Category O

Definition of category O

The objects of category ${\displaystyle \mathcal{O}}$ are ${\displaystyle \mathfrak{g}}$-modules ${\displaystyle M}$ such that:

1. ${\displaystyle M}$ is finitely generated;
2. ${\displaystyle M=\underset{\lambda \in {\mathfrak{h}}^{\ast }}{⨁}{M}_{\lambda }}$;
3. ${\displaystyle M}$ is locally ${\displaystyle \mathfrak{n}}$-finite, i.e. for each ${\displaystyle v\in M}$, the ${\displaystyle \mathfrak{n}}$-submodule generated by ${\displaystyle v}$ is finite-dimensional.

Morphisms in this category are the ${\displaystyle \mathfrak{g}}$-module homomorphisms.

Basic properties

• Each module in category ${\displaystyle \mathcal{O}}$ has finite-dimensional weight spaces.
• Each module in category ${\displaystyle \mathcal{O}}$ is a Noetherian module.
• ${\displaystyle \mathcal{O}}$ is an abelian category.
• ${\displaystyle \mathcal{O}}$ has enough projectives and enough injectives.
• ${\displaystyle \mathcal{O}}$ is closed under taking submodules, quotients, and finite direct sums.
• Objects in ${\displaystyle \mathcal{O}}$ are ${\displaystyle Z(\mathfrak{g})}$-finite: if ${\displaystyle M}$ is an object and ${\displaystyle v\in M}$, then the subspace ${\displaystyle Z(\mathfrak{g})v\subseteq M}$ generated by ${\displaystyle v}$ under the action of the center of the universal enveloping algebra is finite-dimensional.

Koszul duality

A homological feature of category ${\displaystyle \mathcal{O}}$ is that, after choosing graded lifts of blocks, certain blocks can be described by Koszul algebras. In particular, Beilinson, Ginzburg, and Soergel showed that (for suitable graded realizations) the endomorphism algebra of a projective generator of a block (notably the principal block) is a Koszul algebra ${\displaystyle A}$. Equivalently, the corresponding graded block of ${\displaystyle \mathcal{O}}$ is (via a projective generator) equivalent to the category of finite-dimensional graded modules over ${\displaystyle A}$.

In this setting, Koszul duality relates two graded blocks: one block is equivalent to ${\displaystyle \mathrm{g}\mathrm{r}\text{-}A}$, while a second (dual) graded block is equivalent to ${\displaystyle \mathrm{g}\mathrm{r}\text{-}{A}^{!}}$, where ${\displaystyle A^{!}}$ is the Koszul dual algebra of ${\displaystyle A}$. The associated Koszul duality functors induce a triangulated equivalence between the bounded derived categories of these graded realizations, i.e. an equivalence of the form

${\displaystyle D^b({\mathcal{O}}_{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}^{\mathrm{g}\mathrm{r}})\simeq D^b({\mathcal{O}}_{{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}^{\vee }}^{\mathrm{g}\mathrm{r}}),}$

where ${\displaystyle {\mathcal{O}}_{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}^{\mathrm{g}\mathrm{r}}\simeq \mathrm{g}\mathrm{r}\text{-}A}$ and ${\displaystyle {\mathcal{O}}_{{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}^{\vee }}^{\mathrm{g}\mathrm{r}}\simeq \mathrm{g}\mathrm{r}\text{-}{A}^{!}}$.

Koszul duality for category ${\displaystyle \mathcal{O}}$ is closely connected with geometric and combinatorial structures such as the geometry of the flag variety, perverse sheaves, and Kazhdan–Lusztig theory.

Examples

• All finite-dimensional ${\displaystyle \mathfrak{g}}$-modules and their ${\displaystyle \mathfrak{g}}$-homomorphisms are in category ${\displaystyle \mathcal{O}}$.
• Verma modules and generalized Verma modules and their ${\displaystyle \mathfrak{g}}$-homomorphisms are in category ${\displaystyle \mathcal{O}}$.