# Good range and fair range

Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable Cartan subalgebra $\mathfrak{h}$ and a root system $\Delta$. Denote by $\rho$ half the sum of all positive roots. Suppose that $\mathfrak{q}$ is a $\theta$-stable parabolic subalgebra with Levi decomposition $\mathfrak{q}=\mathfrak{l}+\mathfrak{u}$. If $\lambda$ is a weight orthogonal to all the roots of $\mathfrak{l}$, one may define the $A_\mathfrak{q}(\lambda)$ module.

Denote by $\Delta(\mathfrak{u})$ the set of roots in $\mathfrak{u}$, and denote by $\rho(\mathfrak{u})$ half the sum of roots in $\Delta(\mathfrak{u})$. There are two assumptions.

If $(\mathrm{Re}\lambda+\rho,\alpha)\geq0$ for all $\alpha\in\Delta(\mathfrak{u})$, then $\lambda$ is said to be in the weakly fair range.

If $(\mathrm{Re}\lambda+\rho(\mathfrak{u}),\alpha)\geq0$ for all $\alpha\in\Delta(\mathfrak{u})$, then $\lambda$ is said to be in the weakly good range.

It is not clear to me why good is a stronger assumption than fair. Actually, it means that $(\rho(\mathfrak{l}),\alpha)\leq0$ for all $\alpha\in\Delta(\mathfrak{u})$, where $\rho(\mathfrak{l})$ is half the sum of all the positive roots in $\mathfrak{l}$. But if $\alpha$ is the maximal root, this is not true. I shall be grateful if experts here have some comments. Thank you so much!

• you can find a proof that "good implies fair" at the bottom of page 194 of the paper where D.A. Vogan introduced this terminology. – Carlo Beenakker Jul 11 '18 at 9:23
• @CarloBeenakker Thank you so much for your advice, professor Beenakker! The paper does provide a more precise definition for (weakly) fair range. – Hebe Jul 12 '18 at 7:31