ACTION OF HECKE OPERATORS ON SIEGEL THETA SERIES, II

Abstract

We apply the Hecke operators T(p) 2 and [Formula: see text](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 [Formula: see text], 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 [Formula: see text] for j > k when χ(p) = 1, and for j ≥ k when χ(p) = -1, and that θ( gen L) is an eigenform for T(p) 2 .