Let $G$ be a simply connected complex Lie group, $\theta$ an involution,
and $K = G^\theta$ the fixed point subgroup. Pick a $\theta$-invariant
Borel subgroup $B$. Then there is a natural
map $K\backslash G \to G$ taking $Kg \mapsto \theta(g^{-1}) g$,
inducing a map $\varphi : K\backslash G/B \to B\backslash G/B \cong W\_G$.


Most of what I know about this is in papers of [Richardson-Springer],
where they study what one might call the weak Bruhat order on
$K\backslash G/B$. Unfortunately, I want the strong Bruhat order.
In particular, I seek proofs (and of course, references) for the following.



> 
> If $v \gtrdot v'$ is a covering relation in $K\backslash G/B$, then
>  $\ell(\varphi(v)) - \ell(\varphi(v')) = 1,2,3$, and the difference is determined by
>  whether the difference in the Cartan ranks of $v,v'$ is $-1,0$, or $1$.
> 
> 
> If the difference is $2$, then there exists a positive root $\beta$ 
>  such that $\varphi(v) = r\_{\theta\cdot \beta} \varphi(v') r\_\beta$.
> 
> 
> 


I also have a question:



> 
> If the difference is $3$, what is the relation of $\varphi(v)$
>  and $\varphi(v')$?
> 
> 
> 


ADDED: The best references I have found are [[Incitti]](http://fpsac.org/FPSAC03/ARTICLES/40.pdf) for classical groups and [[Hultman]](http://www.mai.liu.se/~axehu73/papper/twist.pdf) in general, but neither quite answers the above, as far as I can tell.



Covering relations in $K\backslash G/B$
The answer to the first question is no.


David Vogan gave me the example of $(Sp(4),GL(4))$, orbits #46 and #76 according to the Atlas numbering here: 
<http://www.liegroups.org/web/atlasInput.html>


Axel Hultman gave me the example of $D\_4$ with generators $a,b,c,d$; $d$ being the non-leaf in the diagram, and ordinary involutions. Then, $abcdcba$ covers $abc$.


