Equidistribution of Hecke eigenforms on the Hilbert modular varieties

Let F be a totally real number field with ring of integers O, and let Γ=SL(2,O) be the Hilbert modular group. Given the orthonormal basis of Hecke eigenforms in S2k(Γ), one can associate a probability measure dμk on the Hilbert modular variety Γ\Hn. We prove that dμk tends to the invariant measure on Γ\Hn weakly as k→∞. This generalizes Luo's result [W. Luo, Equidistribution of Hecke eigenforms on the modular surface, Proc. Amer. Math. Soc. 131 (2003) 21–27] for the case F=Q.