 Iwahori-Hecke algebra* is the $\bbZ[\bfq]$-algebra $\cH_0(\bfq)$ defined by generators $$\cH_0(\bfq):=(\bbZ\varepsilon_0\times\bbZ\varepsilon_0)\otimes_{\bbZ}'\bigoplus_{w\in W} \bbZ[\bfq] T_w,$$ where $\otimes