Seminormal representations of Weyl groups and Iwahori-Hecke algebras

The purpose of this paper is to describe a general procedure for computing analogues of Young's seminormal representations of the symmetric groups. The method is to generalize the Jucys-Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori-Hecke algebras. The combinatorics of these elements allows one to compute irreducible representations explicitly and often very easily. In this paper we do these computations for Weyl groups and Iwahori-Hecke algebras of types $A_n$, $B_n$, $D_n$, $G_2$. Although these computations are in reach for types $F_4$, $E_6$, and $E_7$, we shall, in view of the length of the current paper, postpone this to another work.