Venkatesh has recently formulated a series of conjectures, in collaboration with Galatius, Harris and Prasanna, which aim to explain the presence of the same system of eigenvalues in several cohomological degrees of a bounded symmetric domain.

The simplest non-trivial examples of this phenomenon arise in the scenario of classical modular forms of weight 1, and Bianchi modular forms over an imaginary quadratic field. Venkatesh’s conjectures explain this phenomenon by means of a graded action of a derived Hecke algebra, which can be recast as a Selmer group.