Subsection 3.2.5. (0PF9) · ScalingStacks

3.2.5. Extended affine Hecke algebra

We let $c$ act on the differential graded algebra $H^{\mathrm{nil}}(W_{n})$ by $c(T_{a})=T_{a+1}$ . Let $\hat{H}_{n}=H^{\mathrm{nil}}(W_{n})\rtimes\langle c\rangle$ . For $n\geq 2$ , it is the differential graded ${\mathbf{F}}_{2}$ -algebra generated by $\{T_{a}\}_{a\in{\mathbf{Z}}/n}$ and $c^{\pm 1}$ with relations

T_{a}^{2}=0,\ cT_{a}=T_{a+1}c,\ T_{a}T_{b}=T_{b}T_{a}\text{ if }a\neq b\pm 1

T_{a}T_{a+1}T_{a}=T_{a+1}T_{a}T_{a+1}\ (\text{ for }n>2\ )

and differential $d(T_{a})=1$ , $d(c)=0$ . The element $c$ has degree $0$ , while $T_{a}$ has degree $-1$ . Note that $\hat{H}_{1}={\mathbf{F}}_{2}[\hat{{\mathfrak{S}}}_{1}]={\mathbf{F}}_{2}\langle c\rangle$ , a differential graded algebra in degree $0$ with $d=0$ .

Let $w\in W_{n}$ , $d\in{\mathbf{Z}}$ and $w^{\prime}=wc^{d}$ . We put $T_{w^{\prime}}=T_{w}c^{d}$ . We also put $T_{\sigma}=T_{w}c^{d}$ for $\sigma=wc^{d}$ . The set $\{T_{\sigma}\}_{\sigma\in\hat{{\mathfrak{S}}}_{n}}$ is a basis of $\hat{H}_{n}$ .

0P55

Remark 3.2.6. Define a filtration on ${\mathbf{F}}_{2}[\hat{{\mathfrak{S}}}_{n}]$ with $({\mathbf{F}}_{2}[\hat{{\mathfrak{S}}}_{n}])^{\geq-i}$ the subspace spanned by group elements $w\in\hat{{\mathfrak{S}}}_{n}$ with $\ell(w)\leq i$ . The associated graded algebra is $\hat{H}_{n}$ .

We put $\hat{H}_{0}={\mathbf{F}}_{2}$ .

0P56

Remark 3.2.7. The group $\hat{{\mathfrak{S}}}_{n}$ is more classically described as a semi-direct product ${\mathbf{Z}}^{n}\rtimes{\mathfrak{S}}_{n}$ (cf §3.2.2) coming from its description as the extended affine Weyl group of $\operatorname{GL}\nolimits_{n}$ . The nil affine Hecke algebra of $\operatorname{GL}\nolimits_{n}$ associated with this description (cf e.g. [Rou2, §2.2.2]) is not isomorphic to $\hat{H}_{n}$ . When considering invertible (instead of $0$ ) parameters, the two algebras are isomorphic.

0P57

Example 3.2.8. An element $T_{\sigma}$ of $\hat{H}_{n}$ will be representated by a good strand diagram for $\sigma$ . The multiplication of $T_{\sigma}$ and $T_{\sigma^{\prime}}$ is obtained by concatenating the diagrams of $\sigma$ and $\sigma^{\prime}$ (as in the multiplication of $\sigma$ and $\sigma^{\prime}$ ). If the corresponding diagram is good, then $T_{\sigma}T_{\sigma^{\prime}}=T_{\sigma^{\prime\prime}}$ , where $\sigma^{\prime\prime}$ is represented by the concatenated diagram. Otherwise, $T_{\sigma}T_{\sigma^{\prime}}=0$ . For example: