Projective Verma multiplicities are Kazhdan–Lusztig values at one

Given: The Axiom of Choice (inherited from the projective-lift theory cited below), a regular integral type-$A$ block ${\mathcal{O}}_{\lambda }$ with $W=S_n$, labels $y,w\in W$, and the grading variable $v$ with $q=v^{-2}$.

The standard-rank comparison (Projective Verma multiplicities equal Soergel standard ranks): $(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))={\sum }_d{r}_{y^{-1},d}(B_{w^{-1}})$, where $B_{w^{-1}}$ is the polynomial lift with right reduction $V(P(w\cdot \lambda ))={\bar{B}}_{w^{-1}}$ and the $r_{x,d}$ are the pre-reduction graded standard-support multiplicities in the normalization of Standard graph bimodules, support filtrations and characters.

Soergel's character theorem (Soergel's character conjecture for type A): for every $x\in S_n$ the character of the indecomposable type-$A$ Soergel bimodule $B_x$ is the Kazhdan–Lusztig basis element $\mathrm{c}\mathrm{h}(B_x)={\underset{‾}{H}}_x$, equivalently $[B_x]\mapsto {\underset{‾}{H}}_x$ under the identification of the split Grothendieck group with the Hecke algebra; the scalar-extension comparison identifies the rational and real versions.

The actual standard-character normalization (Standard graph bimodules, support filtrations and characters, The type-A standard character is multiplicative, The split Grothendieck group of the Soergel category is the type-A Hecke algebra, The regular integral dot-orbit labelling convention): $r_{x,d}(B)$ counts the normalized standard layer ${\mathrm{\Delta }}_x(d)=R_x(d-\mathrm{\ell }(x))$ before reduction, so $h_{\mathrm{\Delta }}(B)={\sum }_x{m}_x(B;v){\tilde{T}}_x$ with $m_x(B;v)={\sum }_d{r}_{x,d}(B)v^d$ and ${\tilde{T}}_x=v^{\mathrm{\ell }(x)}{T}_x=H_x$ in the normalized standard Hecke basis of [F4]. The character $h_{\mathrm{\Delta }}$ equals the split-Grothendieck-ring isomorphism. Its simple value is $v(T_i+1)=H_{s_i}+v$, rather than the standard $H_{s_i}$; multiplying these simple values computes a Bott--Samelson character, not in general the character of an indecomposable.

Kazhdan–Lusztig basis (Existence and uniqueness of the Kazhdan–Lusztig basis): ${\underset{‾}{H}}_w={\sum }_{y\le w}{p}_{y,w}{H}_y$ with $p_{w,w}=1$, $p_{y,w}=0$ unless $y\le w$, and simultaneous inversion symmetry $p_{y^{-1},w^{-1}}=p_{y,w}$.

Classical normalization (Kazhdan–Lusztig polynomials in the classical $q$-normalization): $p_{y,w}=v^{\mathrm{\ell }(w)-\mathrm{\ell }(y)}{P}_{y,w}(v^{-2})$ for $y\le w$, with $P_{y,w}=0$ unless $y\le w$ and $P_{w,w}=1$; specialising $v=1$ gives $p_{y,w}(1)=P_{y,w}(1)$.

Block conventions (The regular integral dot-orbit labelling convention): $(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))$ is the Verma-flag multiplicity, $V(P(w\cdot \lambda ))={\bar{B}}_{w^{-1}}$ is the label-inverting projective dictionary, and the support convention $p_{y,w}=0$ unless $y\le w$ is the Bruhat direction fixed there.