Let $G$ be a connected, compact Lie group, $T$ a maximal torus. Let $LG$ be the group of smooth (or polynomial) loops and $X=LG/T$ the affine flag variety ($T$ acts say by right multiplication). In Proposition 8.7.1 of their book *Loop groups*, Pressley–Segal seem to claim that the fixed points of the $S^1$-rotation action on $X$ are given by elements of the affine Weyl-group identified with loops of the form $[\alpha(t)g]$ where $g$ is in the normalizer, $N_G(T)$, of $T$ and $\alpha:S^1 \to T$ is a homomorphism. They say this is a "re-statement of the proof" of Proposition 5.1.2.

However, checking that proof, I can get as far as seeing that it is of the above form, but I think $g$ is not constrained to lie in the normalizer $N_G(T)$. For example if $\alpha$ is the identity, then any such $g$ is fine. In general it seems $g$ should be constrained to lie in the centralizer of the image of the one-parameter subgroup $\alpha$.

Question: What am I misunderstanding?

If one includes the action of $T$ by left-multiplication on $LG$ (or even a generic one-parameter subgroup), then I think the Proposition would be correct. So probably this doesn't actually matter much for the geometry they want to develop. Probably one could even view the stratification from Theorem 8.7.2 as coming from a "Morse–Bott" flow.