Lusztig sheaves, decomposition rule and restriction rule

In this article, we realize the subquotient based modules of certain tensor products or restricted modules via Lusztig’s perverse sheaves on multi-framed quivers, and provide a construction of their canonical bases. As an application, we prove that the decomposition and restriction coefficients of symmetric Kac-Moody algebras equal to the dimensions of top Borel-Moore homology groups for certain locally closed subsets of Nakajima’s quiver varieties.