Semisimplicity of parabolic Hecke algebras (Q1265564)

Let \((W,S)\) be a finite Coxeter system and let \(R:=\mathbb{Z}[r_s;\;s\in S]\) be the polynomial ring on the set \(\{r_s;\;s\in S\}\) of indeterminates chosen so that \(r_s=r_t\Leftrightarrow s\) and \(t\) are conjugate in \(W\) and \(q_s=r^2_s\). The Hecke algebra \(H(S,R)\) is an \(R\)-algebra with basis \(\{T_w;\;w\in W\}\) and multiplication given by \[ T_sT_w=\begin{cases} T_{sw}\quad &\text{if }\ell(sw)>\ell(w)\\ q_sT_{sw}+(q_s-1)T_w\quad &\text{if }\ell(sw)<\ell(w)\end{cases} \] for \(s\in S\); \(w\in W\), where \(\ell\) is the length function on \((W,S)\). The algebra \(H(S,R)\) is called Iwahori-Hecke algebra associated with \((W,S)\). For \(J\subseteq S\), the parabolic Hecke algebra \(H(S,J;R)\) is a subalgebra of \(H(S;R)\) defined as \(H(S,J;R):=\{a\in H(S;R):T_sa=aT_s=q_sa;\;\forall s\in J\}\). This paper determines the semisimplicity of the parabolic Hecke algebra \(H(S,J;R)\). It is an extension of an earlier paper by the author [J. Algebra 183, No. 2, 514-544 (1996; Zbl 0856.20009)] in which he studied the semisimplicity of parabolic Hecke algebras when they have only one parameter \(q\).