Category O Finiteness Duality and Blocks — Examples

Given: The Axiom of Choice (The Axiom of Choice), the setting above and the hypotheses in the example.

The category $\mathcal{O}$ is the categorical direct sum of its generalized central-character subcategories, and each object has only finitely many nonzero character components (Generalized central-character decomposition of O).

Under the Axiom of Choice, ${\chi }_{\lambda }={\chi }_{\mu }$ if and only if $\mu \in W\cdot \lambda$ (Central characters are dot-Weyl orbits).

Under the Axiom of Choice, every object of $\mathcal{O}$ has finite length (Every object of O has finite length), and its simple objects are exactly the distinct $L(\eta )$ (The simple objects of O).

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. Restricted Chevalley duality is an exact contravariant equivalence $D:\mathcal{O}\to {\mathcal{O}}^{\mathrm{o}\mathrm{p}}$, with a natural isomorphism $D^2\cong \mathrm{id}$. It preserves each weight-space dimension, the formal character, and every simple composition multiplicity. (Restricted duality is exact and involutive on O)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. The costandard object $\mathrm{\nabla }(\lambda )$ has a unique simple submodule, isomorphic to $L(\lambda )$. Its socle, the sum of all simple submodules, is that submodule. (The simple socle of a costandard object)

If $⟨\lambda +\rho ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{>0}$, there is an embedding $M(s_i\cdot \lambda )\hookrightarrow M(\lambda )$. (Simple-reflection embeddings of Verma modules)

The proper submodule $J(\lambda )$ which is the sum of all proper submodules is the unique maximal submodule of $M(\lambda )$. The quotient $L(\lambda ):=M(\lambda )\mathrm{/}J(\lambda )$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)

The standard ${\mathfrak{s}\mathfrak{l}}_2$ generators satisfy $[e,f]=h$ and $[h,f]=-2f$ (The special linear Lie algebra sl_2). The induced Verma highest vector satisfies $e{v}_{\eta }=0$ and $h{v}_{\eta }=\eta v_{\eta }$ (Verma modules), and PBW makes $v_k=f^k{v}_{\eta }$ ($k\ge 0$) a basis (The PBW model of a Verma module).

The weights of $M(\lambda )$ are exactly $\lambda -\beta$ for $\beta \in Q^{+}$; every weight space is finite dimensional, and $M(\lambda {)}_{\lambda }=\mathbb{C}{v}_{\lambda }$. (Weights of a Verma module lie below lambda)

For every highest weight $\eta$, $D(L(\eta ))\cong L(\eta )$ as $\mathfrak{g}$-modules. (Restricted self-duality of simple highest-weight modules)

The standard and costandard objects are $\mathrm{\Delta }(\eta )=M(\eta )$ and $\mathrm{\nabla }(\eta )=D(M(\eta ))$. (Standard and costandard objects)

Given: The Axiom of Choice, the setting above, and the hypotheses in the example.

Every object of $\mathcal{O}$ is a direct sum of weight spaces, is finitely generated, and is stable under root operators, which shift weights by elements of the root lattice $Q$ (The classical BGG category O).

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. Every short exact sequence $0\to M(\lambda )\to E\stackrel{p}{\to }M(\lambda )\to 0$ in $\mathcal{O}$ splits. (Verma self-extensions in O split)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. For $M,N\in \mathcal{O}$, ${\mathrm{Hom}}_{\mathcal{O}}(M,N)$ is finite dimensional. For every weight $\lambda$, ${\mathrm{End}}_{\mathcal{O}}(L(\lambda ))=\mathbb{C}\mathrm{id}$. (Finite-dimensional Hom spaces in O)

The Verma PBW model gives the basis $f^r{v}_{\eta }$ ($r\ge 0$) of $M(\eta )$ (The PBW model of a Verma module).

Let $Z=Z(U(\mathfrak{g}))$, let $\chi :Z\to \mathbb{C}$ be a unital complex-algebra character, and put ${\mathfrak{m}}_{\chi }=\ker \chi$. For $M\in \mathcal{O}$, the generalized central-character submodule is $M_{\chi }=\{v\in M:{\mathfrak{m}}_{\chi }^{N}v=0\text{ for some }N\ge 1\}$, and ${\mathcal{O}}_{\chi }$ consists of the objects with $M=M_{\chi }$. (Generalized central-character subcategories)

The simple objects of $\mathcal{O}$ are exactly the modules $L(\mu )$, $\mu \in {\mathfrak{h}}^{\ast }$, and $L(\mu )\cong L(\nu )$ if and only if $\mu =\nu$. (The simple objects of O)

If a cyclic highest-weight module has highest weight $\mu$, then every $z\in Z(U(\mathfrak{g}))$ acts on it by the scalar $\mathrm{pr}(z)(\mu )={\chi }_{\mu }(z)$. (The Harish-Chandra projection computes the highest-weight scalar)

Under the given Choice premise, the highest-weight central characters satisfy ${\chi }_{\mu }={\chi }_{\lambda }$ if and only if $\mu \in W\cdot \lambda$. (Central characters are dot-Weyl orbits)

Under the given Choice premise, every object of $\mathcal{O}$ has a finite composition series and is both Noetherian and Artinian. The length of zero is zero. (Every object of O has finite length)

The central-character subcategories ${\mathcal{O}}_{\chi }$ are categorical direct summands of $\mathcal{O}$, with exact projections (Generalized central-character decomposition of O).

Given: The Axiom of Choice, the setting above, and the hypotheses in the example.

The category $\mathcal{O}$ is the categorical direct sum of its generalized central-character subcategories ${\mathcal{O}}_{\chi }$, with exact component projections (Generalized central-character decomposition of O).

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. For a weight $\lambda$, define ${\mathrm{\Phi }}_{\lambda }=\{\alpha \in \mathrm{\Phi }:⟨\lambda +\rho ,{\alpha }^{\vee }⟩\in \mathbb{Z}\},W_{\lambda }=⟨s_{\alpha }:\alpha \in {\mathrm{\Phi }}_{\lambda }⟩\subset W\mathrm{.}$ Reflections and coroots are those of def-root-reflections-and-the-weyl-group-action, with the shift from def-weyl-vector-rho-for-a-chosen-positive-system. The integral-reflection linkage class through $\lambda$ is $W_{\lambda }\cdot \lambda$. The word integral includes zero and negative integral pairings. If ${\mathrm{\Phi }}_{\lambda }$ is empty the generated group is $\{1\}$. This definition uses generating reflections; no identification with a root-lattice-coset stabilizer is assumed. (The integral Weyl group of a weight)

Under the given Choice premise, let ${\chi }_{\lambda }$ and ${\chi }_{\mu }$ be the central characters obtained from highest weights $\lambda$ and $\mu$. Then ${\chi }_{\lambda }={\chi }_{\mu }\text{if and only if}\mu \in W\cdot \lambda ,$ where $W\cdot \lambda :=\{w(\lambda +\rho )-\rho :w\in W\}$. (Central characters are dot-Weyl orbits)

Let $\mathrm{\Phi }$ be the root system from thm-the-root-set-is-a-reduced-crystallographic-root-system. For a root $\alpha$, define the corresponding coroot by ${\alpha }^{\vee }:=\frac{2H_{\alpha }}{\alpha (H_{\alpha })}\in \mathfrak{h},$ where $H_{\alpha }$ is the vector from def-killing-dual-vector-attached-to-a-root. The associated root reflection is the linear map $s_{\alpha }(\lambda ):=\lambda -\lambda ({\alpha }^{\vee })\alpha (\lambda \in {\mathfrak{h}}^{\ast })\mathrm{.}$ The subgroup of $\mathrm{GL}({\mathfrak{h}}^{\ast })$ generated by these reflections is the Weyl group $W$, acting on ${\mathfrak{h}}^{\ast }$ in the usual way. (Root reflections and the Weyl group action)

Every simple object of $\mathcal{O}$ is $L(\eta )$ for a unique highest weight $\eta$ (The simple objects of O). Its generalized-character membership means each $z-\chi (z)$ acts locally nilpotently (Generalized central-character subcategories), while $z$ acts on any cyclic highest-weight module by the scalar ${\chi }_{\eta }(z)$ (The Harish-Chandra projection computes the highest-weight scalar).

Under the given Choice premise, every object of $\mathcal{O}$ has a finite composition series (Every object of O has finite length). Every Verma module $M(\eta )$ has a unique simple quotient $L(\eta )$ (A Verma module has a unique simple quotient).

A positive integral simple-root pairing gives a nonzero Verma embedding in the reflected direction (Simple-reflection embeddings of Verma modules).

Given: The setting above and the hypotheses in the statement refuted.

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. For an $\mathfrak{h}$-semisimple module $M$ with finite-dimensional weight spaces, its restricted Chevalley dual is $D(M)={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{M}_{\mu }^{\ast },(x\phi )(m)=\phi (\tau (x)m)(x\in U(\mathfrak{g}))\mathrm{.}$ Here each functional is extended by zero on the other weight spaces and $\tau$ is the fixed anti-involution of def-chevalley-contravariant-form. In particular $\tau (h)=h$ and $D(M{)}_{\mu }=M_{\mu }^{\ast }$. A map $f:M\to N$ induces $D(f):D(N)\to D(M)$ by precomposition. The action law follows from $\tau (xy)=\tau (y)\tau (x)$; a root vector of weight $\alpha$ sends $M_{\mu }^{\ast }$ to $M_{\mu +\alpha }^{\ast }$, so the restricted sum is stable. This is a complex-linear algebraic dual, with no conjugation. Ordinary Lie-module duality has a minus sign and reverses weights; twisting that dual by the Lie automorphism $x\mapsto -\tau (x)$ gives the convention used here. (Restricted Chevalley dual)

For $\lambda \in {\mathfrak{h}}^{\ast }$, the Verma module is the induced $\mathfrak{g}$-module $M(\lambda ):=U(\mathfrak{g}){\otimes }_{U(\mathfrak{b})}{\mathbb{C}}_{\lambda },$ where $U(\mathfrak{g})$ is the quotient algebra of def-universal-enveloping-algebra-as-a-tensor-quotient and ${\mathbb{C}}_{\lambda }$ is def-one-dimensional-borel-module-of-weight-lambda. Write $v_{\lambda }:=1\otimes c_{\lambda }$. Thus $M(\lambda )$ is the quotient of $U(\mathfrak{g})$ by the left ideal generated by $x$ for $x\in {\mathfrak{n}}^{+}$ and $h-\lambda (h)$ for $h\in \mathfrak{h}$; in particular, ${\mathfrak{n}}^{+}{v}_{\lambda }=0$ and $h{v}_{\lambda }=\lambda (h)v_{\lambda }$. (Verma modules)

The weights of $M(\lambda )$ are exactly $\lambda -\beta$ for $\beta \in Q^{+}$; every weight space is finite dimensional, and $M(\lambda {)}_{\lambda }=\mathbb{C}{v}_{\lambda }$. (Weights of a Verma module lie below lambda)

Given: The setting above and the hypotheses in the statement refuted.

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. The BGG category $\mathcal{O}$ is the full subcategory of left $U(\mathfrak{g})$-modules $M$ satisfying all three conditions: $M$ is finitely generated; $M={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{M}_{\mu }$ where $M_{\mu }=\{v:hv=\mu (h)v\text{ for all }h\in \mathfrak{h}\}$; and $U({\mathfrak{n}}^{+})v$ is finite dimensional for each $v\in M$. Its morphisms are all $\mathfrak{g}$-linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

The weights of $M(\lambda )$ are exactly $\lambda -\beta$ for $\beta \in Q^{+}$; every weight space is finite dimensional, and $M(\lambda {)}_{\lambda }=\mathbb{C}{v}_{\lambda }$. (Weights of a Verma module lie below lambda)

Given: The setting above and the hypotheses in the statement refuted.

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. The BGG category $\mathcal{O}$ is the full subcategory of left $U(\mathfrak{g})$-modules $M$ satisfying all three conditions: $M$ is finitely generated; $M={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{M}_{\mu }$ where $M_{\mu }=\{v:hv=\mu (h)v\text{ for all }h\in \mathfrak{h}\}$; and $U({\mathfrak{n}}^{+})v$ is finite dimensional for each $v\in M$. Its morphisms are all $\mathfrak{g}$-linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. Let $M$ be a finitely generated $\mathfrak{h}$-semisimple $\mathfrak{g}$-module. Then $M\in \mathcal{O}$ if and only if $\mathrm{supp}M\subset {\bigcup }_{i=1}^r({\lambda }_i-Q^{+})$ for some finite list of weights. In either case every $M_{\mu }$ is finite dimensional. The list may be empty for $M=0$; finite generation is an independent hypothesis. (The support description of category O with finite generation)

Let $x_1,\dots ,x_r$ be an ordered basis of a finite-dimensional complex Lie algebra $\mathfrak{g}$. Then the monomials $x_1^{a_1}{x}_2^{a_2}\cdots x_r^{a_r}(a_i\in {\mathbb{N}}_0)$ form a basis of $U(\mathfrak{g})$. In particular, multiplication identifies $\mathrm{gr}U(\mathfrak{g})$ with the symmetric algebra $S(\mathfrak{g})$ on the symbols of the $x_i$. (PBW gives an ordered monomial basis for the enveloping algebra)

Given: The setting above and the hypotheses in the statement refuted.

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. The BGG category $\mathcal{O}$ is the full subcategory of left $U(\mathfrak{g})$-modules $M$ satisfying all three conditions: $M$ is finitely generated; $M={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{M}_{\mu }$ where $M_{\mu }=\{v:hv=\mu (h)v\text{ for all }h\in \mathfrak{h}\}$; and $U({\mathfrak{n}}^{+})v$ is finite dimensional for each $v\in M$. Its morphisms are all $\mathfrak{g}$-linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

Let $x_1,\dots ,x_r$ be an ordered basis of a finite-dimensional complex Lie algebra $\mathfrak{g}$. Then the monomials $x_1^{a_1}{x}_2^{a_2}\cdots x_r^{a_r}(a_i\in {\mathbb{N}}_0)$ form a basis of $U(\mathfrak{g})$. In particular, multiplication identifies $\mathrm{gr}U(\mathfrak{g})$ with the symmetric algebra $S(\mathfrak{g})$ on the symbols of the $x_i$. (PBW gives an ordered monomial basis for the enveloping algebra)

For $\lambda \in {\mathfrak{h}}^{\ast }$, the Verma module is the induced $\mathfrak{g}$-module $M(\lambda ):=U(\mathfrak{g}){\otimes }_{U(\mathfrak{b})}{\mathbb{C}}_{\lambda },$ where $U(\mathfrak{g})$ is the quotient algebra of def-universal-enveloping-algebra-as-a-tensor-quotient and ${\mathbb{C}}_{\lambda }$ is def-one-dimensional-borel-module-of-weight-lambda. Write $v_{\lambda }:=1\otimes c_{\lambda }$. Thus $M(\lambda )$ is the quotient of $U(\mathfrak{g})$ by the left ideal generated by $x$ for $x\in {\mathfrak{n}}^{+}$ and $h-\lambda (h)$ for $h\in \mathfrak{h}$; in particular, ${\mathfrak{n}}^{+}{v}_{\lambda }=0$ and $h{v}_{\lambda }=\lambda (h)v_{\lambda }$. (Verma modules)

Given: The setting above and the hypotheses in the statement refuted.

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. The BGG category $\mathcal{O}$ is the full subcategory of left $U(\mathfrak{g})$-modules $M$ satisfying all three conditions: $M$ is finitely generated; $M={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{M}_{\mu }$ where $M_{\mu }=\{v:hv=\mu (h)v\text{ for all }h\in \mathfrak{h}\}$; and $U({\mathfrak{n}}^{+})v$ is finite dimensional for each $v\in M$. Its morphisms are all $\mathfrak{g}$-linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. Every $M\in \mathcal{O}$ is a quotient of a module with a finite Verma flag. Consequently it has a finite filtration whose nonzero factors are quotients of Verma modules, and is finitely generated over $U({\mathfrak{n}}^{-})$. The empty filtration is allowed for zero. No truncation hypothesis is needed, and a Verma flag of $M$ itself is not asserted. (Finite filtrations by highest-weight quotients)