We study Ruan's \textit{cohomological crepant resolution conjecture} for
orbifolds with transversal ADE singularities. In the An​-case we compute both
the Chen-Ruan cohomology ring HCR∗​([Y]) and the quantum corrected
cohomology ring H∗(Z)(q1​,...,qn​). The former is achieved in general, the
later up to some additional, technical assumptions. We construct an explicit
isomorphism between HCR∗​([Y]) and H∗(Z)(−1) in the A1​-case,
verifying Ruan's conjecture. In the An​-case, the family
H∗(Z)(q1​,...,qn​) is not defined for q1​=...=qn​=−1. This implies that
the conjecture should be slightly modified. We propose a new conjecture in the
An​-case which we prove in the A2​-case by constructing an explicit
isomorphism.Comment: This is a short version of my Ph.D. Thesis math.AG/0510528. Version
2: chapters 2,3,4 and 5 has been rewritten using the language of groupoids; a
link with the classical McKay correpondence is given. International Journal
of Mathematics (to appear