# Bruhat order and positive roots made negative

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.

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.