Restricting representations to a principal $\mathfrak{sl}(2)$

Question

Let $\mathfrak{g}$ be a semi-simple Lie algebra over $\mathbb{C}$ with simply connected group $G$ and suppose that $$\mathfrak{g} = \bigoplus_i\mathfrak{g}_i$$ is a $\mathbb{Z}$- or $\mathbb{Z}/n\mathbb{Z}$-grading on $\mathfrak{g}$, and consider the subalgebra $\mathfrak{g}_0$ and its representation on $\mathfrak{g}_i$ for $i\neq 0$.

Assume that $\mathfrak{g}_0$ is a non-abelian reductive Lie algebra. Suppose that $X$ is a regular nilpotent element of $\mathfrak{g}_0$ and fix a principal $\mathfrak{sl}(2)$-triple $\mathfrak{s}=\operatorname{span}_{\mathbb{C}}\{F,H,E\}\subset \mathfrak{g}$ associated to $X$. I am interested in understanding the restriction of the representation $\mathfrak{g}_n$ to $\mathfrak{s}$.

I am specifically interested in the following case:

Suppose $\mathfrak{g}_0=\mathfrak{m}$ is the Levi subalgebra of a maximal parabolic Lie algebra $\mathfrak{p}= \bigoplus_{i\geq 0}\mathfrak{g}_i$, where I am considering the grading induced by characters of the center of $\mathfrak{m}$. In this case, the representations $\mathfrak{g}_i$ are irreducible (using maximality) prehomogeneous representations of the associated Levi subgroup $M\subset G$. Working out several cases, it seems that the restriction of $\mathfrak{g}_i$ is always multiplicity free as an $\mathfrak{s}$-module.

Is the restriction of $\mathfrak{g}_i$ always multiplicity free as an $\mathfrak{s}$-module? [EDIT: No, see my answer below.] If not, is there an interpretation of when such a restriction is multiplicity-free?

I recently ran into this problem studying certain distributions in the context of $p$-adic groups. Certain domain properties of these distributions turned out to be equivalent to the multiplicity-free property above on the Langlands-dual Lie algebra. This was surprising to me, and raises the question of whether there exists a broader interpretation of this type of decomposition.

I've seen this crop up a few different places for special representations, such as Kostant's work on the adjoint representation and Gross's work for minuscule representations (refs below), showing that these restrictions often encode deep information about $\mathfrak{g}$ and the representation. However, the general problem is mysterious to me. A vague question is

What does this decomposition (say in terms of those weights arising and their multiplicities) encode about the representation $\mathfrak{g}_i$?

Really, this is just probing to see if there is a nice interpretation out there. The question makes sense for an arbitrary representation, but the additional structure here makes me hope for a meaningful question. Any references would be greatly appreciated.

Kostant, Bertram, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Am. J. Math. 81, 973-1032 (1959). ZBL0099.25603.

Gross, Benedict H., On minuscule representations and the principal SL(2), Represent. Theory 4, 225-244 (2000). ZBL0986.22011.

Answer 1

$\DeclareMathOperator\Sym{Sym}$After further calculation, I realized that the answer to the first question is NO, the restriction of $\mathfrak g$ to $\mathfrak s$ need not be multiplicity free: suppose $\mathfrak{g}=\mathfrak{e}_8$, and suppose that $\mathfrak{p}$ is the maximal parabolic subalgebra associated to the simple root indicated by the $\times$ below: $\qquad\begin{matrix} 0 - 0 - \stackrel{\stackrel{\displaystyle 0}|}{\times} - 0 - 0 - 0 - 0 . \end{matrix}$

Then the Levi subalgebra $\mathfrak{m}=\mathbb{C}\oplus \mathfrak{sl}(3)\oplus \mathfrak{sl}(2)\oplus \mathfrak{sl}(5)$ acts on the $1^\text{st}$ graded piece $\mathfrak{g}_1$ via the tensor product of the standard representations. As a representation of a principal $\mathfrak{sl}(2)$-subalgebra of $\mathfrak{m}$, this decomposes as $$ \Sym^{7}(\mathbb{C}^2)\oplus \Sym^5(\mathbb{C}^2)^{\oplus2}\oplus\Sym^3(\mathbb{C}^2)^{\oplus2}\oplus\Sym^1(\mathbb{C}^2). $$ In fact, this generalizes the simplest counter example of $\mathfrak{g}=\mathfrak{so}(8)$ with the parabolic subalgebra $\mathbb{C}\oplus \mathfrak{sl}(2)\oplus \mathfrak{sl}(2)\oplus \mathfrak{sl}(2)$. In that case, $\mathfrak{g}_1$ decomposes as $\Sym^3(\mathbb{C}^2)\oplus \mathbb{C}^2\oplus \mathbb{C}^2$.

I still would be interested in an understanding of what such decompositions tell us.