Siegel modular form (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, and let $\Gamma \leq \GSp(2g,\Q)$ be an arithmetic subgroup.

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

The arithmetic subgroups of $\GSp(2g,\Q)$ that we focus on are the following ones:

• integral symplectic group: $\Sp(2g,\Z)$
• Congruence subgroups of $\Sp(2g,\Z)$
• paramodular group: $\Gamma^{\mathrm{para}}(d)\subset \Sp(2g,\Q)$

Note in the case of (classical) modular forms, a holomorphicity condition at all the cusps of $\Gamma$ is required in the definition. Due to the so-called Koecher principle such a condition is superfluous for Siegel modular form of degree $g>1$.

For each fixed choice of $\rho$ and $\Gamma$, the set of modular forms of weight $\rho$ on $\Gamma$ is a finite-dimensional complex-vector space denoted by $M_{\rho}(\Gamma)$. Moreover if the representation $\rho$ is the direct sum of two representations $\rho=\rho_1\oplus \rho_2$ then the space $M_{\rho}(\Gamma)$ is isomorphic to $M_{\rho_1}(\Gamma)\oplus M_{\rho_2}(\Gamma)$ and we can restrict ourselves to irreducible representations of $\GL(g,\C)$.

If the highest weight of the representation $\rho$ is $(k,\ldots,k)$ i.e. $\rho=\det^{\otimes k}$ (note that $V$ is one-dimensional in this case) then the previous functional equation takes the form \[ f((a\tau+b)(c\tau+d)^{-1})=\det(c\tau+d)^kf(\tau) \] and $f$ is called a scalar-valued Siegel modular form. The space of such forms is simply denoted by $M_k(\Gamma)$. When $\dim(V)>1$, we usually talk about vector-valued Siegel modular forms.