Parabolic category O for a standard parabolic

Assume the Axiom of Choice (The Axiom of Choice), inherited from the published category-$\mathcal{O}$ theory. Let $\mathfrak{p}={\mathfrak{p}}_J={\mathfrak{l}}_J\oplus {\mathfrak{u}}_J$ be the standard parabolic of Standard parabolic and Levi subalgebras from a subset of simple roots, with reductive parabolic Levi ${\mathfrak{l}}_J$, nilradical ${\mathfrak{u}}_J$ and Weyl group $W_J$; the ambient category is $\mathcal{O}$ of The classical BGG category O with standard objects the Verma modules Verma modules.

(1) Dominance for the Levi. A weight $\lambda$ is $\mathfrak{p}$-dominant if $⟨\lambda ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}$ for all $\alpha \in {\mathrm{\Phi }}_J^{+}$ (this is dominance for the Levi root system; in particular $\lambda$ is integral for ${\mathrm{\Phi }}_J$; see Integral, dominant, and strictly dominant weights for the ambient notion). No integrality condition is imposed on the simple coroots outside $J$, or on the central part ${\mathfrak{z}}_J$ of the Levi.

(2) Generalized Verma modules. For a $\mathfrak{p}$-dominant $\lambda$ let $E(\lambda )$ be the finite-dimensional simple $\mathfrak{p}$-module obtained from the finite-dimensional simple ${\mathfrak{l}}_J$-module of highest weight $\lambda$ (Finite-dimensional simple modules are classified by dominant highest weights) by letting ${\mathfrak{u}}_J$ act by zero. Define the generalized Verma module

$M_{\mathfrak{p}}(\lambda ):=U(\mathfrak{g}){\otimes }_{U(\mathfrak{p})}E(\lambda )\mathrm{.}$

(3) The parabolic category. The parabolic category ${\mathcal{O}}^{\mathfrak{p}}$ is the full subcategory of $\mathcal{O}$ whose objects are the finitely generated $\mathfrak{h}$-semisimple $\mathfrak{g}$-modules on which $U(\mathfrak{p})$ acts locally finitely, that is, on which every vector lies in a finite-dimensional $\mathfrak{p}$-submodule; equivalently, every vector lies in a finite-dimensional ${\mathfrak{l}}_J$-submodule and is annihilated by a power of ${\mathfrak{u}}_J$ (local finiteness for the nilpotent ${\mathfrak{u}}_J$).

The following structural facts are part of the definition; the arguments are given in the Remarks.

(4) Generalized Vermas lie in ${\mathcal{O}}^{\mathfrak{p}}$. For each $\mathfrak{p}$-dominant $\lambda$, $M_{\mathfrak{p}}(\lambda )$ is an object of ${\mathcal{O}}^{\mathfrak{p}}$ with highest weight $\lambda$, and the natural surjection $M(\lambda )\twoheadrightarrow M_{\mathfrak{p}}(\lambda )$ exhibits $M_{\mathfrak{p}}(\lambda )$ as the largest quotient of the Verma module $M(\lambda )$ that lies in ${\mathcal{O}}^{\mathfrak{p}}$.

(5) Simples. The simple objects of ${\mathcal{O}}^{\mathfrak{p}}$ are exactly the simple highest-weight modules $L(\lambda )$ with $\lambda$ $\mathfrak{p}$-dominant (The simple objects of O).

(6) Serre property. ${\mathcal{O}}^{\mathfrak{p}}$ is closed under subobjects, quotients and extensions in $\mathcal{O}$; in particular it is an abelian subcategory of $\mathcal{O}$ stable under finite direct sums.

(7) Costandard objects. For $\mathfrak{p}$-dominant $\lambda$ put ${\mathrm{\nabla }}_{\mathfrak{p}}(\lambda ):=D{M}_{\mathfrak{p}}(\lambda )$, the restricted dual of Restricted duality is exact and involutive on O. These are the costandard objects of ${\mathcal{O}}^{\mathfrak{p}}$; each is naturally a submodule of the ordinary dual Verma $DM(\lambda )$. They need not be quotients of ordinary dual Vermas.

• Finite Levi modules. A finite-dimensional ${\mathfrak{l}}_J$-module with $\mathfrak{h}$-semisimple action splits as the direct sum of its center-weight spaces, and on each such space, which is a finite-dimensional module for the semisimple algebra ${\mathfrak{s}}_J$, Weyl's complete reducibility theorem (Weyl's complete reducibility theorem) applies. Hence such a module is a direct sum of finite-dimensional simple ${\mathfrak{l}}_J$-modules, and those are parametrized by weights integral and dominant for ${\mathrm{\Phi }}_J$, with center ${\mathfrak{z}}_J$ acting through the weight (Finite-dimensional simple modules are classified by dominant highest weights, Standard parabolic and Levi subalgebras from a subset of simple roots (3)).
• Why $M_{\mathfrak{p}}(\lambda )\in {\mathcal{O}}^{\mathfrak{p}}$ (4). Inflating $E(\lambda )$ to $\mathfrak{p}$ by ${\mathfrak{u}}_J\cdot E(\lambda )=0$ is a module structure because ${\mathfrak{u}}_J$ is an ideal of $\mathfrak{p}$ and ${\mathfrak{u}}_J$ acts on $E(\lambda )$ by zero. Poincar'e--Birkhoff--Witt (PBW gives an ordered monomial basis for the enveloping algebra) applied to the triangular decomposition $\mathfrak{g}={\mathfrak{v}}_J\oplus \mathfrak{p}$ (with ${\mathfrak{v}}_J$ the sum of the negative outside root spaces) identifies $M_{\mathfrak{p}}(\lambda )$ with $U({\mathfrak{v}}_J)\otimes E(\lambda )$ as a vector space, and more precisely each bounded negative $\zeta$-degree piece of $U({\mathfrak{v}}_J)$ is finite dimensional (Remark (5) of Standard parabolic and Levi subalgebras from a subset of simple roots) and ${\mathfrak{l}}_J$-stable. Positive ${\mathfrak{u}}_J$ raises the $\zeta$-degree and hence lowers this degree after PBW reordering; since it annihilates the degree-zero Levi part, each such finite piece is $\mathfrak{p}$-stable. This proves $\mathfrak{p}$-local finiteness of $M_{\mathfrak{p}}(\lambda )$ directly, with finite-dimensional $\mathfrak{p}$-stable pieces. The natural surjection $M(\lambda )\twoheadrightarrow M_{\mathfrak{p}}(\lambda )$ (induced by $v_{\lambda }\mapsto 1\otimes e_{\lambda }$) shows that $M_{\mathfrak{p}}(\lambda )$ is a quotient of the Verma module, hence an object of $\mathcal{O}$ (Verma and finite-dimensional weight modules belong to O); it has the same highest weight $\lambda$ and a unique simple head $L(\lambda )$, since every simple quotient is also a simple quotient of $M(\lambda )$ (A Verma module has a unique simple quotient), and all its composition factors lie below $\lambda$ because it is a quotient of $M(\lambda )$ (Every object of O has finite length, The simple objects of O).
• Largest quotient (4). For a parabolic quotient $Q$ of $M(\lambda )$, let $\bar{v}$ be the image of the highest vector. Every positive root vector kills it. Its finite Levi span is, by complete reducibility, a cyclic highest-weight copy of $E(\lambda )$ if $\bar{v}\mathrm{\ne }0$, and zero otherwise. The radical kills this entire span because it kills $\bar{v}$ and $[{\mathfrak{u}}_J,{\mathfrak{l}}_J]\subseteq {\mathfrak{u}}_J$. Tensor induction therefore factors the quotient map through $M_{\mathfrak{p}}(\lambda )$.
• Local finiteness and Serre closure (6). Inside $\mathcal{O}$, $\mathfrak{p}$-local finiteness is equivalent to local nilpotence of the negative simple root vectors $f_j$, $j\in J$. Necessity follows because $f_j$ lowers weights and a finite-dimensional weight module has finite weight support. Conversely, the cyclic Levi module of a weight vector is finitely generated, Cartan-semisimple and locally ${\mathfrak{n}}_J^{+}$-finite, hence is an object of the semisimple Levi's category $\mathcal{O}$ and has finite length. Local nilpotence of $f_j$ descends to its simple highest-weight factors. If $v$ is the top of such a factor and $k$ is least with $f_j^{k}v=0$, the relation $e_j{f}_j^{k}v=k(\lambda (h_j)-k+1)f_j^{k-1}v$ forces $\lambda (h_j)=k-1\in {\mathbb{Z}}_{\ge 0}$ (The root sl_2 triple). The finite-dimensional highest-weight classification makes each factor finite dimensional, so the cyclic Levi module is finite dimensional. A vector with several weights is handled by their finite sum. Moreover $U({\mathfrak{u}}_J)m$ is finite dimensional because ${\mathfrak{u}}_J\subseteq {\mathfrak{n}}^{+}$. PBW gives $U(\mathfrak{p})m=U({\mathfrak{l}}_J)U({\mathfrak{u}}_J)m$; the finitely many basis vectors of that radical span each have a finite Levi span. This proves local finiteness. For extensions, kill a vector's image by one power of $f_j$, then its resulting vector in the submodule by another. Local nilpotence also passes to subobjects and quotients, so this criterion proves the Serre property in the ambient abelian category.
• Simples (5). A simple $\mathfrak{g}$-module $L(\lambda )$ lies in ${\mathcal{O}}^{\mathfrak{p}}$ if and only if $\lambda$ is $\mathfrak{p}$-dominant: necessity holds because the highest weight vector generates a finite-dimensional $\mathfrak{p}$-module, whose finite-dimensional ${\mathfrak{s}}_J$-part has $\lambda$ as a highest weight of a finite Levi simple, forcing $⟨\lambda ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}$ for $\alpha \in {\mathrm{\Phi }}_J^{+}$; sufficiency holds because $L(\lambda )$ is a quotient of $M_{\mathfrak{p}}(\lambda )\in {\mathcal{O}}^{\mathfrak{p}}$ and ${\mathcal{O}}^{\mathfrak{p}}$ is closed under quotients by the previous paragraph.
• Duality and costandards (7). Restricted duality (Restricted duality is exact and involutive on O) is exact and contravariant on $\mathcal{O}$. A locally finite semisimple Levi representation with finite $\mathfrak{h}$-weight spaces is an algebraic direct sum of finite-dimensional Levi simples: taking an increasing sequence of finite-dimensional Levi submodules exhausting the space, complete reducibility splits each inclusion, the successive spans exhaust the space, and every weight meets only finitely many summands (the weight space is finite dimensional); the restricted dual is then the direct sum of the finite duals, again a finite-dimensional-module direct sum. Hence $D$ preserves ${\mathcal{O}}^{\mathfrak{p}}$; the natural map $D{M}_{\mathfrak{p}}(\lambda )\to DM(\lambda )$ is injective (dual of the Verma quotient), so ${\mathrm{\nabla }}_{\mathfrak{p}}(\lambda )$ is naturally a submodule of the ordinary dual Verma. Its simple socle is $L(\lambda )$ by duality from the simple head of $M_{\mathfrak{p}}(\lambda )$. It is not generally a quotient of $DM(\lambda )$: dualizing, that would say $M(\lambda )$ embeds in $M_{\mathfrak{p}}(\lambda )$, which fails as soon as the Verma surjection has a nonzero kernel (for instance $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2$, $J=\{s\}$ and $\lambda =0$, when $M_{\mathfrak{p}}(0)=L(0)$ is one dimensional).
• Choice. The Axiom of Choice is inherited from the published category-$\mathcal{O}$ theory quoted above (restricted duality, finite length, enough structures); no new choice principle is introduced.