Defining Levi subgroups: roots versus centralizer I want to unify two points of view that remain distinct for me, even though they should match.
Let $G$ be a reductive group (I will take examples in $GL(3)$). Let $P_0$ a minimal parabolic, $M_0$ its Levi subgroup, $T_0$ the maximal split torus of the center or $M_0$ and $A_0$ the connected component of the identity of its real points $T_0(\mathbb{R})$. In $GL(3)$, these are for instance
$$
P_0 =
\begin{pmatrix}
* & * & * \\
& * & * \\
& & *
\end{pmatrix}
\quad
M_0 =
\begin{pmatrix}
* &  &  \\
 & * &  \\
& & *
\end{pmatrix}
\quad
A_0(\mathbb{R}_+) =
\begin{pmatrix}
* &  &  \\
& * &  \\
& & *
\end{pmatrix}
\quad
$$
Let $P \supset P_0$ a (standard) parabolic, $M_P$ its Levi component (such that $M_0 \supset M_0$). For instance,
$$
P =
\begin{pmatrix}
* & * & * \\
* & * & * \\
& & *
\end{pmatrix}
\quad
M_P =
\begin{pmatrix}
* & * &  \\
* & * &  \\
& & *
\end{pmatrix}
$$
I want to understand how to define the group $A_P$. I often see two different ways:

*

*define it as the (connected component of the real points of) centralizer of $M_P$, so that
$$
A_P =
\begin{pmatrix}
*_1 &  &  \\
 & *_1 &  \\
& & *
\end{pmatrix}
$$
where the two $*_1$ are the same element.

*define it as the subgroup of $A_0$ killed by the roots in $M_P$. The roots occurring in $M_P$ (under the action of $T_0$) are $\pm \epsilon_1 = \pm e_1 \mp e_2$. The action of $A_0$ on itself by conjugation is trivial, so I suppose this should mean the action of the sought subgroup of $A_0$ by conjugation on $M_P$ should be trivial, i.e. $\varepsilon_1 = 0$ i.e. $t_1^{-1}t_2 = 1$ i.e. matrices of $A_O$ of the above form.

We indeed find the same group. Is this always the case? Is there a way to see why these two notions (killed by some roots and centralizer) coincide in general?
 A: How are you defining the Levi subgroup $M_P$? Here is the chain of definitions I'm used to:
Let $G$ be a linear algebraic group defined over an algebraically closed field (otherwise things get more complicated). Let $P$ be a standard parabolic. As you probably know, these are in bijection with subsets $\Pi_P$ of the set of simple roots $\Pi$ of $(G,A)$ where $A$ is some fixed maximal torus of $G$, for example, you can define $P_{\Pi_P}:=BW_{\Pi_P}B$ where $B$ is a chosen Borel subgroup and $W_{\Pi_P}$ is the subgroup of the Weyl group generated by simple reflections labelled by elements of $\Pi_P$. Then define $A_P=(\bigcap_{\alpha\in\Pi_P}\ker\alpha)^0$. Let $M_P=Z_G(A_P)$ (so for example, when you say the root $e_1-e_2$ is to be trivial, that allows a copy of $\mathrm{GL}_2$ to show up there as the centralizer---you probably know that all Levi subgroups of $\mathrm{GL}_n$ are just products of smaller general linear groups, and this is one way to see why. Then $\Pi_P$ are the simple roots for $M_P$ acting on $A$. It can be shown that $A_P$ is the connected component of the centre of $M_P$ as they've just been defined.
Certainly it isn't correct to have the real numbers appearing anywhere in your definition; these definitions make sense in much greater generality. (It's also good to be a bit careful with the word connected when it comes to real points; $\mathrm{GL}(n,\mathbb{R})$ is not a connected Lie group! In general the functor to real Lie groups has some good properties but can't be totally forgotten.)
You might also be intersted in this related question.
