INTERTWINING-OPERATORS ASSOCIATED TO THE GROUP S-3

For any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections in G. There exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients, for all but a singular set of parameter values (containing only certain negative rational numbers). This paper constructs an integral transform implementing the intertwining operator for the group S-3, the symmetric group on three objects, for parameter value greater than or equal to 1/2. The transform is realized as an absolutely continuous measure on a compact subset of M(2)(R),which contains the group as a subset of its boundary. The construction of the integral formula involves integration over the unitary group U(3).