DEMAZURE OPERATORS FOR COMPLEX REFLECTION GROUPS G(e,e,n)

Abstract

This paper is a continuation of the work in [RS], where we studied Demazure operators for the imprimitive complex reflection group $\tilde{W}=G\left(e,1,n\right)$ and constructed a homogeneous basis of the coinvariant algebra $S_{\tilde{W}}$. In this paper, we study a similar problem for the reflection subgroup $W=G\left(e,e,n\right)$ of $\tilde{W}$. We prove, by assuming certain conjectures, that the operators ${\text{Δ}}_w\left(w\in W\right)$ are linearly independent over the symmetric algebra $S\left(V\right)$. We define a graded space $H_W$ in terms of Demazure operators, and we show that the coinvariant algebra $S_W$ is naturally isomorphic to $H_W$. Then we can define a homogeneous basis of $S_W$ parametrized by $w\in W$.