Dunkl Operators: Theory and Applications

Abstract

Abstract. These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the Dunkl transform. We point out the connection with integrable particle systems of Calogero-MoserSutherland type, and discuss some systems of orthogonal polynomials associated with them. A major part is devoted to positivity results for the intertwining operator and the Dunkl kernel, the Dunkl-type heat semigroup, and related probabilistic aspects. The notes conclude with recent results on the asymptotics of the Dunkl kernel.

The Z2n Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

“…When |A ∩ B| = 1, the result holds by virtue of Lemma 4. Suppose that (20) holds at level n and consider the set B = B ∪{x} with x ∈ A and |A ∩ B| = n. Consider y ∈ A ∩ B; one can write Γ B as in (18). Since x, y ∈ A, Γ A commutes with Γ {x,y} and one can write {Γ A , Γ B } as in (19), the only difference with (19) being that x, y ∈ A.…”

“…Since x, y ∈ A, Γ A commutes with Γ {x,y} and one can write {Γ A , Γ B } as in (19), the only difference with (19) being that x, y ∈ A. Upon applying the induction hypothesis, one finds (20) with B replaced by B after a straightforward calculation.…”

“…Dunkl operators are families of differential-difference operators associated with finite reflection groups. Introduced in [6], these operators appear in many areas: they are at the core of the study of multivariate special functions associated with root systems [7,18], they arise in harmonic analysis and integral transforms [3,20], they are closely related to representations of double affine Hecke algebras [5], they are at the origin of Dunkl processes [14], and they play a central role in the analysis of Calogero-Sutherland systems [22]. Consider the Abelian reflection group Z n 2 = Z 2 × · · · × Z 2 ; the corresponding Dunkl operators T 1 , .…”

AN INTERTWINING OPERATOR FOR THE GROUP B2

“…In [5] it was shown that V exists and is one-to-one as a map on polynomials for any κ except for the set {− m 4 : m ∈ ‫,ގ‬ m 4 / ∈ ‫}ޚ‬ of singular values (for the case of B 2 ). Later Rösler [8] (see also [9]) proved that the functional f → Vf (x) is given by integration with respect to a positive measure, for each x ∈ ‫ޒ‬ 2 . Xu [10] found an intertwining transform for B 2 under restrictive conditions on degree and κ 1 , κ 2 .…”

“…The book of Dunkl and Xu [6] is a reference for differential-difference operators, the intertwining operator and the kernel K(x, y) for any finite reflection group. Asymptotic formulae for the kernel are discussed in [9]. The author thanks the referee for useful remarks which helped improve the introduction and other details in the exposition.…”