Existence and uniqueness of the ordered-group unequal-parameter Kazhdan–Lusztig basis

Fix $w\in W$. By [F3], its strict lower interval $\{y:y<w\}$ is finite and consists only of shorter elements, so construct $C_y$ for those $y$ by induction on length. Their triangular expansions show that the $C_y$ form an $A$-basis of $V_{<w}:={\mathrm{span}}_A\{T_y:y<w\}$: the transition matrix is finite, unitriangular, and ordered by length.

Set $D_w=\overline{T_w}-T_w$. By [F2], there are unique coefficients $a_y\in A$ with $D_w={\sum }_{y<w}{a}_y{C}_y$. Since bar is an involution and each $C_y$ is bar invariant by induction, $\overline{D_w}=-D_w$ implies ${\sum }_{y<w}\overline{a_y}{C}_y=-{\sum }_{y<w}{a}_y{C}_y$. The basis property in step 1.1 gives $\overline{a_y}=-a_y$ for each $y<w$.

Write $a_y={\sum }_{\gamma }{n}_{\gamma }{v}^{\gamma }$ with finite support. The equality $\overline{a_y}=-a_y$ forces $n_0=0$ and $n_{-\gamma }=-n_{\gamma }$. Let $p_y={\sum }_{\gamma <0}{n}_{\gamma }{v}^{\gamma }\in A_{<0}$; then $p_y-\overline{p_y}=a_y$. Define $C_w=T_w+{\sum }_{y<w}{p}_y{C}_y$. Applying bar and using $D_w=\sum a_y{C}_y$ gives $\overline{C_w}=T_w+{\sum }_{y<w}(a_y+\overline{p_y})C_y=C_w$. Expanding the $C_y$ in the standard basis shows that all coefficients below $T_w$ remain in $A_{<0}$, because a sum of two negative exponents is negative. This constructs a normalized bar-invariant $C_w$.

Suppose $C_w^{\mathrm{\prime }}$ is another element with the stated properties. Then $h=C_w^{\mathrm{\prime }}-C_w={\sum }_{y<w}{b}_y{T}_y$ is bar invariant with each $b_y\in A_{<0}$. Choose a nonzero $b_y$ with $\mathrm{\ell }(y)$ maximal. In $\bar{h}$, the coefficient of $T_y$ is $\overline{b_y}$: every other term of maximal length has a distinct leading basis vector, while all lower bar terms have strictly smaller length by [F2, F3]. Bar invariance gives $b_y=\overline{b_y}$, but $A_{<0}\cap \overline{A_{<0}}=\{0\}$, a contradiction. Hence $C_w^{\mathrm{\prime }}=C_w$.

The family $C_w$ is linearly independent: in any finite relation, choose a term of maximal length; its coefficient at the corresponding $T_w$ is unchanged by the unitriangular expansions and must vanish. It spans by induction on $\mathrm{\ell }(w)$, because $T_w=C_w-{\sum }_{y<w}{p}_{y,w}{T}_y$ and all terms on the right other than $C_w$ have smaller length. Thus it is an $A$-basis.

For a simple generator $s$, write $C_s=T_s+p$ with $p\in A_{<0}$. The bar formula in [F2] gives $\overline{T_s}=T_s-v^{L(s)}+v^{-L(s)}$. Bar invariance is therefore equivalent to $p-\bar{p}=v^{-L(s)}-v^{L(s)}$. Splitting into negative and positive exponents gives $p=v^{-L(s)}$ if $L(s)>0$, $p=0$ if $L(s)=0$, and $p=-v^{L(s)}$ if $L(s)<0$, proving the displayed normalization by uniqueness.