The correspondence between the canonical and semicanonical bases

Given any symmetric Cartan datum, Lusztig has provided a pair of inductive key lemmas to construct the perverse sheaves over the corresponding quiver and the functions of irreducible components over the corresponding preprojective algebra respectively. In the present article, we prove that these two inductive algorithms of Lusztig coincide. Consequently we can define two Z 2 × I \mathbb {Z}_{2} \times I -colored graphs and prove that they are isomorhic. This result finishes the proof of the statement that Lusztig’s functions f Z f_{Z} of irreducible components form a basis of the enveloping algebra. We can also deduces its crystal structure from those Lusztig’s functions. As an application, we prove that the transition matrix between the canonical basis (at v = 1 v=1 ) and the semicanonical basis is upper triangular with diagonal entries 1 1 .