The parabolic Hecke module and the antispherical standard basis

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)).