Parabolic Kazhdan–Lusztig polynomials are alternating sums of ordinary ones

The antispherical image. Let $\mathrm{\Phi }:H_v(n)\to N^J=A{\otimes }_{H_J}{H}_v(n)$, $\mathrm{\Phi }(h):=1\otimes h$. It is $A$-linear and satisfies $\mathrm{\Phi }(\bar{h})=\overline{\mathrm{\Phi }(h)}$ by [F3]. Since ${\underset{‾}{H}}_x$ is bar-fixed, so is $\mathrm{\Phi }({\underset{‾}{H}}_x)$; expanding ${\underset{‾}{H}}_x={\sum }_{w\le x}{p}_{w,x}{H}_w$ and using the parabolic factorization $w=zy$ with $z\in W_J$, $y\in {}^{J}W$ and $1\otimes H_w={\phi }_{-}(H_z)n_y=(-v{)}^{\mathrm{\ell }(z)}{n}_y$ gives $\mathrm{\Phi }({\underset{‾}{H}}_x)={\sum }_{y\in {}^{J}W}({\sum }_{z\in W_J}(-v{)}^{\mathrm{\ell }(z)}{p}_{zy,x})n_y$. The top coefficient at $y=x$ is $p_{x,x}=1$ (only $z=1$ contributes, since $zx\le x$ with $z\mathrm{\ne }1$ would give $\mathrm{\ell }(zx)>\mathrm{\ell }(x)$); and for a coefficient at $y\mathrm{\ne }x$, every contributing term has $zy\le x$, hence $y\le zy$ by [F5] and $y<x$. For $y<x$, every contributing ordinary coefficient has $zy<x$: equality $zy=x$ would contradict the unique parabolic factorization of the minimal representative $x$. Thus $p_{zy,x}\in v\mathbb{Z}[v]$, and multiplication by $(-v{)}^{\mathrm{\ell }(z)}$ preserves this property. Hence $\mathrm{\Phi }({\underset{‾}{H}}_x)$ is bar-fixed in $n_x+{\sum }_{y<x}v\mathbb{Z}[v]n_y$, so by uniqueness [F4] it equals $D_x$ and matching coefficients gives (1).

The integral parabolic eigenline. Put $L=\mathrm{\ell }(w_J)$ and $C_J={\sum }_{u\in W_J}{v}^{L-\mathrm{\ell }(u)}{H}_u$. Pairing $u$ with $su$ where $\mathrm{\ell }(su)=\mathrm{\ell }(u)+1$, the left multiplication rule gives $H_s(a_u{H}_u+a_{su}{H}_{su})=a_{su}{H}_u+(a_u+(v^{-1}-v)a_{su})H_{su}$. This equals $v^{-1}(a_u{H}_u+a_{su}{H}_{su})$ iff $a_{su}=v^{-1}{a}_u$. Iterating along reduced words proves that the simultaneous left eigenvectors in $H_J$ are exactly $A{C}_J$: their coefficients satisfy $a_u=v^{-\mathrm{\ell }(u)}{a}_e=v^{L-\mathrm{\ell }(u)}{a}_{w_J}$. The same pairing on the right gives $C_J{H}_s=v^{-1}{C}_J$. Bar preserves the left eigen-equations, since conjugating $H_{s}X=v^{-1}X$ gives $H_s^{-1}\bar{X}=v\bar{X}$ and hence $H_s\bar{X}=v^{-1}\bar{X}$. By [F1] its triangular standard expansion leaves the coefficient of $H_{w_J}$ in $\overline{C_J}$ equal to one. The integral eigenline calculation therefore forces $\overline{C_J}=C_J$. Every $u\in W_J$ is a reduced subword of a reduced expression for the block-reversing longest permutation $w_J$ (sort a block permutation up to the reverse order by adjacent ascents); thus $u\le w_J$. Consequently $C_J$ has the ordinary canonical triangular form, and [F1] identifies it with ${\underset{‾}{H}}_{w_J}$.

Well-definedness and degree bounds for the antispherical family. In the formula of step 1.1, a term with $p_{zy,x}\mathrm{\ne }0$ has $zy\le x$ and $\mathrm{\ell }(zy)=\mathrm{\ell }(z)+\mathrm{\ell }(y)$ by [F5], so $p_{zy,x}\in v^{\mathrm{\ell }(x)-\mathrm{\ell }(zy)}\mathbb{Z}[v^{-2}]$ by [F1]; multiplying by $(-v{)}^{\mathrm{\ell }(z)}$ gives a contribution in $v^{\mathrm{\ell }(x)-\mathrm{\ell }(y)}\mathbb{Z}[v^{-2}]$. Hence $n_{y,x}\in v^{\mathrm{\ell }(x)-\mathrm{\ell }(y)}\mathbb{Z}[v^{-2}]$, so $N_{y,x}^J(q)=v^{\mathrm{\ell }(y)-\mathrm{\ell }(x)}{n}_{y,x}$ lies in $\mathbb{Z}[v^{-2}]=\mathbb{Z}[q]$ with $q=v^{-2}$, and $n_{y,x}$ has parity $\mathrm{\ell }(x)-\mathrm{\ell }(y)$ modulo two. This discharges the well-definedness of the antispherical polynomials.

The integral spherical image. Define $\mathrm{\Theta }:M^J\to H_v(n)$ by $\mathrm{\Theta }(1\otimes h)=C_{J}h$. Step 1.2 proves balancing and bar compatibility. Decompose $H_v(n)={⨁}_{y\in {}^{J}W}{H}_J{H}_y$ as in [F3]. An element $X={\sum }_y{h}_y{H}_y$ satisfies $H_{s}X=v^{-1}X$ for all $s\in J$ iff each $h_y$ does. The integral eigenline calculation of step 1.2 then gives $h_y=c_y{C}_J$ with $c_y\in A$. Thus the image of $\mathrm{\Theta }$ is exactly this integral eigenmodule, and it is injective: the coefficient of $H_{w_{J}y}$ in $C_J{H}_y={\sum }_u{v}^{L-\mathrm{\ell }(u)}{H}_{uy}$ is one, and no other coset contributes to that standard label. By [F2] in the descent case [F6], ${\underset{‾}{H}}_{w_{J}x}$ lies in this image, with unique preimage $X_x={\sum }_y{c}_y{m}_y$.

Triangularity and coefficient comparison. The disjoint coset expansions in step 2.2 give $c_y=p_{w_{J}y,w_{J}x}$. The coefficient $c_x$ is one. If $c_y\mathrm{\ne }0$, then $w_{J}y\le w_{J}x$, so $y\le w_{J}y\le w_{J}x$. Choose a reduced expression for $w_{J}x$ consisting of a word in $J$ followed by a reduced expression for $x$. By [F5] it contains a reduced subword for $y$. Such a subword cannot select any letter from the initial $J$-word: its first selected letter would be in $J$, making it a left descent of $y$, contrary to $y\in {}^{J}W$. Thus the subword lies entirely in $x$, proving $y\le x$. For $y\mathrm{\ne }x$, the ordinary coefficient $p_{w_{J}y,w_{J}x}$ lies in $v\mathbb{Z}[v]$ by [F1]. Bar compatibility and injectivity give $\overline{X_x}=X_x$. Therefore $X_x$ has exactly the defining canonical form in [F4], so $X_x=C_x$ and $m_{y,x}=p_{w_{J}y,w_{J}x}$. This proves (2), including zero coefficients outside the lower ideal.

Degree bounds and conclusion. The ordinary degree bound of [F1] gives $p_{w_0^{J}y,w_0^{J}x}\in v^{\mathrm{\ell }(x)-\mathrm{\ell }(y)}\mathbb{Z}[v^{-2}]$, so $m_{y,x}$ has parity $\mathrm{\ell }(x)-\mathrm{\ell }(y)$ and $M_{y,x}^J(q)=v^{\mathrm{\ell }(y)-\mathrm{\ell }(x)}{m}_{y,x}\in \mathbb{Z}[q]$. Both families are therefore computable from the ordinary coefficients by (1) and (2), and both polynomial families are well defined.