Subsection 3.1. (0PEX) · ScalingStacks

3.1. Differential graded nil Hecke algebras

We discuss here the case of general Coxeter groups. The results will be used only for types $A_{n}$ and $\tilde{A}_{n}$ .

3.1.1. Coxeter groups

We refer to [Hu, §5 and §7.1–7.3] for basic properties of Coxeter groups and Hecke algebras. Recall that a Coxeter group $(W,S)$ is the data of a group $W$ with a subset $S\subset W$ such that $W$ has a presentation with generating set $S$ and relations

s^{2}=1,\ \underbrace{sts\cdots}_{m_{st}\text{ terms}}=\underbrace{tst\cdots}_{m_{st}\text{ terms}}\text{ when }st\text{ has order }m_{st}\ \text{ for }s,t\in S.

A reduced expression of an element $w\in W$ is a decomposition $w=s_{i_{1}}\cdots s_{i_{l}}$ such that $s_{i_{r}}\in S$ for $r=1,\ldots,l$ and such that $l$ is minimal with this property. The integer $l$ is the length $\ell(w)$ of $w$ .

The Chevalley-Bruhat (partial) order on $W$ is defined as follows. Let $w^{\prime},w\in W$ and let $w=s_{i_{1}}\cdots s_{i_{l}}$ be a reduced decomposition. We say that $w^{\prime}\leq w$ if there is $l^{\prime}\leq l$ and an increasing injection $f:\{1,\ldots,l^{\prime}\}\to\{1,\ldots,l\}$ such that $w^{\prime}=s_{i_{f(1)}}\cdots s_{i_{f(l^{\prime})}}$ . This is independent of the choice of the reduced decomposition of $w$ .

3.1.2. Hecke algebras

Let $R={\mathbf{Z}}[\{a_{s},b_{s}\}_{s\in S}]$ where $a_{s}$ and $b_{s}$ are indeterminates with $a_{s}=a_{s^{\prime}}$ and $b_{s}=b_{s^{\prime}}$ if $s$ and $s^{\prime}$ are conjugate in $W$ .

The Hecke algebra $H=H(W)$ of $(W,S)$ is the $R$ -algebra generated by $\{T_{s}\}_{s\in S}$ with relations

T_{s}^{2}+a_{s}T_{s}+b_{s}=0,\ \underbrace{T_{s}T_{t}T_{s}\cdots}_{m_{st}\text{ terms}}=\underbrace{T_{t}T_{s}T_{t}\cdots}_{m_{st}\text{ terms}}\text{ when }st\text{ has order }m_{st}.

Given a reduced decomposition $w=s_{i_{1}}\cdots s_{i_{l}}$ , we put $T_{w}=T_{s_{i_{1}}}\cdots T_{s_{i_{l}}}$ . This element is independent of the choice of the reduced decomposition of $w$ . The set $\{T_{w}\}_{w\in W}$ is a basis of $H$ .

Let $\iota:H\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}H^{\operatorname{opp}\nolimits}$ be the algebra automorphism defined by $T_{s}\mapsto T_{s}$ for $s\in S$ .

Let $I$ be a subset of $S$ . We denote by $W_{I}$ the subgroup of $W$ generated by $I$ . The group $W_{I}$ , together with $I$ , is a Coxeter group and the length function on $W_{I}$ is the restriction of that on $W$ [Hu, §1.10].

We put $R_{I}={\mathbf{Z}}[\{a_{s,I},b_{s,I}\}_{s\in I}]$ where $a_{s,I}$ and $b_{s,I}$ are indeterminates with $a_{s,I}=a_{s^{\prime},I}$ and $b_{s,I}=b_{s^{\prime},I}$ if $s$ and $s^{\prime}$ are conjugate in $W_{I}$ . There is a morphism of rings $R_{I}\to R,\ a_{s,I}\mapsto a_{s},\ b_{s,I}\mapsto b_{s}$ .

We denote by $H_{I}=H_{I}(W)$ the $R$ -subalgebra of $H$ generated by $\{T_{s}\}_{s\in I}$ . There is an isomorphism of $R$ -algebras $R\otimes_{R_{I}}H(W_{I})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}H_{I}(W),\ T_{w}\mapsto T_{w}$ .

We assume for the remainder of §3.1.2 that $W$ is finite. In this case, there is a unique element $w_{S}$ of $W$ with maximal length [Hu, §1.8] and we denote by $N$ its length. We have $w_{S}^{2}=1$ and $w_{S}Sw_{S}=S$ . There is an automorphism of algebras

\iota_{S}:H\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}H,\ T_{v}\mapsto T_{w_{S}\cdot v\cdot w_{S}}.

We denote by $w_{I}$ the longest element of $W_{I}$ and by $N_{I}$ its length. We denote by $W^{I}$ (resp. ${{}^{I}W}$ ) the set of elements $v\in W$ such that $v$ has minimal length in $vW_{I}$ (resp. $W_{I}v$ ). Note that $W^{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}W/W_{I},\ v\mapsto vW_{I}$ [Hu, Proposition 1.10].

3.1.3. Traces

We assume in §3.1.3 that $W$ is finite.

Given $J\subset I$ , we define an $R$ -linear map

t_{I,J}:H_{I}\to H_{J},\ T_{v}\mapsto\begin{cases}T_{w_{J}w_{I}v}&\text{ if }v\in w_{I}\cdot W_{J}\\ 0&\text{ otherwise.}\end{cases}

The next proposition shows this is relative Frobenius form (cf eg [Rou1, §2.3.2]).

0P4L

Proposition 3.1.1. We have $t_{S,J}=t_{I,J}\circ t_{S,I}$ .

Given $h\in H$ and $x\in W_{I}$ , we have

t_{S,I}(hT_{x})=t_{S,I}(h)T_{x},\ t_{S,I}(T_{w_{S}w_{I}\cdot x\cdot w_{I}w_{S}}h)=T_{x}t_{S,I}(h).

Given $h^{\prime}\in H$ commuting with $H_{I}$ , we have $t_{S,I}(hh^{\prime})=t_{S,I}(\iota_{S}(h^{\prime})h)$ .

There is an isomorphism of $R$ -modules

\hat{t}_{S,I}:H\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{H_{I}^{\operatorname{opp}\nolimits}}(H,H_{I}),\ h\mapsto(h^{\prime}\mapsto t_{S,I}(hh^{\prime}))

with

\hat{t}_{S,I}(T_{w_{S}w_{I}\cdot x\cdot w_{I}w_{S}}hT_{y})=T_{x}\hat{t}_{S,I}(h)T_{y}\text{ for }x\in W_{I}\text{ and }y\in W.

0P4M

Proof. Define $w^{I}=w_{S}w_{I}$ and ${{}^{I}w}=w_{I}w_{S}$ , so that ${{}^{I}w}\cdot w^{I}=1$ . We have $w^{I}\in W^{I}$ .

Let $v\in W$ . There is a unique decomposition $v=v^{\prime}v^{\prime\prime}$ where $\ell(v)=\ell(v^{\prime})+\ell(v^{\prime\prime})$ , $v^{\prime\prime}\in W_{I}$ and $v^{\prime}\in W^{I}$ [Hu, Proposition 1.10]. Furthermore, $\ell(v^{\prime})<\ell(w^{I})$ unless $v^{\prime}=w^{I}$ . We have $t_{S,I}(T_{v})=\delta_{v^{\prime},w^{I}}T_{v^{\prime\prime}}$ .

There is a unique decomposition $v^{\prime\prime}=v_{1}v_{2}$ with $\ell(v^{\prime\prime})=\ell(v_{1})+\ell(v_{2})$ , $v_{2}\in W_{J}$ and $v_{1}$ has minimal length in $v^{\prime\prime}W_{J}$ . We have $v=(v^{\prime}v_{1})v_{2}$ where $\ell(v)=\ell(v^{\prime}v_{1})+\ell(v_{2})$ and $v^{\prime}v_{1}$ has minimal length in $vW_{J}$ . Furthermore, $v^{\prime}v_{1}=w^{J}$ if and only if $v^{\prime}=w^{I}$ and $v_{1}=w_{I}w_{J}$ . It follows that

t_{I,J}\circ t_{S,I}(T_{v})=\delta_{v^{\prime},w^{I}}t_{I,J}(T_{v^{\prime\prime}})=\delta_{v^{\prime},w^{I}}\delta_{v_{1},w_{I}w_{J}}T_{v_{2}}=t_{S,J}(T_{v}).

This shows the first statement of the lemma.

We have $T_{v^{\prime\prime}}T_{x}\in H_{I}$ , hence

t_{S,I}(T_{v}T_{x})=t_{S,I}(T_{v^{\prime}}(T_{v^{\prime\prime}}T_{x}))=\delta_{v^{\prime},w^{I}}T_{v^{\prime\prime}}T_{x}=t_{S,I}(T_{v})T_{x}.

This shows the second statement of the lemma.

Let $x^{\prime}=w^{I}\cdot x\cdot{{}^{I}w}$ . We have $\ell(w^{I}\cdot x\cdot{{}^{I}w})=\ell(x)$ . Since $T_{x^{\prime}}T_{v}$ is a linear combination of elements $T_{yz}$ with $y\leq x^{\prime}$ and $z\leq v$ , it follows that if $v^{\prime}\neq w^{I}$ , then $T_{x^{\prime}}T_{v^{\prime}}$ is a linear combination of elements $T_{w^{I}\cdot y\cdot{{}^{I}w}z}$ with $y\in W_{I}$ and $z{\not\in}w^{I}W_{I}$ , hence of elements $T_{u}$ with $u{\not\in}w^{I}W_{I}$ . So, if $v^{\prime}\neq w^{I}$ , then $t_{S,I}(T_{x^{\prime}}T_{v})=0$ .

Assume now $v^{\prime}=w^{I}$ . We have $T_{x^{\prime}}T_{v}=T_{w^{I}\cdot x\cdot{{}^{I}w}}T_{w^{I}}T_{v^{\prime\prime}}=T_{w^{I}\cdot x}T_{v^{\prime\prime}}=T_{w^{I}}T_{x}T_{v^{\prime\prime}}$ because $\ell(x^{\prime}\cdot w^{I})=\ell(w^{I}\cdot x)=\ell(w^{I})+\ell(x)=\ell(x^{\prime})+\ell(w^{I})$ . We deduce that $t_{S,I}(T_{x^{\prime}}T_{v})=T_{x}T_{v^{\prime\prime}}=T_{x}t_{S,I}(T_{v})$ . This shows the third statement of the lemma.

Let $v_{0}\in W^{I}$ . We have $\ell(w^{I})=\ell(w^{I}v_{0}^{-1})+\ell(v_{0})$ . Let $v\in W^{I}$ . Note that $T_{w^{I}v_{0}^{-1}}T_{v}=T_{w^{I}v_{0}^{-1}v}$ or $T_{w^{I}v_{0}^{-1}}T_{v}$ is a linear combination of $T_{w}$ ’s with $\ell(w)<\ell(w^{I}v_{0}^{-1})+\ell(v)$ . It follows that if $t_{S,I}(T_{w^{I}v_{0}^{-1}}T_{v})=0$ if $\ell(v)<\ell(v_{0})$ or $\ell(v)=\ell(v_{0})$ and $v\neq v_{0}$ . We have also $t_{S,I}(T_{w^{I}v_{0}^{-1}}T_{v_{0}})=1$ .

Since $H$ is a free right $H_{I}$ -module with basis $\{T_{v}\}_{v\in W^{I}}$ , we deduce that $\hat{t}_{S,I}$ is surjective. Since $\hat{t}_{S,I}$ is an $R$ -module morphism between free $R$ -modules of the same finite rank, it follows that it is an isomorphism. This shows the fifth statement of the lemma.

Let $s\in S$ and $v\in W$ . Let $s^{\prime}=w_{S}\cdot s\cdot w_{S}\in S$ . If $v{\not\in}\{w_{S},w_{S}\cdot s\}$ , then $t_{S,\emptyset}(T_{v}T_{s})=0$ and $t_{S,\emptyset}(T_{s^{\prime}}T_{v})=0$ . If $v=w_{S}\cdot s$ , then $T_{v}T_{s}=T_{w_{S}}=T_{s^{\prime}}T_{v}$ . If $v=w_{S}$ , then $t_{S,\emptyset}(T_{v}T_{s})=a_{s}=t_{S,\emptyset}(T_{s^{\prime}}T_{v})$ . So, we have shown that $t_{S,\emptyset}(T_{v}T_{s})=t_{S,\emptyset}(T_{s^{\prime}}T_{v})$ . It follows by induction on $\ell(w)$ that $t_{S,\emptyset}(T_{v}T_{w})=t_{S,\emptyset}(T_{w_{S}\cdot w\cdot w_{S}}T_{v})$ for all $w\in W$ .

Consider now $h^{\prime}\in H$ commuting with $H_{I}$ . Let $h^{\prime\prime}\in H_{I}$ . We have

t_{I,\emptyset}(t_{S,I}(hh^{\prime})h^{\prime\prime})=t_{I,\emptyset}(t_{S,I}(hh^{\prime}h^{\prime\prime}))=t_{S,\emptyset}(hh^{\prime\prime}h^{\prime})=t_{S,\emptyset}(\iota_{S}(h^{\prime})hh^{\prime\prime})=\\ =t_{I,\emptyset}(t_{S,I}(\iota_{S}(h^{\prime})hh^{\prime\prime}))=t_{I,\emptyset}(t_{S,I}(\iota_{S}(h^{\prime})h)h^{\prime\prime}).

It follows that $\hat{t}_{I,\emptyset}(t_{S,I}(hh^{\prime}))=\hat{t}_{I,\emptyset}(t_{S,I}(\iota_{S}(h^{\prime})h))$ , hence $t_{S,I}(hh^{\prime})=t_{S,I}(\iota_{S}(h^{\prime})h)$ . This completes the proof of the lemma. ∎

We put $t^{+}_{I,J}=t_{I,J}$ . We define an $R$ -linear map

t^{-}_{I,J}:H_{I}\to H_{J},\ T_{v}\mapsto\begin{cases}T_{vw_{I}w_{J}}&\text{ if }v\in W_{J}\cdot w_{I}\\ 0&\text{ otherwise.}\end{cases}

We have $t^{-}_{I,J}(h)=\iota(t^{+}_{I,J}(\iota(h)))$ .

We put $\hat{t}^{+}_{S,I}=\hat{t}_{S,I}$ . We have an isomorphism of $R$ -modules

\hat{t}^{-}_{S,I}:H\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{H_{I}^{\operatorname{opp}\nolimits}}(H,H_{I}),\ h\mapsto(h^{\prime}\mapsto t^{-}_{S,I}(hh^{\prime}))

with

\hat{t}_{S,I}^{-}(T_{x}hT_{y})=T_{x}\hat{t}_{S,I}^{-}(h)T_{y}\text{ for }x\in W_{I}\text{ and }y\in W.

Consider $I,J\subset S$ with $I\subset J$ or $J\subset I$ . We define an $(H_{I},H_{J})$ -bimodule $L^{\pm}(I,J)$ with underlying $R$ -module $H$ . We put $a=0$ if $\pm=+$ and $a=1$ if $\pm=-$ .

If $I\subset J$ , then the right action of $H_{J}$ is by right multiplication and the left action of $h\in H_{I}$ is by left multiplication by $(\iota_{J}\iota_{I})^{a}(h)$ .

If $J\subset I$ , then the left action of $H_{I}$ is by left multiplication and the right action of $h\in H_{J}$ is by right multiplication by $(\iota_{I}\iota_{J})^{a}(h)$ .

Note that $L^{\pm}(I,J)$ is free of finite rank as a left module and as a right module.

There is an isomorphism of $(H,H_{I})$ -bimodules

L^{\pm}(I,S)^{\vee}=\operatorname{Hom}\nolimits_{H^{\operatorname{opp}\nolimits}}(L^{\pm}(I,S),H)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\pm}(S,I),\ \zeta\mapsto\zeta(1).

The next result follows immediately from Proposition 3.1.1.

0P4N

Corollary 3.1.2. The map $\hat{t}^{\pm}_{S,I}$ is an isomorphism of $(H_{I},H)$ -bimodules

L^{\mp}(I,S)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\pm}(S,I)^{\vee}=\operatorname{Hom}\nolimits_{H_{I}^{\operatorname{opp}\nolimits}}(L^{\pm}(S,I),H_{I}).

The results above can be formulated in terms of dual bases. Note that $\{T_{w}\}_{w\in W^{I}}$ is a basis of the free right $H_{I}$ -module $H$ , while $\{T_{w}\}_{w\in{{}^{I}W}}$ is a basis of the free left $H_{I}$ -module $H$ .

We have

t_{S,I}^{+}(T_{w_{S}w_{I}v^{-1}}T_{w})=\delta_{v,w}\text{ and }t_{S,I}^{-}(T_{v^{\prime}}T_{w^{\prime-1}w_{I}w_{S}})=\delta_{v^{\prime},w^{\prime}}\text{ for }v,w\in W^{I}\text{ and }v^{\prime},w^{\prime}\in{{}^{I}W}.

We deduce that the basis $(T_{w_{S}w_{I}w^{-1}})_{w\in W^{I}}$ when $\pm=+$ (resp. $(T_{w})_{w\in{{}^{I}W}}$ when $\pm=-$ ) of the free left $H_{I}$ -module $L^{\mp}(I,S)$ is dual to the basis $(T_{w})_{w\in W^{I}}$ when $\pm=+$ (resp. $(T_{w^{-1}w_{I}w_{S}})_{w\in{{}^{I}W}}$ when $\pm=-$ ) of the free right $H_{I}$ -module $L^{\pm}(S,I)$ , via the pairing providing the isomorphism of Corollary 3.1.2.

The counit of the adjoint pair $(L^{\mp}(I,S)\otimes_{H}-,L^{\pm}(S,I)\otimes_{H_{I}}-)$ is given by the morphism of $(H_{I},H_{I})$ -bimodules

L^{\mp}(I,S)\otimes_{H}L^{\pm}(S,I)\to H_{I},\ a\otimes b\mapsto t^{\pm}_{S,I}(ab)

while the unit is given by the morphism of $(H,H)$ -bimodules

H\to L^{\pm}(S,I)\otimes_{H_{I}}L^{\mp}(I,S),\ 1\mapsto\begin{cases}\sum_{w\in W^{I}}T_{w}\otimes T_{w_{S}w_{I}w^{-1}}&\text{ if }\pm=+\\ \sum_{w\in{{}^{I}W}}T_{w^{-1}w_{I}w_{S}}\otimes T_{w}&\text{ if }\pm=-.\end{cases}

3.1.4. Nil Hecke algebras

We define the nil Hecke algebra $H_{\mathbf{Z}}^{\mathrm{nil}}(W)$ of $(W,S)$ as the ${\mathbf{Z}}$ -algebra $H(W)\otimes_{R}R/(a_{s},b_{s})_{s\in S}$ . This is the ${\mathbf{Z}}$ -algebra generated by $\{T_{s}\}_{s\in S}$ with relations

T_{s}^{2}=0,\ \underbrace{T_{s}T_{t}T_{s}\cdots}_{m_{st}\text{ terms}}=\underbrace{T_{t}T_{s}T_{t}\cdots}_{m_{st}\text{ terms}}\text{ when }st\text{ has order }m_{st}.

This is a ${\mathbf{Z}}_{\leq 0}$ -graded algebra with $T_{w}$ in degree $-\ell(w)$ for $w\in W$ .

The multiplication is given as follows:

(3.1.1)

T_{w}T_{w^{\prime}}=\begin{cases}T_{ww^{\prime}}&\text{ if }\ell(ww^{\prime})=\ell(w)+\ell(w^{\prime})\\ 0&\text{ otherwise.}\end{cases}

Consider the filtration of the group algebra ${\mathbf{Z}}[W]$ where ${\mathbf{Z}}[W]^{\geq-i}$ is spanned by group elements $w\in W$ with $\ell(w)\leq i$ , for $i\in{\mathbf{Z}}_{\geq 0}$ . The associated ${\mathbf{Z}}_{\leq 0}$ -graded algebra is $H_{\mathbf{Z}}^{\mathrm{nil}}(W)$ and $T_{w}$ is the image of $w\in W$ in the degree $-\ell(w)$ homogeneous component of $H_{\mathbf{Z}}^{\mathrm{nil}}(W)$ .

3.1.5. Differential

Let $H^{\mathrm{nil}}(W)={\mathbf{F}}_{2}\otimes H_{\mathbf{Z}}^{\mathrm{nil}}(W)$ . We define a linear map $d:H^{\mathrm{nil}}(W)\to H^{\mathrm{nil}}(W)$ by

d(T_{w})=\sum_{w^{\prime}<w,\ \ell(w^{\prime})=\ell(w)-1}T_{w^{\prime}}.

0P4P

Proposition 3.1.3. The map $d$ defines a structure of differential graded algebra on $H^{\mathrm{nil}}(W)$ .

0P4Q

Proof. Let $w\in W$ and $s\in S$ with $ws>w$ . We have $d(T_{w}T_{s})=d(T_{ws})=\sum_{w^{\prime}<ws,\ \ell(w^{\prime})=\ell(w)}T_{w^{\prime}}$ . We have [Hu, Theorem 5.10]

\{w^{\prime}\in W\ |w^{\prime}<ws,\ \ell(w^{\prime})=\ell(w)\}=\{w^{\prime\prime}s\ |\ w^{\prime\prime}<w,\ w^{\prime\prime}<w^{\prime\prime}s,\ \ell(w^{\prime\prime})=\ell(w)-1\}\sqcup\{w\}.

It follows that $d(T_{w}T_{s})=d(T_{w})T_{s}+T_{w}=d(T_{w})T_{s}+T_{w}d(T_{s})$ .

Consider now $v\in W$ and $s\in S$ with $vs<v$ . We have $d(T_{v})=d(T_{vs}T_{s})=d(T_{vs})T_{s}+T_{vs}$ by the result above. It follows that $d(T_{v})T_{s}+T_{v}d(T_{s})=T_{vs}T_{s}+T_{v}=0=d(T_{v}T_{s})$ .

We deduce that $d(T_{w}T_{w^{\prime}})=d(T_{w})T_{w^{\prime}}+T_{w}d(T_{w^{\prime}})$ for all $w,w^{\prime}\in W$ .

Since $d^{2}(T_{s})=0$ for $s\in S$ , it follows that by induction that $d^{2}=0$ . ∎

The following corollary shows that the computation of $d(T_{w})$ can be done using the Leibniz rule, given a reduced decomposition of $w$ . The terms that do not vanish are exactly the terms given in the original definition of $d(T_{w})$ .

0P4R

Corollary 3.1.4. Let $w=s_{i_{1}}\cdots s_{i_{l}}$ be a reduced expression of $w\in W$ . We have

d(T_{w})=\sum_{r=1}^{l}T_{i_{1}}\cdots T_{i_{r-1}}T_{i_{r+1}}T_{i_{l}}.

We have $T_{i_{1}}\cdots T_{i_{r-1}}T_{i_{r+1}}T_{i_{l}}\neq 0$ if and only if $s_{i_{1}}\cdots s_{i_{r-1}}s_{i_{r+1}}\cdots s_{i_{l}}$ is reduced, i.e., if and only if $\ell(s_{i_{1}}\cdots s_{i_{r-1}}s_{i_{r+1}}\cdots s_{i_{l}})=\ell(w)-1$ .

Given $r,r^{\prime}$ with $s_{i_{1}}\cdots s_{i_{r-1}}s_{i_{r+1}}\cdots s_{i_{l}}=s_{i_{1}}\cdots s_{i_{r^{\prime}-1}}s_{i_{r^{\prime}+1}}\cdots s_{i_{l}}$ reduced, we have $r=r^{\prime}$ .

0P4S

Proof. The first statement follows from Proposition 3.1.3. The second statement is a property of the multiplication of $T_{w}$ ’s.

For the third statement, let us assume $r<r^{\prime}$ . We have $s_{i_{r+1}}\cdots s_{i_{r^{\prime}}}=s_{i_{r}}\cdots s_{i_{r^{\prime}-1}}$ reduced, hence $s_{i_{r}}s_{i_{r+1}}\cdots s_{i_{r^{\prime}}}$ is not reduced, a contradiction. ∎

0P4T

Remark 3.1.5. Note that the algebra $H^{\mathrm{nil}}(W)$ is acyclic if $S\neq\emptyset$ .

Note also that one can introduce a family of commuting differentials $d_{s}$ for $s\in S$ modulo conjugacy by setting $d_{s}(T_{t})=1$ if $t\in S$ is conjugate to $s$ and $d_{s}(T_{t})=0$ otherwise.

The specialization over ${\mathbf{F}}_{2}$ at $a_{s}=b_{s}=0$ of the bimodules $L^{\pm}(I,J)$ of §3.1.3 acquire a structure of differential graded bimodules, using the differential graded structure of $H^{\mathrm{nil}}(W)$ . We keep the same notation for those differential graded specialized bimodules and for the maps $t$ and $\hat{t}$ .

0P4U

Proposition 3.1.6. If $W$ is finite, then

t_{S,I}:H^{\mathrm{nil}}(W)\to H^{\mathrm{nil}}(W_{I})\langle N-N_{I}\rangle

is a morphism of differential graded ${\mathbf{F}}_{2}$ -modules and Corollary 3.1.2 provides an isomorphism of differential graded $(H^{\mathrm{nil}}(W_{I}),H^{\mathrm{nil}}(W))$ -bimodules

\hat{t}_{S,I}^{\pm}:L^{\mp}(I,S)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\pm}(S,I)^{\vee}\langle N-N_{I}\rangle.

0P4V

Proof. Let $v\in W$ . There is a unique decomposition $v=v^{\prime}v^{\prime\prime}$ where $\ell(v)=\ell(v^{\prime})+\ell(v^{\prime\prime})$ , $v^{\prime\prime}\in W_{I}$ and $v^{\prime}\in W^{I}$ .

We have $d(T_{v})=d(T_{v^{\prime}})T_{v^{\prime\prime}}+T_{v^{\prime}}d(T_{v^{\prime\prime}})$ . If $u\in W$ and $u<v^{\prime}$ , then $u{\not\in}w_{S}W_{I}$ . It follows that

t_{S,I}(d(T_{v}))=t_{S,I}(T_{v^{\prime}}d(T_{v^{\prime\prime}}))=\delta_{v^{\prime},w^{I}}d(T_{v^{\prime\prime}})=d(t_{S,I}(T_{v})).

∎

3.1.6. Differential graded pointed Hecke monoid

Let $W^{\operatorname{nil}\nolimits}$ be the pointed ${\mathbf{Z}}_{\leq 0}$ -graded monoid with underlying pointed set $\{T_{w}\}_{w\in W}\coprod\{0\}$ and multiplication given by (3.1.1). This is the pointed monoid ${\operatorname{gr}\nolimits}W$ associated to the filtration on $W$ given by $W^{\geq-i}=\{w\in W\ |\ \ell(w)\leq i\}$ and there is an identification ${\mathbf{F}}_{2}[W^{\operatorname{nil}\nolimits}]=H^{\operatorname{nil}\nolimits}(W)$ making $W^{\operatorname{nil}\nolimits}$ into a differential graded pointed monoid.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2