Let $ G $ be a connected compact Lie group of rank $ r $ . Let $ T $ be an $ m $ dimensional (proper) closed connected subgroup of $ G $. Let $ N(T) $ be the normalizer in $ G $ of $ T $
$$
N(T):=\{g \in G: gTg^{-1}=T \}
$$
Consider the quotient group
$$
N(T)/T
$$
If $ T $ is abelian can we conclude anything interesting about $ N(T)/T $? By "interesting" I mean things like:
(1) Is $ N(T)/T $ always nontrivial? (in other words N(T) contains elements not in $ T $ )
(2) Is $ N(T)/T $ always disconnected?
(3) What is the dimension of $ N(T)/T $?
Here are my thoughts so far:
Let $ W $ be the Weyl group of $ G $. My guess is that (roughly) $$ N(T)/T \cong Z(T)/T \rtimes N(T)/Z(T) \cong Z(T)/T \rtimes W $$ where $ Z(T) $ is the centralizer of $ T $ in $ G $. So my guess for the first two questions is that $ N(T)/T $ is always nontrivial and always disconnected.
Also curious about the group $ N(T)/T $ for more general subgroups $ T $ of $ G $.