Extended affine Weyl groups and Frobenius manifolds

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t₁, ..., t_n) of WDVV equations of associativity polynomial in t₁, ..., t_n-1, exp t_n.     

A construction of representations of affine Weyl groups

Let G be a complex, semisimple, simply connected algebraic group withLie algebra . We extend scalars to the power series field in one variable C(()), and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of C(()),i.e. fixed point varieties on the full affine flag manifold. We definerepresentations of the affine Weyl group in the homology of these varieties,generalizing Kazhdan and Lusztig's topological construction of Springer'srepresentations to the affine context.     

Rationality of quotients by linear actions of affine groups

LetG = SLn(C)Cn be the (special) affine group. In this paper we study the representation theory of G and in particular the question of rationality for V/G, where V is a generically free G-representation. We show that the answer to this question is positive (Theorem 6.1) if the dimension of V is sufficiently large and V is indecomposable. We explicitly characterize two-step extensions 0 → S → V → Q → 0, with completely reducible S and Q, whose rationality cannot be obtained by the methods presented here (The...     

Representations of affine Hecke algebras and based rings of affine Weyl groups

In this paper we show that the Deligne-Langlands-Lusztig classification of simple representations of an affine Hecke algebra remains valid if the parameter is not a root of the corresponding Poincaré polynomial. This verifies a conjecture of Lusztig proposed in 1989.

     

Abelian ideals of Borel subalgebras and affine Weyl groups

This paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a Borel subalgebra. Our main result is a nice uniform formula for the dimension of maximal ideals in these posets. We also obtain results on ad-nilpotent ideals which complete the analysis started in (J. Algebra 225 (2000) 130, 258 (2002) 112).     

Projective normality of Weyl group quotients

In this note, we prove that for the standard representation Vof the Weyl group W of a semi-simple algebraic group of type A _n , B _n , C _n , D _n , F ₄ and G ₂ over \mathbbC\mathbb{C}, the projective variety \mathbbP(V^m)/W\mathbb{P}(V^m)/W is projectively normal with respect to the descent of O(1)^?|W|\mathcal{O}(1)^{\otimes |W|}, where V ^m denote the direct sum of m copies of V.     

A family of affine Weyl group representations

In this paper we explicitly determine the virtual representations of the finite Weyl subgroups of the affine Weyl group on the cohomology of the space of affine flags containing a family of elementsn _t in an affine Lie algebra. We also compute the Euler characteristic of the space of partial flags containingn _t and give a connection with hyperplane arrangements.This paper forms part of my Ph.D. thesis at M.I.T. I was supported by an NSF Graduate Fellowship and NSF grant DMS 9304580.     

On the determination of Kazhdan–Lusztig cells for affine Weyl groups with unequal parameters

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells which have important applications in representation theory. We study the case where W is an affine Weyl group of type . Using explicit computation with COXETER and CHEVIE, we show that (1) there are only finitely many possible decompositions into left cells and (2) the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures in this case. A key ingredient of the proof is a general result which shows that the Kazhdan–Lusztig polynomials of affine Weyl group are invariant under (large enough) translations.     

Central quotients of biautomatic groups

The quotient of a biautomatic group by a subgroup of the center is shown to be biautomatic. The main tool used is the Neumann-Shapiro triangulation of S ^n-1, associated to a biautomatic structure on . Among other applications, a question of Gersten and Short is settled by showing that direct factors of biautomatic groups are biautomatic. Received: October 4, 1994     

The product of affine orthogonal projections

In the case of a finite number of subspaces in a given Hilbert space, by a theorem of J. von Neumann, the iteration of the product of projectors is always convergent. In a finite dimensional Hilbert space, this theorem has been generalized for affine subspaces. In this paper we construct an example which shows that this result does not hold in the infinite dimensional case.     

Weyl sums over integers with affine digit restrictions

For any given integer q?2, we consider sets N of non-negative integers that are defined by affine relations between their q-adic digits (for example, the set of non-negative integers such that the number of 1's equals twice the number of 0's in the binary representation). The main goal is to prove that the sequence (αn)_n∈N is uniformly distributed modulo 1 for all irrational numbers α. The proof is based on a saddle point analysis of certain generating functions that allows us to bound the corresponding Weyl sums.