Standard bases, right actions and bar involutions on the parabolic Hecke modules

The bases. The $A$-span of $\{H_u:u\in W_J\}$ is exactly $H_J$: it contains the generators and is closed under their left and right multiplication by the quadratic multiplication rule, while every $H_u$ is a reduced product of generators in $J$. By [F1], the standard basis partitions as $\{H_u{H}_y:u\in W_J,y\in {}^{J}W\}$. Hence multiplication is an isomorphism of left $H_J$-modules ${⨁}_{y\in {}^{J}W}{H}_J\to H_v(n)$, $(h_y{)}_y\mapsto {\sum }_y{h}_y{H}_y$: expanding each $h_y$ in the independent $H_u$ proves injectivity and the unique factorization proves surjectivity. Tensoring this isomorphism with either character module $A$ gives $A{\otimes }_{H_J}{H}_v(n)\cong {⨁}_{y\in {}^{J}W}A$. The images of the coordinate generators are $m_y$, respectively $n_y$, proving both bases.

The two minimal branches. Let $x\in {}^{J}W$, $s\in S$ and suppose $xs\in {}^{J}W$. Since $xs=(xs{x}^{-1})x$ is the left translate of $x$ by a transposition and $\mathrm{\ell }(xs)=\mathrm{\ell }(x)\pm 1$, [F4] shows that $xs>x$ holds exactly when $\mathrm{\ell }(xs)=\mathrm{\ell }(x)+1$, and $xs<x$ exactly when $\mathrm{\ell }(xs)=\mathrm{\ell }(x)-1$. If $xs>x$ then [F2] gives $H_x{H}_s=H_{xs}$, so by step 1.1 and the module structure $m_x{C}_s=1\otimes (H_{xs}+v{H}_x)=m_{xs}+v{m}_x$, and likewise $n_x{C}_s=n_{xs}+v{n}_x$. If $xs<x$ then $H_x{H}_s=H_{xs}+(v^{-1}-v)H_x$, so $m_x{C}_s=m_{xs}+(v^{-1}-v)m_x+v{m}_x=m_{xs}+v^{-1}{m}_x$, and likewise $n_x{C}_s=n_{xs}+v^{-1}{n}_x$.

The parabolic property. Let $x\in {}^{J}W$ and $s=s_j$, and suppose $xs\mathrm{\notin }{}^{J}W$; by [F5] $xs$ has a left descent in $J$: some $r\in J$ has $r+1$ to the left of $r$ in $xs$. Since $x\in {}^{J}W$ has no such descent, while $xs$ differs from $x$ only by the swap of the entries at positions $j$ and $j+1$, the only changed relative order is that of the values $a:=x(j)$, $b:=x(j+1)$, so $\{r,r+1\}=\{a,b\}$. The order $a=r+1$, $b=r$ is excluded because it would already be a left descent of $x$. Hence $a=r$ and $b=r+1$ with $r\in J$; the two permutations $xs$ and $s_{r}x$ coincide because both are obtained from $x$ by interchanging the values $r$ and $r+1$. Moreover $\mathrm{\ell }(s_{r}x)>\mathrm{\ell }(x)$ because $r\in J$ and $x\in {}^{J}W$, so $\mathrm{\ell }(xs)=\mathrm{\ell }(x)+1$ and $xs>x$ by F4,(b); thus $H_x{H}_s=H_{xs}=H_{s_{r}x}=H_r{H}_x$ by [F1] and [F2]. Therefore $m_x{C}_s=1\otimes (H_{xs}+v{H}_x)={\phi }_{+}(H_r)m_x+v{m}_x=(v^{-1}+v)m_x$ and $n_x{C}_s={\phi }_{-}(H_r)n_x+v{n}_x=(-v+v)n_x=0$. Uniqueness of $r$ holds because $r$ is the value whose swap produces $xs$; this proves (4) and the third branch of (2).

The bar descent. The map $\beta (a\otimes h):=\bar{a}\otimes \bar{h}$ is well defined on $A{\otimes }_{H_J}{H}_v(n)$: it suffices to check $\overline{a{\phi }_{\pm }(t)}\otimes \bar{h}=\bar{a}\otimes \overline{th}$ for $t=H_s$, $s\in J$, where the left side is $\bar{a}\overline{{\phi }_{\pm }(H_s)}\otimes \bar{h}$ and the right side is $\bar{a}\otimes \overline{H_s}\bar{h}=\bar{a}\otimes (H_s-(v^{-1}-v))\bar{h}=\bar{a}{\phi }_{\pm }(H_s-(v^{-1}-v))\otimes \bar{h}$, and ${\phi }_{+}(H_s-(v^{-1}-v))=v^{-1}+(v-v^{-1})=v=\overline{v^{-1}}$, ${\phi }_{-}(H_s-(v^{-1}-v))=-v+(v-v^{-1})=-v^{-1}=\overline{-v}$ by [F3]; multiplicativity extends this to all $t$. The map is $A$-antilinear, involutive (${\beta }^2=\mathrm{i}\mathrm{d}$ because bar is an involution on $A$ and on $H_v(n)$), and satisfies $\beta (X\cdot h^{\mathrm{\prime }})=\beta (X)\cdot \overline{h^{\mathrm{\prime }}}$ because $(a\otimes h)\cdot h^{\mathrm{\prime }}=a\otimes h{h}^{\mathrm{\prime }}$ maps to $\bar{a}\otimes \overline{h{h}^{\mathrm{\prime }}}=\bar{a}\otimes \bar{h}{\bar{h}}^{\mathrm{\prime }}=(\bar{a}\otimes \bar{h})\cdot {\bar{h}}^{\mathrm{\prime }}$.

Triangular expansion and top coefficient. For $x=s_{i_1}\cdots s_{i_k}$ reduced, [F3] gives $\overline{H_x}={\prod }_{m=1}^k(H_{s_{i_m}}-(v^{-1}-v))$. The term choosing every generator is $H_x$. Every other term is a scalar times a product of fewer than $k$ generators selected from this word. Successive use of [F2] expands such a product into $H_z$ whose group labels are products of subwords of the selected word: each multiplication retains the old label or appends the current generator. Deletion of pairs in [F4] reduces each such group word to a reduced subword of the original reduced expression for $x$. Thus $z\le x$, and $\mathrm{\ell }(z)<k$ because at most $k-1$ letters were retained, so $z<x$. Consequently $\overline{H_x}=H_x+{\sum }_{z<x}{c}_z{H}_z$. For each $z<x$ factor $z=uy$ length-additively with $u\in W_J$, $y\in {}^{J}W$. A reduced expression of $z$ contains a reduced suffix for $y$, so $y\le z<x$. In either module $1\otimes H_z={\phi }_{\pm }(H_u)(1\otimes H_y)$. Applying the bar from step 2.3 therefore gives exactly the triangular expansions in (3), with top coefficient one and all lower coefficients in $A$.

Parts (1), (2), (3) and (4) are proved in steps 1.1, 2.1 together with 2.2, 2.3 together with 3.1, and 2.2 respectively.