In his book *Arithmeticity in the theory of automorphic forms* (http://bookstore.ams.org/surv-82-s) Shimura introduces at page 146 an operator $\Delta_p^q$ which should act on nearly holomorphic modular forms (by raising weight and non-holomorphy degree).

Is this (a version of) the Shimura-Maass differential operator (which in the case of Siegel modular forms can be explicitly written as $(2\pi i)^{-\circ}\det(Y)^\star\det(d/dZ)\det(Y)^{-\star})$?

In particular, I need to understand the action of $\Delta_p^q$ on Hermitian modular forms (for which I mean functions $\tilde{\mathcal{H}_n}\to\mathbb{C}$ modular with respect to $GU(n,n)$, where $\tilde{\mathcal{H}_n}$ is the space of $n\times n$ complex matrices $Z$ such that $i(\overline{Z^t}-Z)$ is positive definite).

Is there an explicit formula that describes $\Delta_p^q$ in my case? If not, what are some properties of the operator in the context of Hermitian modular forms?

It would be great if there was some kind of differential formula, like in the case of Siegel modular forms, since I am trying to study the effect of said operator on the Fourier coefficients (in particular integrality issues).