Is there an analogue for Ramanujan–Serre derivative for Hilbert modular forms?

Question

If $f$ is a modular form of weight $k$, it is well known that $$ D(f)=f' -\tfrac k{12}E_2f $$ is modular of weight $k+2$. Here $E_2$ is the Eisenstein series. I wanted to ask if there is an extension of this fact for Hilbert modular forms.

When looking up Hilbert modular forms (Shimura 1975), I was only able to find the following differential operators: for a Hilbert modular form $f$ on $\mathbb H^n$, we define $$ D_{j,t}(f)=\left(\frac{t}{2i y_j}+\frac{\partial}{\partial z_j}\right)f $$ where $z_j=x_j+iy_j$ is the $j$-th coordinate. I don't think this is the same operator (is it?)

I am really looking for a many variable version of the operator $D(f)=f' -\frac k{12}E_2f$. I don't really have a background in analysis or number theory so this question may be really easy for experts.

Answer 1

I think there's not a good analogue.

The operator $D_{j, k_j}$, acting on Hilbert modular forms of weight $k = (k_1, \dots, k_n)$ where $n = [F: \mathbf{Q}]$, is the "Maass--Shimura differential operator". It doesn't preserve holomorphicity, but it sends holomorphic Hilbert modular forms to "nearly-holomorphic" forms in Shimura's sense.

For $n = 1$ something special happens, which is that the nearly-holomorphic forms are freely generated by $E_2$ as a polynomial ring over the holomorphic ones. So you can "adjust" the differential operator by a multiple of $E_2$ to make it preserve holomorphicity (at a cost of breaking its nice compatibility with Hecke operators), and this gives the Ramanujan--Serre operator. However, for $n > 1$ there isn't a tidy set of generators for nearly-holomorphic forms as an algebra over the holomorphic ones so I don't think there is an analogue.

(Perhaps if you were more precise about what it is that you want this operator to be useful for, then you might get more useful answers in response.)