Lusztig和Vogan介绍了Weyl群的Hecke代数W的u-变形对合模,并且证明了该对合模同构于Hecke代数的一个左理想。申请人和胡峻教授推广了这个模同构。（1）本项目把上述的模同构再次推广到“扩充Weyl群的Hecke代数”的模上。.Lusztig构造了Weyl群的对合到极大环面的正规化子中的一个提升，使得对应的正规化子的元素n在Frobenius映射下等于n的逆。不过，这个证明是按照Weyl群的Lie型分别给出的，并且对于非典型的情况利用电脑做辅助计算。（2）本项目给出Lusztig提升的一个不依赖于计算机程序的统一证明。.Lusztig引入了Weyl群W的特殊不可约表示的概念，并且用渐进Hecke代数J的单模来代替W的不可约表示。（3）本项目研究对应于Weyl群的special表示的J的单双边理想的和的基，看它能否用J的标准基的非负整数集的线性组合表示。

Lusztig and Vogan have introduced a involution module for the Hecke algebra of a Weyl group with basis indexed by the involutions in that Weyl group. Lusztig showed that the involution module above can be realized as a left ideal of the Hecke algebra generated by a very remarkable element of the Hecke algebra. The applicant and Jun Hu extended the module isomorphism. (1) This project will extend the module isomorphism once more to the module over the “extended Hecke algebra”...Lusztig explicitly construct a lifting from the involutions of a Weyl group to the normalizer of a maximal torus in a split reductive group, such that the image of every corresponding element n in the normalizer under the Frobenius map is equal to the inverse of n. However, its proof is based on a case-by-case verification and in the exceptional cases relies in part on a computer calculation.(2) This project will determine a unified proof for Lusztig’s lifting independent on any computer calculation...Lusztig introduced special representations of Weyl group W. It is convenient to replace irreducible representations of W with the corresponding simple modules of the asymptotic Hecke algebra J. (3) This project will show that the two-sided ideal of J, which is the sum of the simple two-sided ideals corresponding to the various special representations of Weyl group W, admits a basis consisting of nonnegative linear combinations of elements of the basis of J.