Action of Hecke Operators on Siegel Theta Series Ii

We apply the Hecke operators T (p) and T ′ j (p) (1 ≤ j ≤ n ≤ 2k) to a degree n theta series attached to a rank 2k Z-lattice L equipped with a positive definite quadratic form in the case that L/pL is regular. We explicitly realize the image of the theta series under these Hecke operators as a sum of theta series attached to certain sublattices of 1 p L, thereby generalizing the Eichler Commutation Relation. We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We explicitly compute the eigenvalues on the average theta series, extending previous work where we had the restrictions that χ(p) = 1 and n ≤ k. We also show that θ(L)|T ′ j (p) = 0 for j > k when χ(p) = 1, and for j ≥ k when χ(p) = −1, and that θ(genL) is an eigenform for T (p). §