on a characterization of parabolic subgroups

2010-12-27T13:30:38Z

Over a base field $k$, linear $k$-groups stand for affine algebraic $k$-groups. For simplicity take $k$ to be a field of characteristic zero, as in this case one has the correspondence between connected linear $k$-groups and finite dimensional Lie $k$-algebras.

Let $\mathfrak{g}$ be the Lie algebra of a connected semi-simple linear $k$-group $G$. It is known that a (non-trivial) parabolic $k$-algebra $\mathfrak{p}$ of $\mathfrak{g}$ can be obtained as follows: there exists a (non-trivial) grading $$\mathfrak{g}=\oplus_{n\in\mathbb{Z}}\mathfrak{g}(n)$$ respecting the Lie bracket $$[\mathfrak{g}(m),\mathfrak{g}(n)]\subset\mathfrak{g}(m+n)$$ and that $\mathfrak{p}=\oplus_{n\geq 0}\mathfrak{g}(n)$. The grading on $\mathfrak{g}$ amounts to an action of $\mathbb{G}_m$ on $G$, which is given by conjugation through a co-character $\mu:\mathbb{G}_m\rightarrow G$, justified by the condition on Lie bracket. For convenience denote also by $\mu$ the grading on $\mathfrak{g}$, and $\mathfrak{p}=\mathfrak{p}^+(\mu)$ the parabolic Lie $k$-subalgebra associated as above, and $P^+(\mu)$ the corresponding parabolic $k$-subgroup. $\mu$ is only unique up to conjugation by the Levi $k$-subgroup $L(\mu)$ of $P^+(\mu)$, whose Lie algebra is $\mathfrak{g}(0)$.

My question is to extend these arguments to $k$-subgroups and Lie $k$-subalgebras. Keep the previous notations, and take $H\subset G$ a connected $k$-subgroup, with Lie $k$-subalgebra $\mathfrak{h}$. Then by restriction one gets a grading $$\mu_H:\mathfrak{h}=\oplus_n\mathfrak{h}(n)$$ with $\mathfrak{h}(n)=\mathfrak{h}\cap\mathfrak{g}(n)$. This grading satisfies $[\mathfrak{h}(m),\mathfrak{h}(n)]\subset\mathfrak{h}(m+n)$, hence is also given by some co-character $\mu_H:\mathbb{G}_m\rightarrow H$.

Assume that $\mu_H$ is non-trivial, and $H$ is reductive itself, how should one compare the two co-characters $\mu_H$ and $\mu$? From the arguments above one sees that $\mathfrak{p}^+(\mu_H)=\mathfrak{h}\cap\mathfrak{p}^+(\mu)$ and $P^+(\mu_H)=H\cap P^+(\mu)$, but if one extends $\mu_H$ to the cocharacter $\mathbb{G}_m\rightarrow H\rightarrow G$, what kind of difference might one find between $\mu_H$ and $\mu$?

Answer by George McNinch for on a characterization of parabolic subgroups

2010-12-27T18:07:47Z

(Edit)

In the given generality, I'm not sure the question has much hope for a tidy answer.

Consider $G=GL(V)$ and write $V = W \oplus W'$ where $W$ has dimension 2. So you get an embedding $H=GL(W) \to G$ in a natural way ($H$ acts trivially on $W'$).

The stabilizer $Q$ in $H$ of a line $L \subset W$ is a parabolic subgroup of $H$. And the parabolic subgroups of $G$ are the stabilizers of flags $F$ in $V$.

Consider a flag $$F = (0 \subset F_1 \subset F_2 \subset \cdots \subset F_r = V)$$ for which (i) $F_1 = L$, (ii) $F_2 = W$, and (iii) $F_i \cap W'$ is a complement in $F_i$ to $F_2=W$ for $i \ge 2$.

If $P$ is the stabilizer of $F$ in $G= GL(V)$, then $P \cap H = Q$. (Note that $H$ is contained in the stabilizer of the flag $F'=(0\subset F_2 \subset \cdots \subset F_r = V)$).

In general there are many such $P$.

on a characterization of parabolic subgroups - MathOverflow

on a characterization of parabolic subgroups

Answer by George McNinch for on a characterization of parabolic subgroups