Let $G$ be a closed, connected Lie group, and let $H$ be a closed (and therefore Lie) subgroup. There is a natural action of $G$ on the space of left cosets $G/H$, for which the stabiliser of $aH$ is the conjugate subgroup ${}^aH:=aHa^{-1}$.
Now let $G$ act diagonally on $G/H\times G/H$. The stabiliser of $(aH,bH)$ is the intersection ${}^aH\cap {}^bH$ of conjugate subgroups. My question is, what can be said about ${}^aH\cap {}^bH$?
I know that ${}^aH={}^bH$ if and only if $ab^{-1}\in N(H)$, the normaliser of $H$ in $G$. Beyond this I couldn't find much via Google. For instance can ${}^aH\cap {}^bH$ be trivial? Or must it be a conjugate of $H$?
Apologies if this is too elementary.
Edit: Thanks for the answers, which show that not much can be said at this level of generality. Now I am looking at a specific example – the icosahedral group $I\cong A_5$ inside $SO(3)$ (where the space of left cosets is the Poincaré sphere). This subgroup has the properties that $I$ is finite, $N(I)=I$ and $I/[I,I]$ is trivial. Does this allow me to conclude that conjugates of $I$ are either equal or intersect in the identity?
More generally, what can be said if I add the assumption $N(H)=H$ to the original question?
Edit 2: I've now asked about the icosahedral group in another question.
Best Answer
Suppose G=SL(2,C) and let H be the stabilizer of a line (so a Borel subgroup). The matrix $$\begin{pmatrix}a&b\\\\c&d\end{pmatrix}$$ in G acts on projective space by $$ z \mapsto \frac{az+b}{cz+d}$$ The stabilizer of ∞ is the matrices with c=0, a Borel subgroup. The stabilizer of both ∞ and 0 is the matrices with b=c=0, a maximal torus. In particular, a two point stabilizer is abelian (the intersection of two Borel subgroups), and a Borel subgroup is non-abelian. Hence they are not isomorphic.
This is just a connected version of Giuseppe's answer.
If you consider the Borel subgroup to be the Lie group itself, then you get an example where the intersection of the conjugates is trivial. If G is the group of 2×2 matrices with c=0 and d=1 acting on projective space, then the stabilizer of 0 has b=0, and the stabilizer of both 0 and 1 has a=1 and b=0. In particular, G=AGL(1) and H is a maximal torus, and the intersection of two conjugates of H is the identity. When the intersection is the identity, this is called being sharply two-transitive or having a regular stabilizer.