Alcove Walks, Hecke Algebras, Spherical Functions, Crystals and Column Strict Tableaux

Abstract

Dedicated to R. MacPherson on the occasion of his 60th birthday IntroductionTogether, Sections 2 and 5 of this paper form a self contained treatment of the theory of crystals and the path model. It is my hope that this will be useful to the many people who, over the years, have told me that they wished they understood crystals but have found the existing literature too daunting. One goal of the presentation here is to clarify the relationship between the general path model and the crystal operators of Lascoux and Schützenberger used in the type A case [LS]. More specifically, Section 2 is a basic pictorial exposition of Weyl groups and affine Weyl groups and Section 5 is an exposition of the theory of (a) symmetric functions, (b) crystals and (c) the path model which is designed for readers whose only background is the material in Section 2. These two sections can be read independently of Sections 3 and 4.Sections 3 and 4 give an exposition of the affine Hecke algebra and recent results regarding the combinatorics of spherical functions on p-adic groups (Hall-Littlewood polynomials) using only the material in Section 2 as background. The q-analogue of the theory of crystals developed in Section 4 specializes to the path model version of the "classical" crystal theory at q −1 = 0. This specialization property is "to be expected" since Macdonald's formula for the spherical function specializes to the Weyl character formula at q −1 = 0. It is my hope that the presentation of the results will illustrate the close connection between the affine Hecke algebra, the path model, and the theory of crystals.The motivation for this paper is the following. The classical path model is a combinatorial tool for working with crystal bases of integrable representations of symmetrizable Kac-Moody algebras, a generalization of column strict Young tableaux. The same combinatorics can be used to study the equivariant K-theory of the flag variety [PR] and this point provides a translation between the path model and the structure of the nil-affine Hecke algebra. Thus, conceptually, crystal bases = the path model = the nil affine Hecke algebra = the T -equivariant K-theory of G/B.The paper of Gaussent-Littelmann [GL] (see also the work of Kapovich-Millson in [KM]) indicated that there is a refined model which removes the 'nil' in this chain of equalities and models 1 the geometry of Kac-Moody groups over a local field. The result is the conceptual chain of equalities positively folded alcove walks = geodesics in the affine building = the affine Hecke algebra = MV cycles in the loop Grassmanian.The connection to the affine Hecke algebra and the approach to spherical functions for a p-adic group in [NR] was made concrete by C. Schwer [Sc] who told me that "the periodic Hecke module encodes the positively folded galleries".The main results of this paper are obtained by viewing the affine Hecke algebra in terms of a new construct, the alcove walk algebra. This combinatorics provides in the same way that the theory of crystal bases gives...

Combinatorics and topology of toric arrangements defined by root systems

“…= W Φ ∨ (see for example [15], [24]). The action of W × Z(Φ) on C 0 (Φ) can be lifted to an action of W .…”

Finite traces and representations of the group of infinite matrices over a finite field

“…This fact follows from the known formulas for these coefficients (see e.g. [Ra, Theorem 4.9], [Sc, Theorem 1.3], [KM] and references therein). Since Hall-Littlewood P -polynomials and Q-polynomials differ by the multiplication by an explicit constant, which is positive for q > 1 (see [M, Section III.2]) we can replace P by Q in any part of (4.5) and the coefficients will be still positive.…”

Alcove walks and nearby cycles on affine flag manifolds

“…In a recent paper [14], Ram has introduced the notion of alcove walk and used it in order to describe the affine Hecke algebra associated to a root datum.…”