Topological properties of $K$ orbits in $G/B$

Question

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!

Answer 1

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.

Answer 2

From the top of my head: This has been studied in the more general context of connected linear reductive $k$-groups, for $k$ any field not in characteristic 2, in Helminck & Wang, On rationality properties of involutions of reductive groups, Adv. in Math. 99 (1993), 26-97 (freely available PDF version here).

But of course the case $k=\mathbb{C}$ (and also $k=\mathbb{R}$) has been well-known and studied long before, so this reference is certainly far from being "optimal". But then it also contains references to lots of previous work, which you can find by skimming its introduction, so it might still help you as a starting point for tracking down other / more / better references. Well, or perhaps somebody will just write up proofs and / or give better references... (in particular, somebody who is not in a hurry due to departing on a trip :-).

Answer 3

For your question 2, the reason is in fact we can prove $$ K\times \mathfrak{p}\xrightarrow{\sim} G\\ (k,p)\mapsto k\exp(p) $$ is an diffeomorphism. Here $\mathfrak{p}$ is the $-1$ eigen space of the involution $\theta$ on the Lie algebra $\mathfrak{g}$. Therefore $K$ is homotopic to $G$, hence connected.

You can see Section I.1 and I.2 of A. Knapp's book "Representation theory of Semisimple Groups".

By the way, I think there should be certain conditions on your involution $\theta$.

Answer 4

This is really a comment, but I want to have tex available. Perhaps this can clear up some of the confusion about involutions, and complex versus compact groups.

If $\theta$ is an arbitrary algebraic (i.e. holomorphic) involution of $G(\mathbb C)$ then $K(\mathbb C)=G(\mathbb C)^\theta$ is a complex, hence noncompact, reductive group. However there is a unique real form $G(\mathbb R)$ of $G(\mathbb C)$ so that $\theta$ normalizes $G(\mathbb R)$, and $K(\mathbb R)=G(\mathbb R)^{\theta}$ is compact. This is Cartan's classification of real groups in terms of their maximal compact subgroups., and accounts for some of the back and forth in the comments about complex versus compact groups.

For example the algebraic involution $\theta(g)={}^tg^{-1}$ of $GL(n,\mathbb C)$ has fixed points $O(n,\mathbb C)$. This is the complexification of $O(n)=GL(n,\mathbb R)^\theta$ - the maximal compact subgroup of $GL(n,\mathbb R)$.