This paper introduces differential-difference operators associated with reflection groups, focusing on their role in the theory of spherical harmonics. The operators are defined for measures invariant under finite reflection groups, which are generated by reflections in the zero sets of linear functions. These measures are products of powers of linear functions and a rotation-invariant measure on the unit sphere in $ \mathbb{R}^n $. The paper constructs a commutative set of first-order differential-difference operators that are analogous to the first-order partial derivatives in the ordinary theory of spherical harmonics. The key result is the definition of the h-Laplacian, a second-order differential-difference operator that acts as an endomorphism on the space of polynomials and is homogeneous of degree -2. It is shown that a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by the h-Laplacian. This leads to the concept of h-harmonic polynomials, which are analogous to harmonic polynomials. The paper also constructs a commutative set of first-order differential-difference operators that are related to the h-Laplacian. These operators are shown to commute and are used to derive recurrence relations and orthogonal decompositions for harmonic polynomials. The paper further explores the Cauchy-Riemann equations associated with dihedral groups, which are analogues of the classical Cauchy-Riemann equations for Gegenbauer and Jacobi polynomials. The analysis includes the determination of adjoints of these operators, their relation to the character table of reflection groups, and the study of degenerate parameter values. The paper concludes with a discussion of the properties of the operators on dihedral groups, including the construction of homogeneous polynomials in the kernel of the operators and their orthogonality properties. The results are illustrated with examples involving specific reflection groups and their associated weight functions.This paper introduces differential-difference operators associated with reflection groups, focusing on their role in the theory of spherical harmonics. The operators are defined for measures invariant under finite reflection groups, which are generated by reflections in the zero sets of linear functions. These measures are products of powers of linear functions and a rotation-invariant measure on the unit sphere in $ \mathbb{R}^n $. The paper constructs a commutative set of first-order differential-difference operators that are analogous to the first-order partial derivatives in the ordinary theory of spherical harmonics. The key result is the definition of the h-Laplacian, a second-order differential-difference operator that acts as an endomorphism on the space of polynomials and is homogeneous of degree -2. It is shown that a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by the h-Laplacian. This leads to the concept of h-harmonic polynomials, which are analogous to harmonic polynomials. The paper also constructs a commutative set of first-order differential-difference operators that are related to the h-Laplacian. These operators are shown to commute and are used to derive recurrence relations and orthogonal decompositions for harmonic polynomials. The paper further explores the Cauchy-Riemann equations associated with dihedral groups, which are analogues of the classical Cauchy-Riemann equations for Gegenbauer and Jacobi polynomials. The analysis includes the determination of adjoints of these operators, their relation to the character table of reflection groups, and the study of degenerate parameter values. The paper concludes with a discussion of the properties of the operators on dihedral groups, including the construction of homogeneous polynomials in the kernel of the operators and their orthogonality properties. The results are illustrated with examples involving specific reflection groups and their associated weight functions.