Rational characteristic classes of bundles with fibre a product of spheres

We prove the existence of many non-trivial characteristic classes of smooth oriented bundles with fibre a product $S^n\times S^n$ of odd-dimensional spheres. We do so by proving injectivity of the map from the ring of rational characteristic classes of oriented fibrations with fibre $S^n\times S^n$; the latter is proven by Berglund–Zeman to be isomorphic to the group cohomology of the symmetric powers of the standard representation of a certain finite-index subgroup $\mathrm{\Gamma }$ of ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$. These characteristic classes of smooth bundles are not generalised Miller–Morita–Mumford classes, and they exist in arbitrarily large cohomological degrees. Inspired by an example given by Morita, we provide a collection of smooth oriented $S^n\times S^n$-bundles, indexed by cyclic subgroups of $\mathrm{\Gamma }$, which detect any given non-zero characteristic class of such fibrations.