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We define $G = \GSp^+(2g,\Q)$ by $$G = \{\gamma\in \GL_{2g}(\Z):\gamma^t J\gamma=r(\gamma)J,\text{ for all }\gamma\in\Bbb{Q}_+\}.$$ Let $L(\Gamma,G)$ be the free $\C$-module generated by the right cosets $\Gamma\alpha$ where $\alpha\in\Gamma \backslash G$. Note $\Gamma$ acts on $L(\Gamma,G)$ by right multiplication and we set $\mathcal{H}_g(\Gamma,G)=L(\Gamma,G)^\Gamma$.

Let $T_1,T_2\in \mathcal{H}_g(\Gamma,G)$ and $T_i = \sum_{\alpha_i \in \Gamma \setminus G} c_i(\alpha) \Gamma\alpha.$ Then $T_1 T_2 = \sum_{\alpha,\alpha'\in \Gamma \setminus G} c_1(\alpha)c_2(\alpha')\Gamma\alpha\alpha'$.

As in the classical case, we pay most attention to the Hecke operators at a prime $p$. It is known that $\mathcal{H}_g = \bigotimes_{p\text{ prime}} \mathcal{H}_{g,p}$ where the construction of the local Hecke algebra $\mathcal{H}_{g,p}$ is the same as before but with $G$ replaced with $G_p = G\cap \text{GL}_{2g}(\Z[p^{-1}])$. The generators of this local algrebra $\mathcal{H}_{g,p}$ are the double cosets $T(p)=\Gamma\text{diag}(I_g;pI_g)\Gamma$ and $T_i(p^2)=\Gamma \text{diag}(I_i,pI_{g-i};p^2I_i,pI_{g-i})\Gamma$ for $1\leq i \leq g$. Some authors also define $T_0(p^2)$, too. The operator $T(p^2)=\sum_{i=1}^g T_i(p^2)$.

The space $\mathcal{H}_g$ acts on Siegel modular forms of degree $g$ and weight $k$ by $F|_k\left(\sum c_i\Gamma\alpha_i\right)=\sum c_i F|_k\alpha_i$ where $\left(F|_k \alpha\right)(Z)=r(\alpha)^{gk-\frac{g(g+1)}{2}}\det(CZ+D)^{-k}F\left(\alpha\cdot Z \right)$. Some authors use a different normalization in this definition. A Hecke eigenform is a form in $M_k(\Gamma)$ which is a simultaneous eigenform for all the operators $T(p)$, $T(p^2)$,...,$T(p^g)$.

Knowl status:
  • Review status: beta
  • Last edited by John Voight on 2018-06-28 01:01:31
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