Some Properties Of Hecke Algebras

This thesis is concerned with Hecke algebras. This paper is composedof four parts.In chapter one, we briefly introduce some related knowledge and listsome known results.In chapter two, we first give some important results about degenerateHecke algebras. After that, we deduce some properties of invertible elementsand unit group, obtain a method to compute invertible elements in degenerateHecke algebras.In chapter three, by learning R-polynomials in Hecke algebras of Coxetergroups, we obtain the special value about R-polynomials in finite Coxetergroups.In chapter four, we learn that G.Lusztig formulates conjectures describ-ing the properties of Hecke algebras with unequal parameters and verifiesthese conjectures in some cases. On this basis, finally, we get some specialbasis of Hecke algebras with unequal parameters.The main result of this thesis is the deep into study under some knownresult, we improved and generalized some former researches. Though work ofthis paper, we have further knowledge for some properties of Hecke algebras.