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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] for classical groups and [Hultman] in general, but neither quite answers the above, as far as I can tell.

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# Covering relations in $K\backslash G/B$

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 $\varphi(v')$?