Definition of reductive homogeneous space

Question

I'm studying homogeneous spaces from the book "An Introduction to Lie Groups and the Geometry of Homogeneous Spaces" by A. Arvanitoyeorgos.

I have some problems understanding the definition of reductive homogeneous space. Given $G$ a Lie group, $K$ a closed Lie subgroup (with respective $\mathfrak{g}$ and $\mathfrak{k}$ Lie algebras), we say that the homogeneous space $G/K$ is reductive if there exist a subspace $\mathfrak{m}$ of $\mathfrak{g}$ s.t. $$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{m}$$ and $Ad(k)\mathfrak{m} \subset \mathfrak{m}$ $\forall k \in K$.

Now, I know that $Ad: G \rightarrow Aut(\mathfrak{g})$ is defined as $Ad(g)=(d I_g)_e$, where $I_g(h)=g^{-1}hg$, but I can't figure out what $Ad(k)\mathfrak{m}\subset \mathfrak{m}$ stand for. I think that I can translate this symbols as "Consider the quotient $\mathfrak{g}/\mathfrak{m}$, with the canonical $\pi:\mathfrak{g}\rightarrow \mathfrak{g}/\mathfrak{m}$. Then for any $k\in K$ $Ad(k) \in \mathfrak{m}$ (but I cannot see why use $Ad(k)\mathfrak{m}\subset \mathfrak{m}$ for saying that $Ad(k) \in \mathfrak{m}$)... but I have the feeling I'm missing something, and my interpretation is not the correct one. I know it's a pretty simple question, but I recently started this subject and I want to understand as much as I can what I read.

Any hint or help would be higly appreciate, thanks in advance.

Answer 1

$Ad (k)$ is an automorphism of the Lie Algebra of G. But you are confusing it with an element of (subalgebra of) Lie (G).

The given condition merely means that under the automorphism $Ad(k)$ of Lie (G) a particular subalgebra is mapped into itself (stabilized).

Answer 2

The condition means that if you consider $K$ acting on $\mathfrak{k}\oplus\mathfrak{m}$ via $Ad\colon K\to \mathrm{GL}(\mathfrak{g})\simeq \mathrm{GL}(\mathfrak{k}\oplus\mathfrak{m})$, then $\mathfrak{m}$ is an invariant subspace.