Let $(\Phi, V)$ be a reduced root system with base $\Delta$ and Weyl group $W$. Let $\ell$ be the length function of $W$ with respect to the set of simple reflections $S = \{s\_{\alpha} : \alpha \in \Delta\}$.


If $\Phi^+$ is the set of positive roots in $\Phi$ with respect to $\Delta$, and for each $w \in W$ we set $\Phi\_w^- = \{ \alpha \in \Phi^+ : w.\alpha < 0 \}$, then $\ell(w) = |\Phi\_w^-|$.


Let $\leq$ be the Bruhat order on $W$. By definition, $w\_1 \leq w\_2$ if and only if every (equivalently, some) reduced decomposition of $w\_2$ contains a subexpression which is a reduced decomposition of $w\_1$. 



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> Q: Is it true that $w\_1 \leq w\_2$ if and only if $\Phi\_{w\_1}^- \subseteq \Phi\_{w\_2}^-$?
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I know that this is true for the subset of elements in $W$ which *support Bessel functions*. We say that $w \in W$ supports a Bessel function if it is equal to $w\_l w\_{l, \theta}$ for some $\theta \subseteq \Delta$, where $w\_l$ is the long element of $W$ and $w\_{l, \theta}$ is the long element of $W\_{\theta} = \langle s\_{\alpha} : \alpha \in \theta \}$. 


This reflects the fact that in a reductive group, the Bruhat order among Weyl group elements which support Bessel functions reverses the order of the Levi subgroups. 



Bruhat order and positive roots made negative
There is a counterexample in $\mathfrak{sl}\_3$. Denote by $\alpha, \beta$ the simple roots, and $s,t$ the corresponding simple reflections. Then $\Phi\_s^- = \{\alpha\}$ and $\Phi\_{st}^- = \{\beta, \alpha+\beta\}$.


It could be that the condition $\Phi\_{w\_1}^- \subseteq \Phi\_{w\_2}^-$ implies $w\_1 \leq w\_2$, I am not sure at the moment.


However, there is a precise condition; see Proposition 3.2.14. in [1]:



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> We have $w\_1 \to w\_2$ if and only if $\langle \Phi\_{w\_2}^- \rangle = \langle \Phi\_{w\_1}^- \rangle + k \alpha$, for some positive root $\alpha$ and $k \in \mathbb{Z}$.
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Here $\langle S \rangle$ denotes the sum of all elements in $S$.


Recall that the Bruhat order $\leq$ is the transitive closure of the relation $\to$.




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[1] A. Čap and J. Slovák. Parabolic Geometries I: Background and General
Theory, volume 154 of Mathematical Surveys and Monographs. American
Mathematical Society, 2009.


