Finite Verma flags and their multiplicities

Assume the Axiom of Choice (The Axiom of Choice). Work in category $\mathcal{O}$ with the conventions of The classical BGG category O, and write $\mathrm{\Delta }(\mu )=M(\mu )$ for the standard objects of Standard and costandard objects, i.e. the Verma modules of Verma modules.

A finite Verma flag of an object $X$ of $\mathcal{O}$ — also called a standard flag or a $\mathrm{\Delta }$-flag — is a finite increasing sequence of subobjects $0=X_0\subseteq X_1\subseteq \cdots \subseteq X_n=X$ such that each quotient $X_i\mathrm{/}{X}_{i-1}$ is isomorphic to a Verma module $\mathrm{\Delta }({\mu }_i)=M({\mu }_i)$, for $i=1,\dots ,n$. An object admitting such a flag is called Verma-filtered. Since the flag is exhausted by its factors, its class in the Grothendieck group $K_0(\mathcal{O})$ of The Grothendieck group and character of O is $[X]={\sum }_{i=1}^n[\mathrm{\Delta }({\mu }_i)]\mathrm{.}$

For a Verma-filtered object $X$ and a weight $\mu$, the multiplicity $(X:\mathrm{\Delta }(\mu ))$ is the number of indices $i$ with ${\mu }_i=\mu$ in a Verma flag of $X$. This number is independent of the chosen flag by Verma-flag multiplicities are independent of the flag ↗, so the notation $(X:\mathrm{\Delta }(\mu ))$ is well-defined for Verma-filtered $X$.

The zero object has the empty flag, and every multiplicity of the zero object is zero; a nonzero Verma-filtered object has at least one factor. A one-step flag of $X$ is exactly an isomorphism $X\cong \mathrm{\Delta }(\mu )$ for a single weight $\mu$, so the objects with a one-step flag are the Verma modules themselves. The flag is a chain of subobjects of $X$ in the module category; it is not required to split, and later examples show that it need not.