Intertwining operators associated to the group 𝑆₃

Dunkl, Charles F.

doi:10.1090/S0002-9947-1995-1316848-8

Intertwining operators associated to the group $S_ 3$
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Trans. Amer. Math. Soc. 347 (1995), 3347-3374 Request permission

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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