It is known that a number of algebraic properties of the braid groups extend to arbitrary finite Coxeter type Artin groups. Here we show how to extend the results to more general groups that we call Garside groups.

Let V be an ℓ-dimensional Euclidean space. Let G ⊂ O(V) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the derivation module D (m) (A) = {θ ∈ DerS | θ(αH) ∈ Sα m H}. The module is known to be a free S-module of rank ℓ by K. Saito (1975) for m = 1 and L. Solomon-H. Terao (1998) for m = 2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D (m) (A). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m − 1)h/2) + mi(1 ≤ i ≤ ℓ) (when m is odd). Here m1 ≤ · · · ≤ mℓ are the exponents of G and h = mℓ + 1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G.) Some new results concerning the primitive derivation D are obtained in the course of proof of the main result.

Abstract. We prove a highest weight classification of the finite-dimensional irreducible representations of a quantum affine algebra, in the spirit of Cartan’s classification of the finite-dimensional irreducible representations of complex simple Lie algebras in terms of dominant integral weights. We also survey what is currently known about the structure of these representations. 1.

Abstract not found

Abstract. A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painlevé type are discussed.

Abstracts This paper is based on the author’s talk at 1997 Taniguchi Symposium “Integrable Systems and Algebraic Geometry”. We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumptions, these two objects are identical. The flat pencils of contravariant metrics on a manifold M appear naturally in the classification of bihamiltonian structures of hydrodynamics type on the loop space L(M). This elucidates the relations between Frobenius manifolds and integrable hierarchies. 1

The purpose of this paper is two-fold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [5], [4] and [10]) on the structure of the descent algebras of the Coxeter groups of type An and Bn. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/WJ, where the WJ are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra. 1.

. In this paper, we prove the degenerations of Schubert varieties in a minuscule G=P , as well as the class of Kempf varieties in the flag variety SL(n)=B, to (normal) toric varieties. As a consequence, we obtain that determinantal varieties degenerate to (normal) toric varieties. Introduction In this paper, we carry out the proof of the results announced in [21]. Let G be a semisimple, simply connected algebraic group defined over an algebraically closed field k. Fix a maximal torus T in G, a Borel subgroup B oe T . Let W be the Weyl group of G relative to T . Let Q ' B be a parabolic subgroup of classical type, say Q = T r i=1 P k i , where P k i , 1 i r, is a maximal parabolic subgroup of classical type (see [26] for the definition of a parabolic subgroup of classical type). Let W (Q) be the Weyl group of Q. For w 2 W=W (Q), let X(w)(= BwQ (mod Q) with the canonical reduced structure of a scheme) denote the Schubert variety in G=Q, associated to w. Given m = (m 1 ; : : : ; m ...

We derive summation formulas for a specic kind of multidimensional basic hypergeometric series associated to root systems of classical type. We proceed by combining the classical (one-dimensional) summation formulas with certain determinant evaluations. Our theorems include A r extensions of Ramanujan's bilateral 1 1 sum, C r extensions of Bailey's very-well-poised 6 6 summation, and a C r extension of Jackson's very-well-poised 8 7 summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A r , B r , C r , and D r , respectively, of Chu's bilateral transformation formula for basic hypergeometric series of Gasper-Karlsson-Minton type. Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation theorems of the classical theory of basic hypergeometric series.

We analyse the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the C ∗-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pairs of points in a symmetric space of unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with 8 points in common. We show