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 $\rho$ 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: $\Vert \rho (Y^{1/2})F(Z)\Vert \ll {\prod }_{i=1}^n({\mu }_i(Y{)}^{{\lambda }_1/2}+{\mu }_i(Y{)}^{-{\lambda }_1/2})$ where ${\lambda }_1\ge \dots \ge {\lambda }_n$ is the highest weight of $\rho$ and ${\mu }_i(Y)$ are the eigenvalues of the matrix $Y$.

7 pages