The problem of classification of perverse sheaves on the quotient h/W for a semisimple Lie algebra g has an explicit answer which turns out to be related to the algebraic properties of induction and restriction operations for parabolic subalgebras in g. For g=gl(n) we get a relation between factorizing systems of perverse sheaves on the symmetric products of the complex line and braided Hopf algebras. Joint work with V. Schechtman.