A Remark on the Dunkl Differential—Difference Operators

Abstract

Let E be a Euclidean vector space of dimension n with inner product (·, ·). For α ∈ E with (α, α) = 2 we write (1.1) r α (λ) = λ − (α, λ)α, λ ∈ E for the orthogonal reflection in the hyperplane perpendicular to α. Definition 1.1. A normalized root system R in E is a finite set of non zero vectors in E, normalized by (α, α) = 2 ∀α ∈ R, such that r α (β) ∈ R ∀α, β ∈ R. Let R ⊂ E be a normalized root system. We write W = W (R) for the group generated by the reflections r α , α ∈ R. Denote by C[E] the algebra of C-valued polynomial functions on E. For w ∈ W , ξ ∈ E, α ∈ R introduce the operators (1.2) w, ∂ ξ , ∆ α : C[E] −→ C[E] by (1.3) (wp)(λ) = p(w −1 λ) (1.4) (∂ ξ p)(λ) = d dt {p(λ + tξ)} t=0 (1.5) (∆ α p)(λ) = p(λ) − p(r α λ) (α, λ). Remark 1.2. The operators ∆ α , α ∈ R were studied by Bernstein, Gel'fand and Gel'fand and are related to the Schubert cells and the cohomology of G/P [BGG]. They are the infinitesimal analogues of the Demazure operators [De 1,2]. Let R + = {α ∈ R; (α, λ) > 0} for some fixed generic λ ∈ E be a positive subsystem of R.