  We formulate a conjectured orthogonality relation between the Fourier
coefficients of Maass forms on PGL(N) for N>=2. Based on the work of
Goldfeld-Kontorovich and Blomer for N=3, and on our conjecture for N>=4, we
prove a weighted vertical equidistribution theorem (with respect to the
generalized Sato-Tate measure) for the Satake parameter of Maass forms at a
finite prime. For N=3, the rate of convergence for the equidistribution theorem
is obtained.

Number Theory