# Complexifying Lie group actions

In Atiyah-Pressley's paper Convexity and Loop Groups, it is claimed that the Bruhat cells $C_\lambda$ are invariant under the natural $S^1 \times T$-action and that the authors will prove this claim. To my knowledge, they do not prove this result, but rather the following:

The Bruhat cells $C_\lambda$ are invariant under the action of the complexified torus action $T_{\mathbb C} \times \mathbb C^\times$.

This brings about several other questions:

1. The proof involves extending the action of a compact Lie group, say $K$ acting on $M$, to an action of its complexification $K_\mathbb C$ on $M$, given that $M$ is endowed with an almost complex structure.

How is this done in general? Let $J$ be the almost complex structure. I suspect that one differentiates the group action $K \times M \to M$ to a Lie algebra action $\phi:\mathfrak k \to \text{Vec}(M)$, then extends to an action $\mathfrak k_\mathbb C \to \text{Vec}(M)$ by $\phi(X+iY) = \phi(X)+J\phi(Y)$. However, I do not believe this preserves the brackets unless add more structure (maybe $J$ integrable?).

1. Even if we knew that the $C_\lambda$ were invariant under $T_\mathbb C\times \mathbb C^\times$, why would this imply that the $C_\lambda$ are invariant under the real $T\times S^1$-action?