Stabilizers of the action of Levi on abelianization of nilpotent radical

Question

$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical of $P$.

Then $M$ naturally acts on the abelianization $U/[U, U]$. Its Lie algebra $V_U:=\Lie U / [\Lie U, \Lie U]$ is naturally a linear representation of $M$.

Choose a vector $v \in V_U$, what do we know about the stabilizer $M_v$ of $M$? In the case $P$ is a Borel, such construction is often used in the definition of a generic character of $U$ for Whittaker models in representation theory. In the case $P=G$, everything is trivial.

Here are my questions:

1. Is $M_v$ always connected? If not, could we determine the component group of $M_v$ (at least for generic $v$)?
2. Is $M_v$ reductive (at least for generic $v$)?
3. Do we know the GIT quotient $V_U // M$? What are closed orbits and maximal dimensional orbits in $V_U$?

Answer 1

I am posting this as an answer since it is too long for a comment.

Consider the case that $G$ equals $\textbf{SL}(W)$ for a finite dimensional vector space $W$. Let $W=S\oplus Q$ be a direct sum decomposition of $W$ by subspaces, and let $P$ be the parabolic subgroup of $\textbf{SL}(W)$ of all linear automorphisms that map $S$ into $S$. Thus $M$ is isomorphic to an extension of $\mathbb{G}_m$ by $\textbf{SL}(S)\times \textbf{SL}(Q)$.

The unipotent radical is isomorphic to $\text{Id}_W + \text{Hom}(Q,S)$, e.g., for every element $f$ in $\text{Hom}(Q,S)$ and for every element $(s,q)$ in $S\oplus Q$, the corresponding action is $$ f\ast (s,q) = (s+f(q),q).$$ The unipotent radical is already Abelian, namely $U=\text{Hom}(Q,S)$ as an additive group. Also $V_U$ equals $\text{Lie}(U)$ equals $U$ equals $\text{Hom}(Q,S)$. The action of $\textbf{SL}(S)\times \textbf{SL}(Q)$ is the evident action: $\textbf{SL}(Q)$ acts by precomposition by the inverse and $\textbf{SL}(S)$ acts by postcomposition. The $\mathbb{G}_m$ factor acts by scaling.

The orbits of this action are the determinantal loci $D_{r}\subset \text{Hom}(Q,S)$ for integers $$0\leq r \leq r_{\text{max}}=\text{min}(\text{dim}(S),\text{dim}(Q))$$parameterizing linear transformations of rank $r$. These orbits are affine if and only either $r$ equals zero or $r$ equals $\text{dim}(S)$ equals $\text{dim}(Q)$. Thus, by the Matsushima-Richardson criterion, the stabilizer subgroups are reductive if and only if either $r$ equals zero, in which case the stabilizer equals $M$, or $r$ equals $\text{dim}(S)$ equals $\text{dim}(Q)$, in which case the stabilizer subgroup is isomorphic to $\textbf{SL}(S)$, which also is isomorphic to $\textbf{SL}(Q)$.

Answer 2

I don't know the answer to your questions in this generality, but let me mention a rich class of examples. Let $P$ be the maximal parabolic corresponding to the highest root of $G$, and let $V = V_U$ in your notation. In this case, $\mathrm{Sym}(V^\ast)^M$ is isomorphic to $\mathbf{C}[q]$, where $q$ is possibly a constant polynomial. For instance, if $G = \mathrm{PSp}_{2n}$, then $M = \mathrm{Sp}_{2n}$ and $V$ is the standard representation; the invariant quotient is just a point. This class of examples shows that it's also possible that the stabilizer of $v\in V$ in the Levi quotient $M$ is disconnected. For example, when $G = G_2$, the Levi quotient is $\mathrm{SL}_2$ and $V \cong \mathrm{Sym}^3(\mathbf{A}^2)$. The stabilizer of a general point is $\mathbf{Z}/3$ (and $V/\!\!/M \cong \mathbf{A}^1$ in this case, via the discriminant).

Answer 3

This is an example of a "internal Chevalley module," which has been studied in a bunch of contexts. A good reference is ``On the structure of parabolic subgroups of algebraic groups'' by Rohrle.

For your questions,

1. No, as some examples have shown. Connectedness already fails for the Borel of $\mathrm{SL}_2$ as the generic stabilizer is $\mathbb{Z}/2\mathbb{Z}$. Rohrle includes a lot of material on the component groups.
2. An important point is that $V$ is always pre-homogenous as an $M$-representation. This implies that there is an open $M$-orbit, so any notion of generic should refer to vectors in this orbit. Some times $M_v$ is reductive (think $M=GL_n\times GL_n$ acting on $V=M_n$ for the $(n,n)$-parabolic in $GL_{2n}$. Then $M_v\simeq GL_n$), but need not be (think of $M=GL_n\times GL_1$ acting on $V=\mathbb{A}^n$ for the $(n,1)$-parabolic in $GL_{n+1}$. Here the generic stabilizer is $M_v\simeq P_{n-1,1}$). Stabilizers in smaller (non-zero) orbits tend to be non-reductive, as the example by @JasonStar shows.
3. $V//M=\{*\}$ since it is prehomogeneous, hence has finitely many orbits. The example given by @skd computes the quotient by $M^{der}$, so there is a discriminant.