Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$.

Denote by $\Phi$ the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}_\alpha$ the root subspace of $\mathfrak{g}$ corresponding to a root $\alpha$.

We fix a choice of the set of positive roots $\Phi^+$, and let $\Delta$ be the corresponding subset of simple roots in $\Phi^+$. Note that each subset $I\subseteq\Delta$ generates a root system $\Phi_I\subseteq\Phi$, with positive roots $\Phi_I^+=\Phi_I\cap \Phi^+$.

There are a number of subalgebras of $\mathfrak{g}$ associated with the root system $\Phi_I$. Let $ \mathfrak{l}_I=\mathfrak{h}\oplus\sum_{\alpha\in\Phi_I}\mathfrak{g}_\alpha$ be the Levi subalgebra and let $ \mathfrak{u}_I=\sum_{\alpha\in\Phi^+\backslash\Phi_I^+}\mathfrak{g}_\alpha$ be the nilpotent radical. Then $\mathfrak{p}_I=\mathfrak{l}_I\oplus \mathfrak{u}_I$ is the standard parabolic subalgebra.

We note that once $I$ is fixed, there is little use for other subsets of $\Delta$. Therefore, we omit the subscript if a subalgebra is obviously associated to $I$.

The category $\mathcal{O}^\mathfrak{p}$ is the full subcategory of $U(\mathfrak{g})$-Mod such that every object $M$ in category $\mathcal{O}^\mathfrak{p}$ satisfies the following conditions.

1. $M$ is a finitely generated $U(\mathfrak{g})$-module.

2. $M$ is a direct sum of finite-dimensional simple $U(\mathfrak{l})$-modules.

3. $M$ is locally finite as a $U(\mathfrak{p})$-module.

I have search the notation: locally finite.

I get the following:

Locally finite modules $M$ are modules whose finitely generated submodules are finite (as sets).

I would like to know is that the correct definition for the locally finite in the definition of $\mathcal{O}^\mathfrak{p}$, secondly, I would like to know what is the intuitions/motivations for one to define $\mathcal{O}^\mathfrak{p}$ in such a way. Other two conditions seems more natural than the third one to me.

  • $\begingroup$ For the definition of locally finite you should replace 'finite (as sets)' by 'finite dimensional (as vector spaces)' in your boldface definition. Equivalently every element of $M$ lives in a $U(\mathfrak{p})$-submodule that is finite dimensional as a vector space. $\endgroup$ – Simon Wadsley Dec 19 '18 at 12:08
  • $\begingroup$ I think that, as often, the motivation comes from the examples. i.e. all three conditions hold in a large family of interesting examples of $U(\mathfrak{g})$-modules. $\endgroup$ – Simon Wadsley Dec 19 '18 at 12:11

Locally finite in this context means that every element (equivalently every finitely generated submodule) is finite-dimensional vector space. One reason to consider this condition is to see that it holds for (parabolic) Verma modules which are one of the basic modules that you can study (i.e. modules algebraically induced from a finite-dimensional representation). Another point of view can be to use the second condition and replace the third one with the following which however gives you the same category of modules

3b. the nilradical of $\mathfrak{p}$ acts locally nilpotently on $M$

This is a natural weakening of finite-dimensionality.


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