Zeta functions of orthogonal groups of integral positive-definite quadratic forms

This survey concerns representations of Hecke–Shimura rings of integral positive-definite quadratic forms on spaces of polynomial harmonic vectors, and the question of simultaneous diagonalization of the corresponding Hecke operators. Explicit relations are deduced between the zeta functions of the quadratic forms in 2 and 4 variables corresponding to the harmonic eigenvectors of genera 1 and 2, and the zeta functions of Hecke and Andrianov of theta series weighted by these eigenvectors, respectively. Similar questions for single-class quadratic forms were considered earlier in the paper [1]. The general situation is discussed in the paper [2].