Decomposition of real quasimodular forms of depth 1

Question

Let $\widetilde{M}_k^{\leq \ell}$ be the space of weight $k$ depth $\leq \ell$ quasimodular forms, and $\widetilde{M}_{k,\mathbb R}^{\leq \ell}$ be a subspace of $\widetilde{M}_k^{\leq \ell}$ whose elements have real Fourier coefficients. Let $D=\frac{1}{2\pi i}\frac{d}{d\tau}=q\frac{q}{dq} : \widetilde{M}_k^{\leq \ell} \to \widetilde{M}_{k+2}^{\leq \ell+1}$ be the differential operator. If $k>2$, then we have:

Theorem. $\widetilde{M}_k^{\leq 1}=M_k \oplus DM_{k-2}$.

It is obvious that $M_{k,\mathbb R} \oplus DM_{k-2,\mathbb R} \subseteq \widetilde{M}_{k,\mathbb R}^{\leq 1}$, and I believe that $M_{k,\mathbb R} \oplus DM_{k-2,\mathbb R} = \widetilde{M}_{k,\mathbb R}^{\leq 1}$. More generally, we have

Theorem. $\widetilde{M}_k^{\leq \frac{k}{2}}=\bigoplus_{j=0}^{\frac{k}{2}-2} D^j M_{k-2j} \oplus \mathbb C D^{\frac{k}{2}-1}E_2$.

and I hope to prove $\widetilde{M}_{k,\mathbb R}^{\leq \frac{k}{2}}=\bigoplus_{j=0}^{\frac{k}{2}-2} D^j M_{k-2j,\mathbb R} \oplus \mathbb R D^{\frac{k}{2}-1}E_2$ but I'm having trouble. Does anyone have an idea?

Answer 1

There is an involution of $\widetilde M_k^{\le l}$, $$i : \; f(\tau) \mapsto \overline{f(-\overline{\tau})}, \quad \sum_{n=0}^{\infty} a_n q^n \mapsto \sum_{n=0}^{\infty} \overline{a_n} q^n$$ which commutes with $D$, whose fixed points are exactly the quasimodular forms with real Fourier coefficients. So if $f \in \widetilde M_{k, \mathbb{R}}^{\ell}$ is decomposed $$f = \sum_{j=0}^{\ell - 2} D^j f_j + c \cdot D^{\ell - 1} E_2, \quad f_j \in M_{k - 2j}, \; c \in \mathbb{C}$$ then $$f = i(f) = \sum_{j=0}^{\ell - 2} D^j i(f_j) + \overline{c} \cdot D^{\ell - 1} E_2.$$

Since the sum $\widetilde M_k^{\le \ell} = \bigoplus_{j = 0}^{\ell - 2} D^j M_{k - 2j} \oplus \mathbb{C} D^{\ell - 1} E_2$ is direct, we have $$f_j = i(f_j) \; \text{for all} \; j \; \text{and} \; c = \overline{c},$$ so $f_j \in M_{k - 2j, \mathbb{R}}$ and $c \in \mathbb{R}$.