| In this thesis, we prove the embedding theorem of generalized Verma modules in the stratified exact categories associated to certain finite posets in Coxeter groups. This theorem insures that any two homomorphisms from one generalized Verma module to another differ only by a constant multiple and that any non-zero homomorphism between two generalized Verma modules must be a monomorphism. Then we give two applications of the embedding theorem of generalized Verma modules to certain stratified exact categories associated to finite Coxeter systems. Firstly, we prove a base change property for the homomorphism groups between projective objects in these stratified exact categories. Secondly, assuming that Vogan's conjecture holds, we prove the rigidity of generalized Verma modules in these stratified exact categories. Finally, we establish the equivalence between Vogan's conjecture and the Kazhdan-Lusztig conjecture. Thus, Vogan's conjecture for generalized Verma modules in certain stratified exact categories associated to Weyl groups is true. |
