Sato-Tate equidistribution for families of Hecke-Maass forms on SL(n,R)/SO(n)

We establish the Sato-Tate equidistribution of Hecke eigenvalues on average for families of Hecke–Maass cusp forms on SL(n,R)/SO(n). For each of the principal, symmetric square and exterior square L-functions we verify that the families are essentially cuspidal and deduce the level distribution with restricted support of the low-lying zeros. We also deduce average estimates toward Ramanujan.