The Kazhdan–Lusztig Character and Multiplicity Theorem

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.

The comparison of [F1] expresses the Verma-flag multiplicity of the indecomposable projective $P(w\cdot \lambda )$ as the total graded standard-support rank of the polynomial lift $B_{w^{-1}}$: $(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))={\sum }_d{r}_{y^{-1},d}(B_{w^{-1}})$.

The character theorem [F2] gives $h_{\mathrm{\Delta }}(B_{w^{-1}})={\underset{‾}{H}}_{w^{-1}}={\sum }_x{p}_{x,w^{-1}}{H}_x$. By the coefficient formula of [F3], the left side is ${\sum }_x{m}_x(B_{w^{-1}};v)H_x$, where $m_x(B_{w^{-1}};v)={\sum }_d{r}_{x,d}(B_{w^{-1}})v^d$ uses the normalized polynomial standard layers before reduction. Comparing coefficients in the standard Hecke basis of [F4] yields $m_x(B_{w^{-1}};v)=p_{x,w^{-1}}(v)$ for every $x$, as an identity of Laurent polynomials. This comparison uses the character formula; it does not identify an indecomposable class with a product of simple classes.

Evaluate step 1.2 at the label $x=y^{-1}$ and at $v=1$: the left side becomes ${\sum }_d{r}_{y^{-1},d}(B_{w^{-1}})$, which equals the multiplicity by step 1.1, and the right side becomes $p_{y^{-1},w^{-1}}(1)$; the inversion symmetry $p_{y^{-1},w^{-1}}=p_{y,w}$ of [F4] and the specialization $p_{y,w}(1)=P_{y,w}(1)$ of [F5] turn this into $(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))=P_{y,w}(1)$ for all $y,w\in W$. The vanishing clause is the support statement $p_{y,w}=0$ unless $y\le w$ of [F4] and [F5], and only the total value at one is used: no individual coefficient of $P_{y,w}$ is interpreted as a category-$\mathcal{O}$ multiplicity, and no grading on $\mathcal{O}$ is constructed — the grading in play is the auxiliary grading of the polynomial lift, normalized on the Soergel side.

Apply BGG reciprocity [F1] with the two weights $w\cdot \lambda$ and $y\cdot \lambda$: because $P(w\cdot \lambda )$ is the projective cover of $L(w\cdot \lambda )$ and $\mathrm{\Delta }(y\cdot \lambda )=M(y\cdot \lambda )$ by [F3], reciprocity reads $(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))=[\mathrm{\Delta }(y\cdot \lambda ):L(w\cdot \lambda )]=[M(y\cdot \lambda ):L(w\cdot \lambda )]$.

The projective-Verma theorem [F2] evaluates the projective-flag multiplicity in the regular integral block: $(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))=P_{y,w}(1)$, the value at $q=1$ of the classical Kazhdan–Lusztig polynomial, and this value is zero unless $y\le w$ in the Bruhat order.

Substituting the value of step 1.2 into the chain of equalities of step 1.1 gives $[M(y\cdot \lambda ):L(w\cdot \lambda )]=(P(w\cdot \lambda ):\mathrm{\Delta }(y\cdot \lambda ))=P_{y,w}(1)$ for all $y,w\in W$, and the vanishing clause transfers along the same chain: the multiplicity is zero unless $y\le w$, in agreement with the Bruhat direction recorded in [F4]. Since $P_{y,w}(1)$ is a combinatorial quantity of the symmetric group alone, this multiplicity statement — the Kazhdan–Lusztig conjecture's multiplicity assertion — holds with the same answer for every dominant regular integral weight.

By [F1] the class of every standard in $K_0({\mathcal{O}}_{\lambda })$ expands in the simple basis as $[M(y\cdot \lambda )]={\sum }_w[M(y\cdot \lambda ):L(w\cdot \lambda )][L(w\cdot \lambda )]={\sum }_w{A}_{y,w}[L(w\cdot \lambda )]$, where $A_{y,w}=P_{y,w}(1)$; by [F4] and [F7] the matrix $A$ is unitriangular for the Bruhat order: $A_{w,w}=1$ and $A_{y,w}=0$ unless $y\le w$.

The matrix $A$ is the specialization at $v=1$ of the Kazhdan–Lusztig coefficient matrix $P=(p_{x,w})$, because $p_{y,w}(1)=P_{y,w}(1)$ by [F4]; specialisation $v\mapsto 1$ is a ring homomorphism $\mathbb{Z}[v^{\pm 1}]\to \mathbb{Z}$, so the inverse matrix of $A$ is $A^{-1}=Q^{\mathrm{\prime }}(1)=(q_{x,w}^{\mathrm{\prime }}(1))$ by the two-sided inverse statement $Q^{\mathrm{\prime }}=P^{-1}$ of [F3]. By the support clause of [F3], $q_{w,x}^{\mathrm{\prime }}(1)=0$ unless $w\le x$, and all entries are integers; in particular the sum in the formula is supported on $x\ge w$, and every coefficient $q_{w,x}^{\mathrm{\prime }}(1)$ is the value at one of an integer Laurent polynomial.

The specialization is not the Möbius function in general; the following finite witness is computed inside $S_4$ with the recursion [F6]. Put $b:=1324=s_2$ and $T:=3412=s_2{s}_1{s}_3{s}_2$; the reduced subwords of this reduced word for $T$ give the interval $[b,T]=\{1324;1342,1423,2314,3124;1432,2413,3142,3214;3412\}$, ten elements, and all comparable pairs of gap at most two have $P=1$ by the degree bound of [F4], so the only pair that can be nonconstant is the endpoint $(b,T)$, of gap $3$. Apply [F6] with the left descent $s=s_2$ of $T$: $s_{2}T=2413$, $s_{2}b=1234$ and $c=1$, so $P_{b,T}=P_{1234,2413}+q{P}_{b,2413}-{\sum }_z\mu (z,2413)q^{(4-\mathrm{\ell }(z))\mathrm{/}2}{P}_{b,z}$, where $z$ runs over the elements of $[b,2413]=\{1324,1423,2314,2413\}$ with $s_{2}z<z$; left multiplication by $s_2$ swaps the values $2$ and $3$, turning these four elements into $1234,1432,3214,3412$ of lengths $0,3,3,4$, so only $z=b$ has the required descent, and there $\mu (b,2413)=0$ because the gap $2$ is even by [F4]. Hence the correction sum vanishes, and with $P_{b,2413}=1$ (gap $2$) the recursion reduces to $P_{b,T}=P_{1234,2413}+q$. For the remaining value apply [F6] at $(1234,2413)$ with the left descent $s_1$: $s_{1}2413=1423$ and $c=0$, so $P_{1234,2413}=q{P}_{2134,1423}+P_{1234,1423}-{\sum }_z\mu (z,1423)q^{(3-\mathrm{\ell }(z))\mathrm{/}2}{P}_{1234,z}$; the first term vanishes because $1423=s_3{s}_2$ has no reduced subword $s_1$, so $2134\le ̸1423$ and $P_{2134,1423}=0$ by [F4] and [F5], and the correction sum is empty because the four elements of $[1234,1423]=\{1234,1324,1243,1423\}$ all have lengths $0,1,1,2$ raised to $1,2,2,3$ by left multiplication with $s_1$, so none is a left descent. Thus $P_{1234,2413}=1$, and altogether $P_{b,T}(q)=1+q$, while $P_{y,z}=1$ for every other comparable pair in $[b,T]$.

Since the classes $[L(w\cdot \lambda )]$ and $[M(x\cdot \lambda )]$ are the two $\mathbb{Z}$-bases of the Grothendieck group of the block by [F2], step 1.1 exhibits the transition matrix from simples to standards, and its inverse from step 1.2 performs the reverse change of basis: for every $w\in W$ one has $[L(w\cdot \lambda )]={\sum }_x{q}_{w,x}^{\mathrm{\prime }}(1)[M(x\cdot \lambda )]$, the sum being finite and supported on $x\ge w$ by [F5] and step 1.2. Applying the additive character map of [F2] termwise gives the equivalent formal-character identity $\mathrm{c}\mathrm{h}L(w\cdot \lambda )={\sum }_x{q}_{w,x}^{\mathrm{\prime }}(1)\mathrm{c}\mathrm{h}M(x\cdot \lambda )$. In the signed normalization $q_{x,w}=\mathrm{s}\mathrm{g}\mathrm{n}(x)\mathrm{s}\mathrm{g}\mathrm{n}(w)q_{x,w}^{\mathrm{\prime }}$ of [F3], substituting $q_{w,x}^{\mathrm{\prime }}(1)=(-1{)}^{\mathrm{\ell }(w)+\mathrm{\ell }(x)}{q}_{w,x}(1)$ and using that $\mathrm{\ell }(w)+\mathrm{\ell }(x)$ and $\mathrm{\ell }(w)-\mathrm{\ell }(x)$ differ by the even number $2\mathrm{\ell }(x)$ gives the signed inverse-Kazhdan–Lusztig sum $\mathrm{c}\mathrm{h}L(w\cdot \lambda )={\sum }_{x\ge w}(-1{)}^{\mathrm{\ell }(w)-\mathrm{\ell }(x)}{q}_{w,x}(1)\mathrm{c}\mathrm{h}M(x\cdot \lambda )$.

The chain formula of [F3] expresses the inverse coefficients of step 2.1 as the weighted incidence-algebra inverse of $A$: $q_{x,w}^{\mathrm{\prime }}(1)={\sum }_{m\ge 0}(-1{)}^m{\sum }_{x=z_0<z_1<\cdots <z_m=w}{P}_{z_0,z_1}(1)\cdots P_{z_{m-1},z_m}(1)$, a finite sum over the strictly increasing chains of the finite interval $[x,w]$ described in [F5]. If every comparable-pair value $P_{y,z}(1)$ in $[x,w]$ equals $1$, each product in the $m$-th term equals $1$ and the alternating chain count ${\sum }_m(-1{)}^m\mathrm{\#}\{\text{strictly increasing chains }x=z_0<\cdots <z_m=w\}$ is exactly the ordinary Bruhat Möbius value $\mu (x,w)$; so on intervals with all comparable-pair values equal to $1$ the coefficients are ordinary Bruhat Möbius numbers.

Consequently the specialized multiplicity matrix on the interval $[b,T]$ is $A=Z+E$, where $Z$ is the zeta matrix of the Bruhat poset and $E$ has the single nonzero entry $1$ at $(b,T)$. Since $b$ is the minimum and $T$ the maximum of the interval, $Z_{x,b}=1$ only for $x=b$ and $Z_{T,x}=1$ only for $x=T$, so $E{Z}^{-1}=E=ZE$ and $(Z+E)(Z^{-1}-E)=I-ZE+E{Z}^{-1}=I$, that is $A^{-1}=Z^{-1}-E$; at the endpoint the inverse entry is therefore $\mu (b,T)-1=-1-1=-2$, whereas the ordinary Bruhat Möbius value of the interval is $\mu (b,T)=-1$ (the rank sizes are $1,4,4,1$ and the Möbius recursion gives $1$ at the bottom, $-1$ at the four atoms, $+1$ at the four rank-two elements and $-(1-4+4)=-1$ at the top). Using the endpoint polynomial from step 1.3, this proves the claim that the coefficients specialize to ordinary Bruhat Möbius values when all comparable-pair values are $1$ but not in general, and exhibits them as integers that may be negative; by step 2.1 these are exactly the coefficients of the character formula, and only their value at $q=1$ occurs.