On interaction of symplectic and orthogonal Hecke–Shimura rings of one-class quadratic forms

Transformation formulas of theta-series with harmonic polynomials of one-class quadratic forms under Hecke operators are interpreted as a result of interaction of standard representation of symplectic Hecke–Shimura ring on theta-series with natural representation of orthogonal Hecke–Shimura ring on the same theta-series considered as invariants of quadratic forms. Properties of the interaction maps and their relations with action of Hecke operators are considered.