On diagonalization of singular Frobenius operators on Siegel modular forms

A Frobenius operator ║Π(m) maps a Siegel modular form with Fourier coefficients f (A), where A runs over matrices of nonnegative definite integral quadratic forms of given order, to the function with Fourier coefficients f (mA). In cases of Siegel modular forms or cusp forms of integral weights k for the groups Γⁿ₀(q) with Dirichlet characters χ modulo q, the Frobenius operators with m dividing a power of q (singular operators) can be interpreted as Hecke operators on the corresponding spaces and so map the spaces into themselves. It is proved, in particular, that if q is square-free and the character χ² is primitive modulo q, then each space of cusp forms has a basis of common eigenfunctions for all regular Hecke operators and all singular Frobenius operators, and the absolute values of all eigenvalues of ║Π(m) are equal to m^{½(nk-n(n+1)/2)}. The result for n = 1 is due to W. W. Li; n = 2 and a prime level q was obtained in an earlier paper by the author and A. A. Panchishkin.