Abstract. Let Φ be a finite crystallographic irreducible root system and P Φ be the convex hull of the roots in Φ. We give a uniform explicit description of the polytope P Φ , analyze the algebraic-combinatorial structure of its faces, and provide connections with the Borel subalgebra of the associated Lie algebra. We also give several enumerative results.

In 1979 Kazhdan and Lusztig defined, for every Coxeter group W , a family of polynomials, indexed by pairs of elements of W , which have become known as the Kazhdan-Lusztig polynomials of W , and which have proven to be of importance in several areas of mathematics. In this paper, we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these polynomials. Our results also imply, and generalize, the recent one in [Adv. in Math. 180 (2003) 146-175] on the combinatorial invariance of Kazhdan-Lusztig polynomials.

Abstract. We prove that the combinatorial concept of a special matching can be used to compute the parabolic Kazhdan-Lusztig polynomials of doubly laced Coxeter groups and of dihedral Coxeter groups. In particular, for this class of groups which includes all Weyl groups, our results generalize to the parabolic setting the main results in [Advances in Math. 202 (2006), 555-601]. As a consequence, the parabolic Kazhdan-Lusztig polynomial indexed by u and v depends only on the poset structure of the Bruhat interval from the identity element to v and on which elements of that interval are minimal coset representatives.

In this paper we present a unifying approach to study the homotopy type of several complexes arising from forests. We show that this method applies uniformly to many complexes that have been extensively studied.

We study the root polytope P Φ of a finite irreducible crystallographic root system Φ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system Φ. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of P Φ and analyze its relation with the facets of P Φ . For Φ of type A n or C n , we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of P Φ . We show that this triangulation restricts to a triangulation of the positive root polytope P + Φ .