(Reference) A Shimura operator acting on Hermitian modular forms

Question

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).

Answer 1

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.

Answer 2

Although this does not literally answer your question, there are some points that may be useful to you that are not really mentioned at all in the source you mention. That is, it is useful (in my opinion) to take a somewhat more structured, rather than formulaic, approach to Maass-Shimura operators, to understand why they do what they do, and then produce the formulaic aspects as corollaries.

First, I think it is important to understand that, for sufficiently high lowest $K$-type ... let's say one-dimensional... holomorphic discrete series (of symplectic groups, unitary groups $U(p,q)$, and others) are free modules over the enveloping algebra $U{\mathfrak n}^{\mathrm opp}$ of the unipotent radical $\mathfrak n^{\mathrm opp}$ of the opposite parabolic to the Siegel-type parabolic. This is not so hard to prove, e.g., http://www.math.umn.edu/~garrett/m/v/holo_disc.pdf

Thus, any rational structure on that unipotent radical gives a rational structure to the holomorphic discrete series.

Then, if one wants to map to a nearby one-dimensional $K$-type inside that holomorphic discrete series, it is a linear algebra problem to determine what element of $U{\mathfrak n}^{\mathrm opp}$ does so.

Then there are both determination of these operators in coordinates, and specifying/normalizing their rationality properties.

In the sorts of applications I'm familiar with, the subsequent representation-theoretic issue is exactly the decomposition of the tensor product of two holomorphic discrete series as direct sum of holomorphic discrete series (including rationality discussion). This is straightforward in the case of sufficiently high lowest $K$-types, due to the free-ness mentioned above.