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).



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> 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?
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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).



(Reference) A Shimura operator acting on Hermitian modular forms
It turns out that the answer to my question was in a previous paper by Shimura, namely Differential operators, holomorphic projection, and singular forms (lemma 4.2, <https://projecteuclid.org/euclid.dmj/1077286742>).


The operator is (up to a constant factor)
$$\Delta\_q=\det(Y)^{\kappa-1-q}\det(d/dZ)\det(Y)^{q-\kappa+1}$$
where
$$\kappa=\begin{cases}\frac{n+1}{2} & \mbox{Siegel case}
\\\ \ n & \mbox{Hermitian case}\end{cases}$$
and then $\Delta\_q^p=\Delta\_{q+2p-2}\cdots\Delta\_{q+2}\Delta\_q$ as usual.


