Let $$Z(\mathfrak{g})$$ be the centre of $$U(\mathfrak{g})$$ and let $$\chi_\lambda$$ be an algebra homomorphism $$Z(\mathfrak{g}) \to \mathbb{C}$$ such that $$z \cdot v = \chi_\lambda(z)v$$ for all $$z \in Z(\mathfrak{g})$$, $$v \in M(\lambda)$$, then $$M_I(\lambda)$$ and its subquotients including $$L(\lambda)$$ have the same infinitesimal character $$\chi_\lambda$$.

Why we call the infinitesimal character $$\chi_\lambda$$ of $$L(\lambda)$$ regular if $$\langle \lambda + \rho,\alpha^\lor\rangle \neq 0$$ for all $$\alpha\in\Phi$$ instead of $$\langle \lambda,\alpha^\lor\rangle \neq 0$$ for all $$\alpha\in\Phi$$?

I know that $$\chi_\lambda=\chi_\mu$$ iff $$\lambda=w\cdot\mu$$ for some $$w\in W$$. It makes more sense to define in the former way. But is there any other/fundamental reason for the definition?

• Could you please introduce your notation and assumptions ? – Paul Broussous Dec 17 '18 at 17:20
• Note that the language "dot-regular" is also used, to clarify the use of the $\rho$-shift. The unmodified "regular" is unfortunately ambiguous. (By the way, I agree with Vit that it's best to use the dot-notation rather than try to modify the highest weight as people earlier did.) " – Jim Humphreys Dec 17 '18 at 22:39

People sometimes abuse notation and talk about character $$\lambda$$ when in fact they mean $$\chi_\lambda$$. Sometimes you can find definitions of Verma modules that incorporates $$\rho$$ shift, e.g. $$M(\lambda) = \mathfrak{U(g)\otimes_{U(p)}} \mathbb{F}_{\lambda - \rho}$$. This can be quite confusing to newcomers. I prefer to work without the shift for weights and for calculations instead of affine action I just add $$\rho$$ use normal action and call the resulting numerical vector character.