Singular and Parabolic Kazhdan–Lusztig Theory

Work with the normalized type-$A$ Hecke algebra $H_v(n)$ over $A=\mathbb{Z}[v^{\pm 1}]$ of The normalized type-A Hecke algebra and its bar involution: generators $H_s$ with $H_s^2=1+(v^{-1}-v)H_s$, standard basis $\{H_w:w\in S_n\}$, bar involution $\overline{H_s}=H_s^{-1}=H_s-(v^{-1}-v)$ and Kazhdan--Lusztig basis $\{{\underset{‾}{H}}_w\}$ of Existence and uniqueness of the Kazhdan–Lusztig basis, so that ${\underset{‾}{H}}_s=H_s+v$. The underlying Coxeter data are those of The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition, Permutation Weyl group and inversion length and The generic type-A Hecke algebra.

Fix $J\subseteq S=\{s_1,\dots ,s_{n-1}\}$, let $W_J=⟨s:s\in J⟩\subseteq S_n$ be the standard parabolic subgroup and let ${\mathrm{\Phi }}_J=\mathrm{\Phi }\cap {\mathrm{span}}_{\mathbb{R}}\{{\alpha }_s:s\in J\}$ be its standard type-$A$ root subsystem, all in the conventions of Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups.

The two characters. Let $H_J\subseteq H_v(n)$ be the subalgebra generated by $\{H_s:s\in J\}$. For $s\in J$ the quadratic relation makes $v^{-1}$ and $-v$ the two roots of $x^2-(v^{-1}-v)x-1=0$, so there are well-defined algebra homomorphisms ${\phi }_{+},{\phi }_{-}:H_J\to A$ with

${\phi }_{+}(H_s)=v^{-1},{\phi }_{-}(H_s)=-v(s\in J)\mathrm{.}$

Viewing $A$ through ${\phi }_{+}$ (resp. ${\phi }_{-}$) as a right $H_J$-module gives the spherical parabolic module $M^J:=A{\otimes }_{H_J}{H}_v(n)$ and the antispherical parabolic module $N^J:=A{\otimes }_{H_J}{H}_v(n)$, right $H_v(n)$-modules under right multiplication on the second factor.

Coset representatives. Throughout, ${\mathrm{\Phi }}^{+}$ is the positive system of $\mathrm{\Phi }$ and ${\mathrm{\Phi }}_J^{+}={\mathrm{\Phi }}_J\cap {\mathrm{\Phi }}^{+}$, and

${}^{J}W:=\{w\in S_n:\mathrm{\ell }(sw)>\mathrm{\ell }(w)\text{ for all }s\in J\},W^J:=\{w\in S_n:\mathrm{\ell }(ws)>\mathrm{\ell }(w)\text{ for all }s\in J\}$

are the left-minimal and right-minimal representatives of the cosets $W_{J}w$ and $w{W}_J$, respectively. By Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification and Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives every $w\in S_n$ factors uniquely as $w=ly$ with $l\in W_J$, $y\in {}^{J}W$ and $\mathrm{\ell }(w)=\mathrm{\ell }(l)+\mathrm{\ell }(y)$, and inversion interchanges ${}^{J}W$ and $W^J$.

Standard bases. For $x\in {}^{J}W$ define $m_x:=1\otimes H_x\in M^J,n_x:=1\otimes H_x\in N^J\mathrm{.}$ The families $\{m_x:x\in {}^{J}W\}$ and $\{n_x:x\in {}^{J}W\}$ are the spherical standard basis and antispherical standard basis, respectively. Their basis property is proved in Standard bases, right actions and bar involutions on the parabolic Hecke modules ↗ (1), by the free left $H_J$-module decomposition $H_v(n)={⨁}_{x\in {}^{J}W}{H}_J{H}_x$ and tensoring with the character modules.

• Well-definedness of ${\phi }_{\pm }$. Assign every generator $H_s$ of the full presented algebra $H_v(n)$ the same scalar $v^{-1}$, respectively $-v$. Direct expansion verifies the quadratic relation; both sides of each braid or commutation relation have the same scalar value. Thus the assignments define algebra maps $H_v(n)\to A$. Restricting these maps to $H_J$ gives ${\phi }_{\pm }$, without assuming a presentation for the embedded subalgebra.
• Module structure. For a right $H_J$-module $B$, the tensor product $B{\otimes }_{H_J}{H}_v(n)$ is a right $H_v(n)$-module by right multiplication on the second factor; the two modules $M^J$ and $N^J$ differ only in the $H_J$-action on $A$, which is trivial ($v^{-1}$) for $M^J$ and sign ($-v$) for $N^J$. The standard basis vectors are defined above; their basis property, right actions and compatible bars are proved in Standard bases, right actions and bar involutions on the parabolic Hecke modules ↗.
• Normalization. Spherical is ${\phi }_{+}(H_s)=v^{-1}$ (the trivial character at the parameter of the anti-spherical reference) and antispherical is ${\phi }_{-}(H_s)=-v$ (the sign character). The two are not exchanged by any unit, and the small parabolic $S_3\mathrm{/}{S}_2$ exhibits different canonical coefficients; this is why both modules are recorded. The singular-block multiplicities of this page are read in the spherical normalization.
• Root subsystem. ${\mathrm{\Phi }}_J$ is a closed subsystem of the type-$A$ root system and ${\mathrm{\Phi }}_J^{+}={\mathrm{\Phi }}_J\cap {\mathrm{\Phi }}^{+}$ is its positive system (Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups, Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives (2)).

Retain the notation of The parabolic Hecke module and the antispherical standard basis and put $m_x:=1\otimes H_x$, $n_x:=1\otimes H_x$ for $x\in {}^{J}W$, and $C_s:={\underset{‾}{H}}_s=H_s+v$ for $s\in S$. Then:

(1) (Bases.) The sets $\{m_x:x\in {}^{J}W\}$ and $\{n_x:x\in {}^{J}W\}$ are $A$-bases of $M^J$ and $N^J$ respectively. Indeed every $w\in S_n$ factors uniquely as $w=ly$ with $l\in W_J$, $y\in {}^{J}W$, length-additively, and ${\phi }_{+}(H_l)=v^{-\mathrm{\ell }(l)}$ and ${\phi }_{-}(H_l)=(-v{)}^{\mathrm{\ell }(l)}$ are units of $A$, so $1\otimes H_w={\phi }_{\pm }(H_l)(1\otimes H_y)$ in both modules.

(2) (Right action.) For $s\in S$ and $x\in {}^{J}W$, $m_x{C}_s=\{\begin{array}{ll}{m}_{xs}+v{m}_x,& xs\in {}^{J}W,xs>x,\\ m_{xs}+v^{-1}{m}_x,& xs\in {}^{J}W,xs<x,\\ (v+v^{-1})m_x,& xs\mathrm{\notin }{}^{J}W,\end{array}{n}_x{C}_s=\{\begin{array}{ll}{n}_{xs}+v{n}_x,& xs\in {}^{J}W,xs>x,\\ n_{xs}+v^{-1}{n}_x,& xs\in {}^{J}W,xs<x,\\ 0,& xs\mathrm{\notin }{}^{J}W\mathrm{.}\end{array}$

(3) (Bar.) The assignment $a\otimes h\mapsto \bar{a}\otimes \bar{h}$ (bar on $A$ and the bar involution of $H_v(n)$) descends to $A$-antilinear involutions of $M^J$ and $N^J$ that satisfy $\overline{X\cdot h}=\bar{X}\cdot \bar{h}$ and $\overline{m_x}=m_x+{\sum }_{y<x}{a}_{y,x}{m}_y,\overline{n_x}=n_x+{\sum }_{y<x}{b}_{y,x}{n}_y$ with $a_{y,x},b_{y,x}\in A$ and top coefficient $a_{x,x}=b_{x,x}=1$.

(4) (Parabolic property.) If $x\in {}^{J}W$ and $s\in S$ with $xs\mathrm{\notin }{}^{J}W$, then $xs=rx$ for a unique simple $r\in J$ and $xs>x$.

The independence of $\{m_x\}$ uses the free left $H_J$-basis $\{H_y\}$ of $H_v(n)$ and not merely the spanning property; a unit coefficient alone would not prove independence. The top coefficient one in (3) is the coefficient of $H_x$ in the expansion of the inverse generators; it is not a claim that all lower coefficients lie in $v\mathbb{Z}[v]$. The bar is multiplicative; the balancing check in step 2.3 is exactly where the multiplicative convention (rather than an anti-automorphism convention) is used.

Retain the notation of The parabolic Hecke module and the antispherical standard basis and Standard bases, right actions and bar involutions on the parabolic Hecke modules. For every $x\in {}^{J}W$ there is a unique $C_x\in M^J$ and a unique $D_x\in N^J$ with (i) $\overline{C_x}=C_x$ and $\overline{D_x}=D_x$, and (ii) $C_x\in m_x+{\sum }_{y<x}v\mathbb{Z}[v]m_y$ and $D_x\in n_x+{\sum }_{y<x}v\mathbb{Z}[v]n_y$, the sums over the lower Bruhat ideal of $x$ inside ${}^{J}W$. The sets $\{C_x{\}}_{x\in {}^{J}W}$ and $\{D_x{\}}_{x\in {}^{J}W}$ are $A$-bases of $M^J$ and $N^J$. No coefficientwise positivity is asserted.

The construction is the ordinary Kazhdan--Lusztig triangular induction run over the finite poset $({}^{J}W,\le )$ with the induced bar-duality matrix; the only inputs are the bar involution, the top coefficient one of the induced bar, and the lower-triangular support. Only the listed facts, hence no coefficientwise positivity, degree or parity statement, enter; those are not asserted here.

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.

Retain the notation of The parabolic Kazhdan–Lusztig bases and polynomials.

(1) (Antispherical alternating sum.) For all $x,y\in {}^{J}W$, $n_{y,x}={\sum }_{z\in W_J}(-1{)}^{\mathrm{\ell }(z)}{v}^{\mathrm{\ell }(z)}{p}_{zy,x},$ the sum over all $z\in W_J$ (the product $zy$ taken in $S_n$, and $p_{\cdot ,x}$ the ordinary coefficient of the Kazhdan--Lusztig basis of Existence and uniqueness of the Kazhdan–Lusztig basis, so the inner coefficients vanish unless $zy\le x$).

(2) (Spherical reduction.) If $W_J$ is finite with longest element $w_0^J$ then for all $x,y\in {}^{J}W$, $m_{y,x}=p_{w_0^{J}y,w_0^{J}x}\mathrm{.}$

In particular both families are computable from the ordinary Kazhdan--Lusztig coefficients of $S_n$, with no separate recursion.