# BGG resolution and representations of parabolic subalgebras

Everything here is over $\mathbb{C}$. Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra and let $\mathfrak{p}$ be a parabolic subalgebra (relative to some fixed Borel subalgebra that is unimportant for this question). Then $\mathfrak{p}$ has a decomposition $$\mathfrak{p} = \mathfrak{l} \oplus \mathfrak{u_+},$$ where $\mathfrak{l}$ is a reductive subalgebra (the Levi factor of $\mathfrak{p}$) and $\mathfrak{u}_+$ is a nilpotent ideal (the nilradical of $\mathfrak{p}$). Finally, we can decompose $\mathfrak{g}$ as

$$\mathfrak{g} = \mathfrak{u}_- \oplus \mathfrak{l} \oplus \mathfrak{u}_+$$ (as $\mathfrak{l}$-modules), where $\mathfrak{u}_-$ and $\mathfrak{u}_+$ are dual to each other via the Killing form of $\mathfrak{g}$.

Assume further that the following (equivalent) conditions hold:

1. $\mathfrak{g}/ \mathfrak{p}$ is irreducible as a $\mathfrak{p}$-module;

2. $\mathfrak{u}_-$ is irreducible as an $\mathfrak{l}$-module;

3. $\mathfrak{u}_-$ is an abelian Lie algebra;

4. 2 and 3 with $\mathfrak{u}_-$ replaced by $\mathfrak{u}_+$.

Buzzwords here are "Hermitian symmetric space" and "generalized flag variety." There is a classification of these in terms of root systems but I don't want to use that.

I need to understand the decomposition of ${\bigwedge}^2 \mathfrak{u}_-$ into irreducible modules for $\mathfrak{l}$. Using the classification of these parabolics, you can just see explicitly what the highest weight of $\mathfrak{u}_-$ is, and then it's not too hard to compute what ${\bigwedge }^2 \mathfrak{u}_-$ is, but I would like a more elegant way to see what's going on here.

I have been informed that there is some version of the BGG resolution that will be helpful for this - this evidently gives the highest weights of ${\bigwedge }^2 \mathfrak{u}_-$ in terms of the dotted action of some elements of the Weyl group on the highest weight of $\mathfrak{u}_-$, but at this point I'm stuck. I don't know enough (ok, anything really) about the BGG resolution to know where to look for this stuff. Either an explanation or a reference would be much appreciated.

Edit: I would also be happy with a pointer to a nice reference for the BGG resolution in general.

• You should look at two papers: Garland-Lepowsky, Invent. Math. 34 (1976) (online at gdz.sub.uni-goettingen.de), and Lepowsky, J. Algebra 49 (1977). The classical BGG resolution and resulting Bott theorem on cohomology of the nilradical are treated in my 2008 AMS graduate text. By the way, your "affine action" must refer to what is usually called the "dot action" of the Weyl group, with the origin shifted by $-\rho$. Jul 29, 2011 at 21:33
• Yes, that's what I meant by affine action. I will edit accordingly. Thank you for clarifying and thanks for the references!
– MTS
Jul 30, 2011 at 1:06