parabolic subalgebras and Cartan decomposition

Question

Let $\mathfrak{g}$ be a complex simple Lie algebra and $\mathfrak{k}$ its complex subalgebra such that $(\mathfrak{g},\mathfrak{k})$ is a Hermitian symmetric pair; $\mathfrak{g}= \mathfrak{k}\oplus\mathfrak{p}$ is the corresponding Cartan decomposition subject to some Cartan involution $\theta$. Moreover, there is a splitting $\mathfrak{p} = \mathfrak{p}^- \oplus \mathfrak{p}^+$.

Problem: Classify all $\theta$-stable parabolic subgroups $\mathfrak{q}=\mathfrak{l}\oplus\mathfrak{u}$ of $\mathfrak{g}$ such that $\mathfrak{l}\subseteq\mathfrak{k}$ and $\mathfrak{p}^+\subseteq\mathfrak{u}$.

Motivation: In the article Dirac operators and Lie algebra cohomology. Represent. Theory 10 (2006), the authors prove that in such a case there is a Hodge decomposition for $\mathfrak{u}$-homology of a unitarizable $(\mathfrak{g},K)$-module. I am interested for which real parabolic subalgebras of some real form of $\mathfrak{g}$ there is a Hodge decomposition.

Answer 1

The question needs to be made a bit precise. To talk of a Cartan decomposition in your sense, what we need to start with is a real Lie subalgebra ${\mathfrak g}_0$ with a "maximal compact sub-algebra" ${\mathfrak k}_0$ (i.e. Lie algebra of a maximal compact subgroup, assuming that the underlying Lie group $G$ is linear) whose complexifications are $\mathfrak g$ and $\mathfrak k$ respectively. Now there is the extension to $\mathfrak g$ of the Cartan involution $\theta $.

Given this, $\theta $ stable parabolic subalgebras $\mathfrak q$ of the complex Lie algebra $\mathfrak g$ whose nilradical $\mathfrak u$ contains ${\mathfrak p}^+$, are not many! These are exactly the ones such that ${\mathfrak q} \supset {\mathfrak p}^+$ and whose Levi $\mathfrak l$ (which is defined over $\mathbb R$- recall that $G$ is the group of real points of $G({\mathbb C})$) is such that $L\subset K$ (i.e. ${\mathfrak l}\subset {\mathfrak k}$). In terms of the (Vogan-Zuckerman) cohomological representations $A_{\mathfrak q}(0)$, the representation $A_{\mathfrak q}(0)$ is the unique holomorphic discrete series with trivial infinitesimal character.

If you ask that the whole parabolic sub-algebra ${\mathfrak q}$ contains ${\mathfrak p}^+$, this is equivalent to asking that ${\mathfrak u}\cap {\mathfrak p}= {\mathfrak u}\cap {\mathfrak p}^+$, and hence the corresponding $A_{\mathfrak q}(0)$ is a representation of holomorphic type.

I believe this is implicitly contained in a well known paper of Vogan and Zuckerman in Compositio (1984?) on unitary representations with cohomology.

Not too fussed about the "bounty". But this is the "final" answer: every such $q$ is of the form $$ q=m\oplus {\mathfrak p}^+$$ where $m$ is a parabolic subalgebra of $k$.