Context
Let $G$ be a Lie group, $K$ a proper subgroup of $G$ and $H$ a proper subgroup of $K$ i.e. $H\subset K\subset G$.
The Lie group $G$ acts on the left on a vector space $V$. Let $x$ be a distinguished element of $V$ such that $K$ is the stabiliser of $x$ i.e. $\text{Stab}(x):=\{g\in G | g^*x=x\}=K$.
I am interested in elements in the $G$-orbit of $x$ which are $H$-invariant i.e. elements $y\in V$ such that there exists $g\in G$ such that $y=g^*x$ and $h^*y=y$ for all $h\in H$. This boils down to study the set:
$N=\{g\in G | ghg^{-1}\in K \text{ for all }h\in H\}$.
Whenever $K=H$, then $N$ identifies with the notion of normalizer of $H$ in $G$.
Note that $K$ is not assumed to be normal hence $N$ is not a group in general. However, it can be shown that:
- $N$ is a left $K$-space.
- $C_G(H)\subset N$ where $C_G(H)=\{g\in G| gh=hg \text{ for all }h\in H\}$ is the centralizer of $H$ in $G$.
Questions:
-Is there a name for $N$?
-I'm interested more precisely in the quotient $N/K$. Is it possible to describe this quotient e.g. in terms of the quotient $C_G(H)/C_K(H)$?
-Let us denote $n:=\{ \lambda\in\mathfrak{g} | [\lambda,\xi]\in\mathfrak{k}\text{ for all }\xi\in \mathfrak{h}\}$ where $\mathfrak{h}\subset\mathfrak{k}\subset\mathfrak{g}$ are the Lie algebras associated to $H\subset K\subset G$, respectively. Assuming that both $H$ and $K$ are connected, what are necessary or sufficient conditions to have $N/K\simeq n/\mathfrak{k}$?
-Are there any references discussing a similar problem and containing a similar definition for this $N$?