Positivity of Shimura operators

In [16] G. Shimura introduced a family of invariant differential operators that play a key role in the study of nearly holomorphic automorphic forms, and he asked for a determination of their \textquotedblleft domain of positivity\textquotedblright. In this paper we relate the eigenvalues of Shimura operators to certain polynomials introduced by A. Okounkov, which leads to an explicit answer to Shimura's questions.