Leading Coefficients Of The Kazhdan-lusztig Polynomials For Affine Weyl Groups Of Type (?)?(?) And Some Related Research

The polynomial of Kazhdan-Lusztig is a very crucial object in Kazhdan-Lusztig theory.When we considerate a Weyl group or affine Weyl group, their coefficients of Kazhdan-Lusztig polynomials play important roles in representation theory and Li theory.Thus the leading coefficients of Kazhdan-Lusztig polynomials of Weyl group or affine Weyl group will be of great significance for us. On the one hand, they may help us to realize all the coefficients of Kazhdan-Lusztig polynomials. On the other hand, they show importance in Lie theory and representation theory. In recent years, the research in leading coefficients of Kazhdan-Lusztig polynomials attract more attention of scholars. While the calculation of leading coefficients is also very hard to compute, the study in this aspect is not too much actually. The main results of this paper are as follows:1. For affine Weyl group of type ??? , we get some leading coefficients??y,w?,where ?y,w? ? c₀×c2 , c₀ and c2 are different type two-sided cells by using Lusztig's semi-linear equations. Because of the complexity of calculation, we study with the help of computer which showed in appendix.Beside we get several relevant properties by left and right string.2. Due to the importance of distinguished involutions in Kazhdan-Lusztig theory and the moderate association with decomposition of two-sided cells of Coxeter groups, we depict the decomposition of two-sided cells and left cells of affine Weyl group W of type ??? in the second part of this article .Then we give the graphs of left cells and the graphs of distinguished involutions of w. Meanwhile, we check the distinguished involutions obtained by some known Kazhdan-Lusztig polynomials .3. In order to apply semi-linear equations to W of type ??? , we give the representations of the elements in extended affine Weyl group of ??? corresponding to the fundamental dominant weights in the third part. In addition, we get some quantities among the calculation of leading coefficients of W by computer ,which make a foundation for the further study of leading coefficients of W.