Action of Hecke Operators on Siegel Theta Series I

We apply the Hecke operators T (p) and T̃j(p 2) (1 ≤ j ≤ n, p prime) to a degree n theta series attached to a rank 2k Z-lattice L, n ≤ k, equipped with a positive definite quadratic form in the case that L/pL is hyperbolic. We show that the image of the theta series under these Hecke operators can be realized as a sum of theta series attached to certain closely related lattices, thereby generalizing the Eichler Commutation Relation (similar to some work of Freitag and of Yoshida). We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We show the eigenvalue for T (p) is (k − n, n), and the eigenvalue for T ′ j(p 2) (a specific linear combination of T0(p), . . . , Tj(p 2)) is pj(k−n)+j(j−1)/2β(n, j) (k − j, j) where β(∗, ∗), (∗, ∗) are elementary functions (defined below).