Let $F$ be a nearly holomorphic vector-valued Siegel modular form of weight $\rho$ with respect to some congruence subgroup of $\mathrm{ Sp}_{ 2n}(\mathbb Q)$. In this note, we prove that the function on $\mathrm{ Sp}_{ 2n}(\mathbb R)$ obtained by lifting $F$ 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 $Y$.
Categories: math.NT