A basis of eigenfunctions of Hecke operators in the theory of modular forms of genus $n$

Let $\mathfrak M^n_k(\Gamma,\mu)$, where $n,k>0$ are integers, $\Gamma$ is some congruence subgroup of $\Gamma^n=\operatorname{Sp}_n(\mathbf Z)$ and $\mu\colon\Gamma\to\mathbf C^*$ is a congruence-character of $\Gamma$, be the space of all Siegel modular forms of genus $n$, weight $k$ and character $\mu$ with respect to $\Gamma$. In this paper, for a very broad class of congruence subgroups $\Gamma$ of $\Gamma^n$, including all those previously investigated and practically all those groups encountered in applications, the author constructs a sufficiently large commutative ring of Hecke operators, acting on $\mathfrak M^n_k(\Gamma,\mu)$, a canonical decomposition
\begin{equation} \mathfrak M^n_k(\Gamma,\mu)=\bigoplus^n_{r=0}\mathfrak M^{n,r}_k(\Gamma,\mu) \tag{1} \end{equation}
and a canonical inner product $(\,{,}\,)_\Gamma$ on $\mathfrak M^n_k(\Gamma,\mu)$. It is shown that the Hecke operators preserve the canonical decomposition (1) and that they are normal with respect to the canonical inner product $(\,{,}\,)_\Gamma$.
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