Kazhdan–Lusztig polynomials in the classical $q$-normalization

Identify $\mathbb{Z}[q]$ with $\mathbb{Z}[v^{-2}]$ by $q\mapsto v^{-2}$. For $y\le w$ in $S_n$, put $d=\mathrm{\ell }(w)-\mathrm{\ell }(y)$ and define the Kazhdan–Lusztig polynomial $P_{y,w}(q)\in \mathbb{Z}[q]$ by $P_{y,w}(v^{-2})=v^{-d}{p}_{y,w},$ where ${\underset{‾}{H}}_w={\sum }_{y\le w}{p}_{y,w}{H}_y$ is the Kazhdan–Lusztig basis of Existence and uniqueness of the Kazhdan–Lusztig basis. The parity clause of that theorem makes the right side a polynomial in $v^{-2}$. The conventions are: $P_{y,w}=0$ unless $y\le w$, $P_{w,w}=1$, $P_{y,w}$ has constant term 1, and for $y<w$ its degree is at most $(\mathrm{\ell }(w)-\mathrm{\ell }(y)-1)\mathrm{/}2$. The $\mu$-coefficient is $\mu (y,w):=\text{coefficient of }{q}^{(\mathrm{\ell }(w)-\mathrm{\ell }(y)-1)\mathrm{/}2}\text{ in }{P}_{y,w},$ defined to be 0 when $\mathrm{\ell }(w)-\mathrm{\ell }(y)$ is even; equivalently $\mu (y,w)$ is the coefficient of $v$ in $p_{y,w}$. One writes $\mu (y\mathrm{\mid }w)\mathrm{\ne }0$ when $y\mathrm{\ne }w$ and ($y<w$ and $\mu (y,w)\mathrm{\ne }0$) or ($w<y$ and $\mu (w,y)\mathrm{\ne }0$). The dictionary with the literature is recorded for use: with the classical parameter $T_s^2=(q-1)T_s+q$ one has $H_w=v^{\mathrm{\ell }(w)}{T}_w$, the element ${\underset{‾}{H}}_w$ is the basis element $C_w^{\mathrm{\prime }}$ of [EW], and $h_{y,w}=p_{y,w}$ with $v^{\mathrm{\ell }(w)-\mathrm{\ell }(y)}{P}_{y,w}(v^{-2})=h_{y,w}$ [EW, Remark 3.2].

The coefficient rescaling, support, constant term and degree bound use the locally proved coefficient, parity and degree clauses of Existence and uniqueness of the Kazhdan–Lusztig basis. Coefficientwise positivity is not required.