An Intertwining Operator for the Group B2

There is a commutative algebra of di¤erential-di¤erence operators, acting on polynomials on R2, associated with the re‡ection group B2. 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 4F3-type. These properties are in the form of recurrence and contiguity relations and are proved herein. 1. Introduction 1.1. Overview. We construct an integral for the intertwining operator V associated to the re‡ection group of type B2 (order 8) acting on R, with one parameter . For polynomials or adequately smooth functions in x = (x1; x2) de…ne the di¤erential-di¤erence operators: T1f (x) = @ @x1 f (x) + 1 f (x) f ( x1; x2) x1 (1.1) + 2 f (x) f (x2; x1) x1 x2 + 2 f (x) f ( x2; x1) x1 + x2 ; T2f (x) = @ @x2 f (x) + 1 f (x) f (x1; x2) x2 + 2 f (x) f (x2; x1) x2 x1 + 2 f (x) f ( x2; x1) x1 + x2 : These operators are special cases of those de…ned by the author in [2]. Their key property is commutativity, T1T2 = T2T1. We deal only with the restricted case 1 = 2 = . The intertwining operator V preserves the degree of homogeneous polynomials and satis…es V @ @xi f (x) = TiV f (x) for i = 1; 2, and V 1 = 1. In [5] it was shown that V exists and is one-to-one as a map on polynomials for any except for the set m4 : m 2 N; m 4 = 2 Z of singular values. The de…nition and existence of V for > 0 was shown in [3]. Later Rösler [8] (see also [9]) proved that the functional f 7! V f (x) is given by integration with respect to a positive measure, for each x 2 R. An explicit integral was found for the group S3 (symmetric group on 3 objects) in [4]. Xu [10] found an intertwining transform for B2 under 2000 Mathematics Subject Classi…cation. Primary 33C80, 33C20; Secondary 33C70, 43A80.