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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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2 Answers 2

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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 $abc$.

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  • $\begingroup$ $(Sp(4),GL(4))$ should be $(Sp(8),GL(8))$ ($n=4$ in Atlas notation)? $\endgroup$
    – Michael Joyce
    Commented Mar 23, 2012 at 16:59
  • $\begingroup$ Yes $n=4$. The real groups are $Sp({\mathbb R}^8) \supset U(4)$. $\endgroup$
    – Allen Knutson
    Commented Mar 29, 2012 at 3:59
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While the paper of Richardson-Springer does study the weak order, it also has useful results on the usual (strong) Bruhat order. In particular, Theorem 7.11 says that Bruhat order is characterized as the weakest partial order on $K \backslash G / B$ that contains the weak order and satsifies the obvious geometric condition $Y \subseteq Y' \Rightarrow P_s \cdot Y \subseteq P_s \cdot Y'$ where $Y$ and $Y'$ are arbitrary $B$-stable subvarieties and $P_s$ denotes the minimal parabolic subgroup associated to the simple reflection $s$.

I don't know a specific reference in the literature for the results you want, but I suspect they can be proven using the generalized reduced decomposition that is introduced in R-S.

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