Purity for big Galois representations

In 1980s, Hida constructed p-adic families of ordinary cusp forms. He showed that the Fourier expansions of the ordinary normalized Hecke eigen cusp forms can be p-adically interpolated. Moreover, their associated Galois representations can also be interpolated via a big Galois representation. The Galois representations associated to cusp forms are known to be pure. This suggests a notion of purity for big Galois representations. In this talk, we will discuss this notion and explain its role in the study of variation in p-adic families.