Length functions and Demazure operators for G( e,1, n), I

Abstract

This note is the first part of consecutive two papers concerning with a length function and Demazure operators for the complex reflection group W = G( e, 1, n). In this first part, we study the word problem on W based on the work of Bremke and Malle [BM]. We show that the usual length function ℓ( W) associated to a given generator set S is completely described by the function n( W), introduced in [BM], associated to the root system of W. In the second part, we will study the Demazure operators of W on the symmetric algebra. We define a graded space H W in terms of Demazure operators, and show that H W is isomorphic to the coinvariant algebra S W , which enables us to define a homogeneous basis on S W parametrized by w ϵ W.