On factorization of Hecke polynomials for the sympletic group of genus $n$

We consider the problem of finding relations between the Fourier coefficients of Siegel modular forms which are eigenfunctions for the Hecke operators, and the corresponding eigenvalues. We obtain a qualitative solution of the problem (Theorem 8.1). The method is based on studying connections between representations of the Hecke rings of the symplectic group and the special linear group on spaces of Fourier coefficients of Siegel modular forms.
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