Quasi-coxeter Elements Of Type Bn And Kazhdan-Lusztig Cells

Let W=(W,S) be an affine Weyl group. D. Kazhdan and G. Lusztig defined the concepts of cell decomposition in the Weyl group and the affine Weyl group W. which play an important role in the representation theory of the associated Hecke algebra, Lie group and Lie algebra. W can be decomposed into the union of the disjoint two-sided cells, while each two-sided cell can be decomposed into the union of the disjoint left cells, and each left cell has the unique distinguished involution. If all the distinguished involutions can be found, the problem of decomposition of the cells of W will be solved. In this paper, the definition of Quasi-coxeter elements of type Bn is given and the set E(W) is defined in the set of Coxeter elements, and the relation between quasi-Coseter and oriented graph is established, by the oriented graph, lusztig polynomials and a-valved functions, exact description for all the distinguished involutions of W is obtained. it is sure that some Quasi-Coxeter elements are precisely left cells by use of the set E(W).