Projective Verma multiplicities equal Soergel standard ranks

The object $P:=P(w\cdot \lambda )$ is a nonzero indecomposable projective of ${\mathcal{O}}_{\lambda }$, namely a projective cover of the simple $L(w\cdot \lambda )$, and by the projective dictionary [F2] its Soergel value is $V(P)={\bar{B}}_{w^{-1}}$, the right reduction of the single polynomial lift $B_{w^{-1}}$.

Every decomposition of $P$ into indecomposables is a one-summand decomposition. Indeed, by [F5] a decomposition $P\cong {⨁}_{v}P(v\cdot \lambda {)}^{m_v}$ exists and any two such decompositions agree up to isomorphism and permutation of summands; since the nonzero object $P$ is an indecomposable projective cover by [F4], only one summand can be present, so the decomposition is $P\cong P(v\cdot \lambda )$ for a single label $v$. Applying the equivalence $V$ of [F2] to this isomorphism gives ${\bar{B}}_{w^{-1}}\cong {\bar{B}}_{v^{-1}}$; as $V$ is a bijection on isomorphism classes of indecomposables, $v=w$. Hence in the dictionary of [F3] the polynomial lift assigned to $P$ is the single term $\mathsf{B}(P)=B_{w^{-1}}$, with no further summands.

Substitute $\mathsf{B}(P)=B_{w^{-1}}$ from step 1.2 into the general rank comparison [F3] applied to the projective $P=P(w\cdot \lambda )$: its left side is the Verma-flag multiplicity $(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))$ of [F1] and [F6], and its right side is ${\sum }_d{r}_{y^{-1},d}(B_{w^{-1}})$, the sum of the graded standard-support multiplicities of the polynomial lift $B_{w^{-1}}$. This is the displayed identity for all $y,w\in W$. The ranks on the right are pre-reduction ranks of the polynomial lift, by the final clause of [F3] and by the reduction convention [F2]; no separated filtration of $V(P(w\cdot \lambda ))$ is produced or used.