On Some Free Algebras of Automorphic Forms

It is proved that, for $n=8,9,10$, the natural algebra of automorphic forms of the group $O^+_{2,n}(\mathbb{Z})$ acting on the $n$-dimensional symmetric domain of type IV is free, and the weights of generators are found. This extends results obtained in the author's previous paper for $n\le 7$. On the other hand, as proved in a recent joint paper of the author and O. V. Shvartsman, similar algebras of automorphic forms cannot be free for $n>10$.