The parabolic Kazhdan–Lusztig bases and polynomials

Retain the notation of The parabolic Hecke module and the antispherical standard basis and write the parabolic Kazhdan--Lusztig bases of Existence and uniqueness of the parabolic Kazhdan–Lusztig bases as

$C_x={\sum }_{y\le x,y\in {}^{J}W}{m}_{y,x}{m}_y,D_x={\sum }_{y\le x,y\in {}^{J}W}{n}_{y,x}{n}_y,x\in {}^{J}W,$

with coefficients $m_{y,x},n_{y,x}\in \mathbb{Z}[v]$ supported on the lower Bruhat ideal, $m_{x,x}=n_{x,x}=1$ and $m_{y,x},n_{y,x}\in v\mathbb{Z}[v]$ for $y<x$.

The spherical parabolic Kazhdan--Lusztig polynomials are $M_{y,x}^J(q):=v^{-\mathrm{\ell }(x)+\mathrm{\ell }(y)}{m}_{y,x},q=v^{-2},$ and the antispherical (sign) parabolic Kazhdan--Lusztig polynomials are $N_{y,x}^J(q):=v^{-\mathrm{\ell }(x)+\mathrm{\ell }(y)}{n}_{y,x}\mathrm{.}$ The parity and degree bounds established in Parabolic Kazhdan–Lusztig polynomials are alternating sums of ordinary ones ↗ make these polynomials in $q$; that reduction theorem discharges the well-definedness asserted here and is recorded in justified_by. Values at $v=1$ are written $m_{y,x}(1)$, $n_{y,x}(1)$.

Normalization dictionary. With $P_{y,x}^J(u)$ the parabolic polynomials of Deodhar (1987) attached to the subset $J$, one has $M_{y,x}^J=P_{y^{-1},x^{-1}}^J$ for $u=-1$ and $N_{y,x}^J=P_{y^{-1},x^{-1}}^J$ for $u=q$, in the index convention of Deodhar and of the anti-spherical category. Thus the spherical family is the $u=-1$ normalization and the antispherical family the $u=q$ normalization; the two are not related by a unit.

Spherical reduction to ordinary coefficients. If $W_J$ is finite with longest element $w_0^J$, then $m_{y,x}=p_{w_0^{J}y,w_0^{J}x}\text{for all }y,x\in {}^{J}W,$ where $p$ is the coefficient of the ordinary Kazhdan--Lusztig basis of Kazhdan–Lusztig polynomials in the classical $q$-normalization and ${\underset{‾}{H}}_w={\sum }_{y\le w}{p}_{y,w}{H}_y$ in the normalization of The normalized type-A Hecke algebra and its bar involution.

• Well-definedness. The coefficients $m_{y,x},n_{y,x}$ exist and are unique by the parabolic basis theorem; the triangular support and the vanishing of $m_{y,x},n_{y,x}$ for $y\le ̸x$ are part of that theorem. The assertions that the shifted coefficients are polynomials in $q=v^{-2}$ (parity and degree bound) are not part of the existence theorem: they are proved in the alternating-sum reduction theorem Parabolic Kazhdan–Lusztig polynomials are alternating sums of ordinary ones ↗, which is the recorded justification of this definition. No parity clause is read back into the existence theorem.
• Both families are canonical-basis coefficients, not chosen representatives. $C_x$ and $D_x$ are the unique bar-fixed elements with triangular support of the parabolic basis theorem; the symbols $m_{y,x}$ and $n_{y,x}$ are their standard-basis coefficients in $M^J$ and $N^J$ respectively.