I'll be working over the complex numbers.
Let $G$ be a connected reductive group, $\theta\colon G\to G$ an involution. Let $K=G^{\theta}$ be the fixed point subgroup. I am trying to track down references for some facts about $K$-orbits on the flag variety $G/B$.


**Added later:** to clarify, $G$ is a complex algebraic group, $\theta$ is an automorphism of group schemes over $Spec(\mathbb{C})$ of order $\leq 2$.


**Added later:** Here are some standard examples. The nicest one is as follows. Let $H$ be a connected reductive group and take $G=H\times H$. Define $\theta\colon G\to G$ by $\theta(x,y) = (y,x)$. So the fixed point group is just isomorphic to the original $H$. The flag variety of $G$ is the 2-fold product of the flag variety of $H$ and we are looking at $H$ acting on this diagonally. From the equivariant geometry point of view this is the `same' as looking at Borel (of $H$) orbits on the flag variety of $H$. Note that in this case my questions are rather trivial, since all the orbits are linear affine spaces (in particular, they are contractible).


For a more interesting example, take $G=SL\_2(\mathbb{C})$. For $\theta$ take conjugation by the matrix $\left( \begin{matrix} 1 & 0 \\ 0& -1\end{matrix} \right)$. Then $K$ is the usual (algebraic) torus. The flag variety is $\mathbb{P}^1 = \mathbb{C}\sqcup{\infty}$, and $K$ acts on it by $\lambda\cdot x = \lambda^2x$. So 3 orbits $0$, $\infty$ and $\mathbb{C}^\*$. The component group of $\mathbb{C}^\*$ is $\mathbb{Z}/2\mathbb{Z}$. And as soon as I write this I see that I messed up a bit with my question 3) and 3') and need to fix it.


The reason to use the symbol $K$ and evoke images of maximal compacts is as follows: with the above setup there is a real form $G\_{\mathbb{R}}$ of $G$ such that $K\_{\mathbb{R}}= K\cap G\_{\mathbb{R}}$ is a maximal compact of both $G\_{\mathbb{R}}$ and $K$. Once again I do not know of a reference for this `fact', but I haven't really thought about it (so this might be quite simple).


**Question 1)** I have seen it stated in several places (for instance in Lusztig and Vogan's paper `Singularities of closures of $K$-orbits on flag manifolds') that the component group $K\_x/K\_x^0$ for any $x\in G/B$ has exponent $2$. Does anyone know a precise reference for this fact? Or better, have some intuition/argument why this is at least morally true?


**Question 2)** If $G$ is semisimple and simply connected, then $K$ is connected. Again this is supposedly true, but I would love a reference/argument/intuition.


**Question 3)** Assume to be in the situation of 2) so that $K$ is connected. ~~Is it possible to find a covering group $K\_1$ of $K$~~ Does the action of $K$ always factor through another group $K\_1$ such that the component group $(K\_1)\_x/(K\_1)\_x^0$ is connected for all $x\in G/B$, and $K\_1$-orbits coincide with $K$-orbits. Note: I am not saying that this is true. Just wondering if it's possible. But I have no intuition for this. The motivation for this is that I would like to untwist monodromy occuring in $K$-equivariant local systems by working with $K\_1$ instead.


**Question 3')** Instead of $K$-orbits on $G/B$, I can instead work with $B$-orbits on $G/K$ (these have the benefit of being connected without any additional assumptions on $G$). Same sort of question as 3). ~~Is it possible to find a covering group of $B$~~ Does the action of $B$ factor through a $B\_1$ such that the component groups of the $B\_1$-action at all points of $G/K$ are trivial, and the $B\_1$-orbits coincide with $B$-orbits?


Yep, my knowledge of groups is weak!



Topological properties of $K$ orbits in $G/B$
1. If $\theta$ is an involution of a complex linear algebraic group $G$ and if $K=G^\theta$ is its fixed-point set, then $K/K^0$ will always have exponent 2. This follows from a generalized "Cartan decomposition". The argument goes as follows. Let $\mathfrak p$ denote the (-1)-eigenspace of $d\theta$ on $\mathfrak g$. Then, morally at least, we ought to be able to express everything in $K$ as $k=k\_0\exp X$, with $k\_0 \in K^0$ and $X \in \mathfrak p$. But then $k^{-1} = (k\_0\exp X)^{-1} = \exp(-X) k\_0^{-1} = k' \exp(-X)$, for some $k' \in K^0$ (as $K^0$ is normal in $K$). On the other hand, $\exp(-X) = \exp(d\theta(X)) = \theta(\exp X) = \exp X$. Consequently, $k = k^{-1}$ in $K/K^0$, as desired. I've glossed over some details, which you can find in Loos, *Symmetric Spaces*, vol. 1, Benjamin (1969).
2. This follows from Theorem 8.1 in Steinberg, *Endomorphisms of linear algebraic groups*, Mem. Amer. Math. Soc. **80** (1968). An older reference---for compact Lie groups, at least---is Theorem 3.4 in Borel, *Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes*, Tôhoku Math. J. **13** (1961), 216–240.


