A note on the growth of nearly holomorphic vector-valued Siegel modular forms

Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $\rho$ with respect to some congruence subgroup of $$\mathrm{S}\mathrm{p}${2n}(\mathbb Q)$\mathrm{.}Inthisnote,weprovethatthefunctionon$$\mathrm{S}\mathrm{p}${2n}(\mathbb R)$obtainedbylifting$F$ has the moderate growth (or “slowly increasing”) property. This is a consequence of the following bound that we prove: $|\rho(Y^{1/2})F(Z) | \ll \prod_{i=1}^n (\mu_i(Y)^{\lambda_1/2} + \mu_i(Y)^{-\lambda_1/2})$where$ \lambda_1 \ge \ldots \ge \lambda_n$ is the highest weight of $\rho$ and ${\mu }_i(Y)$ are the eigenvalues of the matrix $Y$.