Multiplicative arithmetic of theta-series of odd quadratic forms

We study the action of the operators of symplectic Hecke rings of arbitrary degree on the theta-series of positive definite quadratic forms in an odd number of variables with vector-valued spherical coefficients corresponding to irreducible representations of the unitary group. We find a correspondence between generators of the Hecke rings and generalized Eichler–Brandt matrices. We apply these results to obtain conditions for linear dependence of theta-series, necessary conditions for lifting automorphic eigenforms on the orthogonal group to Siegel modular eigenforms, and an Euler expansion for symmetric Dirichlet series as a product of local zeta-functions with coefficients computed explicitly in terms of Eichler–Brandt matrices.