Are Wigner Functions eigenfunctions of J^2 and Jz?

[Oojee]

Homework Statement

I have a question related to representation of rotation operator R in the basis spanned by the eigenvectors of J² and J_z. I am studying from Quantum Mechanics by Zettili. The development of Wigner D-matrix and its elements D^j (Wigner functions) is clear. But the book goes on to say that the Wigner functions are joint eigenfunctions of J² and J_z. How can the matrix elements be the eigenfunctions?

 

[NateD]

Homework EquationsThe Wigner D-matrix is given byDjm(α,β,γ) = (−1)m ∑ℓ (−1)ℓ exp(−imγ)C∗ℓ m (α)Cℓ m (β) where Cℓ m (α) are the Wigner C-functions.The Attempt at a SolutionI think that the Wigner functions are not the eigenfunctions themselves, but rather the matrix elements of the rotation operator which can be used to calculate the eigenfunctions. The Wigner D-matrix gives us the connection between the eigenvalues of J2 and Jz and the angular momentum basis states |jm>. This connection can then be used to calculate the eigenfunctions.