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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 $w_1$.

Q: Is it true that $w_1 \leq w_2$ if and only if $\Phi_{w_1}^- \subseteq \Phi_{w_2}^-$?

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.

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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]:

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 $k \in \mathbb{Z}$.

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 $\to$.


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

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