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

In Kazhdan-Lusztig theory,Kazhdan-Lusztig(hereinafter referred to as K-L)polynomial is its core research object.The first term coefficient of K-L polynomials(hereinafter referred to as K-L coefficient)plays an irreplace-able role in studying the representation theory of all K-L polynomials,Hecke algebra and the study of cell.Irreducible mode in the theory of Lie is crucial for solving the K-L coefficients.This thesis mainly calculates the K-L coefficien in the lowest two-sided cell of the C3 type affine Weyl group.The specific findings are as follows:1.Calculate the partial tensor product decomposition expression of the C3 type Lie algebra.First of all,according to the knowledge of the Lie algebra,the weight set ∏(λ)of V(λ)is obtained,let λ be an arbitrary dominant weight and V(λ)be an irreducible module;secondly,we can calculate the weight multiplicity by using the Freudenthal formula;finally,we can get the part of the C3 type Lie algebra product decomposition expression according to the weight set ∏(λ)and the weight multiplicity m(μ).2.Calculate Cw0du-1Cdu’w0 and δw0du-1,du’w0,z1w0.In the first place,in terms of the expression form of the elements in the lowest two-sided cell,we get du;in the second place,we get the results of Cw0du-1Cdu’w0.It relies on the recursion formula of the K-L polynomial Py,w;in the end,from the expression of δw0du-1,du’w0,z1w0,we know that δw0du-1,du’w0,z1w0 is coefficient of the[2]8.It is the coefficient of q4,where q is a variable So we can find such a coefficient from the expression form of Cw0du-1Cdu’w0.Considering the two results,we also obtain the conclusion that the K-L coefficients are all less than or equal to 1 in the lowest two-sided cell of the W.