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Let $g$ a positive integer greater than 2, $\rho: \GL(g,C)\to \GL(V)$ a finite-dimensional complex representation of $\GL(g,\mathbb{C})$ and $\Gamma$ an arithmetic subgroup of $\GSp(2g,\Q)$.

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 any $\begin{pmatrix} a&b\\c&d\end{pmatrix}\in \Gamma$ and for any $\tau\in\mathcal{H}_g$.

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

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 $M_{\rho}(\Gamma)$. 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 fuctional 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.

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  • Last edited by Fabien Cléry on 2021-05-06 16:23:40
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