Generic Hecke algebra for Renner monoids

We associate with every Renner monoid~$R$ a \emph{generic Hecke algebra}~$\H(R)$ over~$\mathbb{Z}[q]$ which is a deformation of the monoid $\mathbb{Z}$-algebra of $R$. If $M$ is a finite reductive monoid with Borel subgroup~$B$ and associated Renner monoid~$R$, then we obtain the associated Iwahori-Hecke algebra~$\H(M,B)$ by specialising~$q$ in~$\H(R)$ and tensoring by~$\mathbb{C}$ over~$\mathbb{Z}$, as in the classical case of finite algebraic groups. This answers positively to a long-standing question of L.~Solomon