Sur les quotients discrets de semi-groupes complexes

Let $X=G\mathrm{/}K$ be an irreducible Hermitian symmetric space of the non-compact type and let $S\in G^{\text{mbb}}C$ be the associated compression semi-group. Let $\mathrm{\Gamma }$ be a discrete subgroup of $G$. We give a sufficient condition for $\mathrm{\Gamma }\mathrm{\backslash }S$ to be a Stein manifold. Moreover, we show that in general $\mathrm{\Gamma }\mathrm{\backslash }S$ is not Stein, which disproves a conjecture by Achab, Betten and Kr"otz.