors among the $L(A)$, and the projective covers have finite $M(A)$-flags; the simple and standard classes are bases of the block Grothendieck group with the stated unitriangular character matrix. By [F3] the block is a single linkage block, so all its labels are connected by actual indecomposable-tilting edges, including repeated-entry singular central characters; the direct sum over the admitted multisets is ${\mathcal{O}}^{\mathrm{\Lambda }}$.

Opposite blocks. Reversing the Levi order and reading tableaux by columns in the reverse direction swaps the blocks and reverses the prefix order of the labels; by [F4] the reversed labels are the same tableaux with reversed column readings, and the corresponding central characters are the reversed-coordinate multisets. This is the opposite-block dictionary.