The Eichler Commutation Relation for theta series with spherical harmonics

It is well known that classical theta series which are attached to positive definite rational quadratic forms yield elliptic modular forms, and linear combinations of theta series attached to lattices in a fixed genus can yield both cusp forms and Eisenstein series whose weight is one-half the rank of the quadratic form. In contrast, generalized theta series—those augmented with a spherical harmonic polynomial—will always yield cusp forms whose weight is increased by the degree of the spherical harmonic. A recent demonstration of the far-reaching importance of generalized theta series is Hijikata, Pizer and Shemanske’s solution to Eichler’s Basis Problem [4] (cf. [2]) in which character twists of such theta series are used to provide a basis for the space of newforms. In this paper we consider theta series with spherical harmonics over a totally real number field. We show that such theta series are Hilbert modular cusp forms whose weight is integral or half-integral, depending on the rank of the associated lattice. We explicitly describe the action of the Hecke operators on these theta series in terms of other theta series, yielding a generalization of the well-known Eichler Commutation Relation. Finally, we use these theta series to construct Hilbert modular forms which are invariant under a subalgebra of the Hecke algebra. We are able to show that if the quadratic form has rankm and the spherical harmonic has degree l, then the theta series attached to the genus of a lattice is identically zero whenever l is small relative to m; in particular, the associated collection of theta series are linearly dependent.