Consider the vector space of symmetric matrices. O(n) acts on it by conjugation $gMg^t$.

Question informally: consider invariant differential operators what is known about their radial parts ? (Is there something nice like Harish-Chandra's homomorphism (this MO quest.))

More precisely: a) Consider O(n) invariant differential operator with CONSTANT COEFFICIENTS D. It can be identified with some symmetric polynomial. Can we write down its radial part in terms of this polynom ? (In Harish-Chandra's case the answer is this polynom conjugated by the Vandermonde (this MO quest.)).

b) Very particularly take $D=det(\partial_{ij}) $ what is its radial part ? (This operator enters Capelly and Cayley identities).

c) If it is not with constant coefficients what can be said ?