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Intertwining operators associated to the group $ S\sb 3$

Author: Charles F. Dunkl
Journal: Trans. Amer. Math. Soc. 347 (1995), 3347-3374
MSC: Primary 22E30; Secondary 20B30, 33C50, 33C80
MathSciNet review: 1316848
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Abstract: 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 $ \geqslant \frac{1} {2}$. The transform is realized as an absolutely continuous measure on a compact subset of $ {M_2}({\mathbf{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)$.

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Keywords: Dunkl operators, intertwining operator, reflection groups, integral transform
Article copyright: © Copyright 1995 American Mathematical Society

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