I was reading Luna's paper *Toute variété magnifique est sphérique* and stumbled on a few facts about Bialynicki-Birula decompositions and fixed points that I don't understand.

Here is the setup. Let $G$ be a connected reductive group over an algebraically closed field $k$ (of characteristic 0, though I'm not certain if that matters), and fix a Borel subgroup $B \subset G$ and a maximal torus $T \subset B$. Denote by $B^-$ the opposite Borel group to $B$ containing $T$. Let $X$ be an irreducible, normal, complete $G$-variety, and suppose that $X$ has finitely many $G$-orbits. Given any one-parameter subgroup $\lambda: \mathbb{G}_m \to T$ and any $y \in X^T$, we write $$X(\lambda,y) = \{x \in X\ |\ \lim_{t \to 0} \lambda(t)x = y\}.$$

Here are the claims Luna makes that I don't understand:

(1) The fixed point set $X^T$ is finite.

(2) We are mainly interested in the case where $\lambda$ is in the Weyl chamber of $B$ (i.e.\ where $\langle \lambda, \alpha \rangle > 0$ for all positive roots $\alpha$), so that $X(\lambda,y)$ is $B$-stable. Luna states that for a "sufficiently general" such $\lambda$, we will have $X^{\mathbb{G}_m} = X^T$, where $\mathbb{G}_m$ acts on $X$ via its image under $\lambda$. He also states that in this case, the $X(\lambda,y)$ for various $y \in X^T$ form the Bialynicki-Birula decomposition of $X$.

(3) With $\lambda$ satisfying the conditions in (2), if $X(\lambda,y)$ is open, then $y$ is fixed by the opposite Borel subgroup $B^-$. (Luna doesn't say anything about this, but I'm also curious: is it true that if $y$ is fixed by $B^-$, then $X(\lambda,y)$ is open?)

All of these statements seem pretty reasonable to me, and I've worked them out in the case where $X = \mathbb{P}(V)$, $G = \mathrm{SL}(V)$, and $B$ (resp. $T$) is the subgroup of upper triangular (resp. diagonal) matrices. In this case, everything is clear using projective coordinates, but I don't know how to make these types of arguments without appealing to coordinates like that. Any proofs (or references to proofs) would be much appreciated!