The Kazhdan-lusztig Coefficients In The Lowest Two-sided Cells For An Affine Weyl Group Of Type (?)₃

The Kazhdan-Lusztig(hereinafter referred to as K-L)polynomial is the core research object in K-L theory.In Weyl groups or affine Weyl groups,the leading coefficient of K-L polynomial(hereinafter referred to as K-L coefficient)plays an important role in understanding all K-L polynomials.The tensor product of irreducible modules is an important issue in the representation theory of Lie algebras.It plays a key role in calculating the K-L coefficient in the lowest two-sided cell.Some important results of K-L theory and tensor product decomposition of some irreducible modules are needed to calculate the K-L coefficient in the lowest two-sided cell for the affine Weyl Group.The results are as follows:1.Let L be a simple Lie algebra of type A₃,? be an arbitrary dominant weight,and V(?)be an irreducible module with ? as the highest weight.First,the weight set ?(?)of V(?)is obtained.For any ? ??(?),the weight multiplicity m(?)of the type A₃ Lie algebras are calculated according to the Freudenthal formula.For the fundamental dominant weight ? ?H,the weight set ?(?)and the weight multiplicity m(?)are calculated.The tensor product of Lie algebras of type A₃ is calculated by the related knowledge of the tensor product decomposition of the Lie algebras.2.Through the analysis of the structure of the affine Weyl Group of type A₃ W,we obtain the expression of the group elements in the corresponding extended affine Weyl group W.Then according to the expression form of the elements in the lowest two-sided cell,we get du;By using the recursion formula of the K-L polynomial Py,w,some results of tensor product for the Lie algebra of type A₃,the results of (?) and (?),we obtain the conclusion that the K-L coefficients are all ?1 in the lowest two-sided cell of the W.