BGG resolution and representations of parabolic subalgebras

Question

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.

Answer 1

The question involves a fairly long history, going back at least to Bott's Annals paper and two long follow-up papers by Kostant, "Lie algebra cohomology and the generalized Borel-Weil theorem", Ann. of Math. (2) 74 (1961), 329–387, plus the 1963 paper mentioned by rObOt. Those papers predate the work on Verma modules and BGG resolution in the 1970s, which however had Bott's theorem as a byproduct. The BGG arguments made a comparison with Lie algebra cohomology, which in turn involved an easy computation of the 1-dimensional representations of a Cartan subalgebra (Levi subalgebra of a Borel subalgebra) on exterior powers of the nilradical; see 6.4 in my book on the BGG category. This gets less elementary in the parabolic case considered here.

The special situation with Hermitian symmetric spaces is surveyed at the end of Chapter 9 in my book, along with references to what I think are the main developments in that direction. Again Kostant's papers are a key starting point, but the later approaches using the parabolic BGG resolution were explored by people like Lepowsky (a student of Kostant), Boe, Collingwood, Irving, Shelton. Eventually this work enters the more complicated realm of infinite dimensional representations and Kazhdan-Lusztig-Vogan theory. I'm not sure what the best answer is to the narrower question raised here, but most work on Hermitian symmetric spaces has required case-by-case study using the standard classification.

Answer 2

Maybe you should look at the paper Lie Algebra Cohomology and Generalized Schubert Cells by Bertram Kostant. He computes the decomposition via some sort of algebraic laplacian operator.