Siegel modular form with character (awaiting review)

Let $g\ge 2$ be an integer, let $\rho\colon \GL(g,\mathbb{C})\to \GL(V)$ be a finite-dimensional complex representation, $\Gamma \leq \GSp(2g,\Q)$ be an arithmetic subgroup and $\psi$ a character of $\Gamma$.

A holomorphic map $f$ on the Siegel upper half space $\mathcal{H}_g$ taking values in $V$ is called a Siegel modular form with character $\psi$ and weight $\rho$ on $\Gamma$ if \[ f((a\tau+b)(c\tau+d)^{-1})=\psi(\gamma)\rho(c\tau+d)f(\tau) \]
for all $\gamma=\begin{pmatrix} a&b\\c&d\end{pmatrix}\in \Gamma$ (with $a,b,c,d \in \mathrm{M}(g,\mathbb{Q})$) and all $\tau\in\mathcal{H}_g$.

The space of Siegel modular forms with character $\psi$ and weight $\rho$ on $\Gamma$ is denoted by $M_{\rho}(\Gamma,\psi)$ and its subspace of cusp forms by $S_{\rho}(\Gamma,\psi)$.