Hecke operators and Lambert series.

Discrete Series Characters and the Lefschetz Formula for Hecke Operators

This paper consists of three independent but related parts. In the ̄rst part (xx1{ 6) we give a combinatorial formula for the constants appearing in the \numerators" of characters of stable discrete series representations of real groups (see x3) as well as an analogous formula for individual discrete series representations (see x6). Moreover we give an explicit formula (Theorem 5.1) for certain ...

Hecke Operators

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 Commuta...

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 generali...

Genus Theta Series, Hecke Operators and the Basis Problem for Eisenstein Series

We derive explicit formulas for the action of the Hecke operator T (p) on the genus theta series of a positive definite integral quadratic form and prove a theorem on the generation of spaces of Eisenstein series by genus theta series. We also discuss connections of our results with Kudla’s matching principle for theta integrals.