A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues.  In a series of papers, Venkatesh and his collaborators proposed an arithmetic reason for this: a hidden degree-shifting action of a certain motivic cohomology group. In this talk, we will explain the setup of this conjecture for coherent cohomology of Hilbert modular forms over $\mathbb{C}$, and give some recent result supporting it. This is joint work with Alex Horawa.