We study the intertwining operator and $h$-harmonics in Dunkl's theory on $h$-harmonics associated with reflection groups. Based on a biorthogonality between the ordinary harmonics and the action of the intertwining operator $V$ on the harmonics, the main result provides a method to compute the action of the intertwining operator $V$ on polynomials and to construct an orthonormal basis for the space of $h$-harmonics.

Keywords:

$h$-harmonics, intertwining operator, reflection group $h$-harmonics, intertwining operator, reflection group

MSC Classifications:

33C50, 33C45 show english descriptions Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 33C50 - Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

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