AN INTERTWINING OPERATOR FOR THE GROUP B₂

Abstract

Abstract. There is a commutative algebra of differential-difference operators, acting on polynomials on ‫ޒ‬ 2 , associated with the reflection group B 2 . This paper presents an integral transform which intertwines this algebra, allowing one free parameter, with the algebra of partial derivatives. The method of proof depends on properties of a certain class of balanced terminating hypergeometric series of 4 F 3 -type. These properties are in the form of recurrence and contiguity relations and are proved herein.2000 Mathematics Subject Classification. Primary 33C80, 33C20; Secondary 33C70, 43A80. Introduction.1.1. Overview. We construct an integral for the intertwining operator V associated to the reflection group of type B 2 (order 8) acting on ‫ޒ‬ 2 , with one parameter κ. For polynomials or adequately smooth functions in x = (x 1 , x 2 ) define the differential-difference operators:These operators are special cases of those defined by the author in [2]. Their key property is commutativity, T 1 T 2 = T 2 T 1 . We deal only with the restricted case κ 1 = κ 2 = κ. The intertwining operator V preserves the degree of homogeneous polynomials and satisfies V ( ∂ ∂x i f )(x) = T i Vf (x) for i = 1, 2, and V 1 = 1. The definition and existence of V for κ > 0 was shown in [3].The easy example for this operator is furnished by the ‫ޚ‬ 2 action (x −→ −x) on ‫.ޒ‬

Reflection groups in analysis and applications

“…We have found two other explicit formulae for V , for A 2 [16] and the equal parameter case for B 2 [20]; however these formulae do not exhibit positive measures. The method involved integration over the associated compact Lie groups and a formula of Harish-Chandra.…”

“…It may be possible to do this by using integration over the associated compact Lie groups (unitary and compact symplectic groups); the approach used in [16], [20] for Ë 3 and B 2 .…”

Dunkl kernel associated with dihedral groups

“…For instance, when the root system is of type B 1 , the Dunkl kernel is a combination of the modified Bessel function of the first kind and of its first derivative. For the rank-two root systems of types A 2 , B 2 , multiple integral representations were derived in [2,11,12]: the key tool in the first of these papers is the so-called shift principle [11,Proposition 1.4] while the last ones rely heavily on Harish-Chandra integral representations for the unitary and the symplectic groups respectively. In [18], a multiple integral representation of the Dunkl intertwining operator associated with an arbitrary orthogonal root system was proved and subsequently exploited in [6] in order to get the corresponding generalized translation operator.…”

Dunkl kernel and Dunkl translation for a positive subsystem of orthogonal roots

“…One of the most challenging problem in the Dunkl analysis is to find explicit formulae for the kernel of this transform, the well-known Dunkl kernel. Some significant results exist (such as asymptotic behavior for instance, see [23]), but an explicit formula is still lacking except in the cases where the root system is of A 2 -type ( [12]), of B 2 -type ( [13]) and where the reflection group is Z d 2 ( [11,26]). Another key tool associated with Dunkl operators is the generalized translation or Dunkl translation.…”

“…; y/ D V W Ä .e h ;yi /. Unfortunately, the Dunkl kernel is explicitly known only in some special cases; when the root system is of A 2 -type ( [12]), of B 2 -type ( [13]) and when the reflection group is Z d 2 (see [11,26]). Nevertheless we know that this kernel has many properties in common with the classical exponential to which it reduces when Ä D 0.…”