Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and let $\mathfrak{U}(\mathfrak{g})$ be its universal enveloping algebra. Fix a Cartan subalgebra $\mathfrak{h} \subset \mathfrak{g}$, and denote the dual space by $\mathfrak{h}^{\ast}$. The associated root system is denoted by $\Delta$. Fix a positive system $\Delta^{+}$ with associated Borel $\mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}^{+}$, where $\mathfrak{n}^{+} := \sum_{\alpha \in \Delta^{+}}\mathfrak{g}^{\alpha}$. Let $\omega$ be an anti-linear involution on $\mathfrak{g}$, and let $\mathfrak{g}^{\mathbb{R}}$ be the associated real form. We extend $\omega$ naturally to $\mathfrak{U}(\mathfrak{g})$.
I am interested in unitarizable highest weight modules over $\mathfrak{g}$ with respect to the real form $\mathfrak{g}^{\mathbb{R}}$. That is, a $\mathfrak{g}$-module $M$ such that there exists a non-trivial $v_{\Lambda} \in M$ for which: $$ (1) \ \mathfrak{U}(\mathfrak{g})v_{\Lambda} = M; \qquad (2) \ \mathfrak{n}^{+}v_{\Lambda} = 0; \qquad (3) \ Hv_{\Lambda} = \Lambda(H)v_{\Lambda} \ \text{for all} \ H \in \mathfrak{h}, $$ and there exists a positive definite Hermitian form such that $$ \langle Xv,w \rangle = \langle v,\omega(X)w\rangle $$ for all $X \in \mathfrak{U}(\mathfrak{g})$ and $v,w \in M$. Without loss of generality, we can assume that the positive definite Hermitian form on $M$ is the Shapovalov form induced from the Verma module with highest weight $\Lambda$.
Take a non-trivial element $v \in M$, as well as $X \in \mathfrak{g}^{\alpha}$ and $Y \in \mathfrak{g}^{\beta}$ for $\alpha,\beta \in \Delta^{+}$. Note that $\Delta^{+}$ has a natural ordering of roots.
Question: Is there an ordering of norms such that $\|Xv\| \leq \|Yv\|$ if and only if $\alpha \leq \beta$? In other words, for a fixed $v \in M$, are these norms ordered by the ordering of the roots?
