BGP-reflection functors and Lusztig.s symmetries of modified quantized enveloping algebras

Let U be a quantized enveloping algebra and U. its modified form. Lusztig gives some symmetries on U and U.. In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig.s symmetries on U and their important properties, for instance, the braid relations. In this paper, we define a modified form H of the Ringel-Hall algebra and realize the Lusztig.s symmetries on U. by applying the BGP-reflection functors to H.