Quasiparabolic Subgroups of Coxeter Groups and Their Hecke Algebra Module Structures
It is well known that the R polynomial can be defined for the Hecke algebra of Coxeter groups, and the Kazhdan Lusztig theory can be developed to understand the representations of Hecke algebra. There is also a generalization for the existence of R polynomial and Kazhdan Lusztig theory for the Hecke algebra module of standard parabolic subgroups of Coxeter groups. In recent work of Rains and Vazirani, a generalization of standard parabolic subgroups, called quasiparabolic subgroups, are introduced, and the corresponding Hecke algebra module is well defined. However, the existence of the analogous involution (Kazhdan Lusztig bar operator) on the Hecke algebra module of quasiparabolic subgroups is unknown in general. Assuming the existence of the bar operator, the corresponding R polynomials and Kazhdan Lusztig polynomials can be constructed. We prove the existence of the bar operator for the corresponding Hecke algebra modules of quasiparabolic subgroups in finite classical Coxeter groups with a case by case verification (Chapter 4) As preparation, we classify all quasiparabolic subgroups of finite classical Coxeter groups. The approach is to first find all rotation subgroups of finite classical Coxeter groups (Chapter 2) Then we exclude the non quasiparabolic subgroups and confirm the quasiparabolic subgroups (Chapter 3)