Maximally commuting structure inside a nonabelian Lie group [SU($N$), SO($N$)]

Question

Consider a non-abelian compact Lie group $G$. Pick up any generator $T$ of Lie algebra of this $G$. This generator $T$ generates a U(1) subgroup inside $G$.

Question: What is the structure (group?) inside this $G$ that maximally commuting with this U(1) generated by $T$? Let us call this as a maximally commuting structure inside $G$ respect to U(1) from $T$.

1. What is the maximally commuting structure for $G=$SU($N$)? and for $G=$U($N$)?

2. What is the maximally commuting structure for $G=$SO($N$)? and for $G=$O($N$)?

3. What is the maximally commuting structure for $G=$Sp($N$)?

Thank you for the helps from experts!

p.s. Naively, the maximally commuting structure for $G=$SU($N$) should at least contains U(1)$^{N-1}$. But I am not sure what will it be in general.

p.s.2. please also answer the specific examples given above.

Answer 1

You can easily show that it's a subgroup, at least. Let's call it $Z_{U(1)}(G)$ for want of a better notation. Then $e\in Z_{U(1)}(G)$ is trivial, $a, b\in Z_{U(1)}(G)\implies ab\in Z_{U(1)}(G)$ is almost trivial (just move the element of $U(1)$ past each one in turn), and for $a\in Z_{U(1)}(G)\implies a^{-1}\in Z_{U(1)}(G)$, consider the element $(a^{-1}u)^{-1}$ for some $u\in U(1)$ and compare it to $(ua^{-1})^{-1}$...

ETA: More generally this is known as the centralizer of your $U(1)$ and you may be able to find more information about it under that name.

Answer 2

The examples that interest you are simple groups. I assume the group is simply connected. The maximal commutative subgroup of a semi-simple Lie subgroup is known as the Cartan subgroup. Its algebra is the Cartan subalgebra.

https://en.wikipedia.org/wiki/Semisimple_Lie_algebra#Definition_of_a_Cartan_subalgebra