Coxeter groups: statistics and Kazhdan-Lusztig polynomials

Among the Coxeter groups most studied there are finite Coxeter groups and Weyl affine groups. For each of them, a combinatorial interpretation is known as subgroups of generalized permutation groups. This thesis introduces new statistics on these affine permutation groups which extend the well-known concepts of excedance and major index widely studied for finite Coxeter groups. At the same time, this work introduces new techniques that allows us to compute some families of Kazhdan-Lusztig polynomials for Coxeter groups and, where possible, try some link with the combinatorial statistics. In particular we give an explicit formula that computes KL polynomials on pairs of Boolean elements and finally we compute KL polynomials for all the quasi-minuscule parabolic quotients, showing that in this context all non-zero polynomials are monic monomials.