A Note on the Growth of Nearly Holomorphic Vector-Valued Siegel Modular Forms

Abstract

Let F be a nearly holomorphic vector-valued Siegel modular form of weight ρ 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 λ ₁ ≥… ≥ λ _n is the highest weight of ρ and μ _i (Y ) are the eigenvalues of the matrix Y .