Geometric realizations of Lusztigs symmetries on the whole quantum groups

In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztigs perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of Lusztigs symmetries on the positive part of a quantum group, we shall give geometric realizations of Lusztigs symmetries on the whole quantum group.