ScalingStacks

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.