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

TL;DR

This paper proves that nearly holomorphic vector-valued Siegel modular forms exhibit moderate growth by establishing a specific eigenvalue bound, which has implications for their analytic behavior on symplectic groups.

Contribution

It introduces a new bound on vector-valued Siegel modular forms, demonstrating their moderate growth property through eigenvalue estimates.

Findings

Established a growth bound involving eigenvalues of Y.

Proved the moderate growth property for lifted modular forms.

Connected growth estimates to the highest weight of the representation.

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: $\|\rho(Y^{1/2})F(Z) \| \ll \prod_{i=1}^n (\mu_i(Y)^{\lambda_1/2} + \mu_i(Y)^{-\lambda_1/2})$ where $ \lambda_1 \ge \ldots \ge \lambda_n$ is the highest weight of $\rho$ and $\mu_i(Y)$ are the eigenvalues of the matrix $Y$.