EIGENVALUES OF HECKE OPERATORS ON HILBERT MODULAR GROUPS

Let F be a totally real field, let I be a nonzero ideal of the ring of integers O-F of F, let Gamma(0)(I) be the congruence subgroup of Hecke type of G = Pi(d)(j=1) SL2(R) embedded diagonally in G, and let chi be a character of Gamma(0)(I) of the form chi ((a)(b) (c)(d)) = chi(d), where d -> x(d) is a character of O-F modulo I. For a finite subset P of prime ideals p not dividing I, we consider the ring H-I, generated by the Hecke operators T(p(2)), p is an element of P (see 3.2) acting on (T, chi)-automorphic forms on G. Given the cuspidal space L-xi(2,cusp) (Gamma(0)(I)\G, chi), we let V-omega, run through an orthogonal system of irreducible G-invariant subspaces so that each V-omega is invariant under H-I. For each 1 <= j <= d, let lambda(omega) = (lambda(omega,j)) be the vector formed by the eigenvalues of the Casimir operators of the d factors of G on V-omega, and for each p is an element of P, we take lambda(omega,p) is an element of J(p) := [0,1 + N(p)) boolean OR i(0, root 1 N(p)(2)]) so that lambda(2)(pi,p) - N(p) is the eigenvalue on V-pi of the Hecke operator T(p(2)). If for some prime p the Hecke operator T(p) can be defined then its eigenvalue on V-pi is real and equal to lambda(pi,p) or -lambda(omega,p). For each family of expanding boxes t -> Omega(t), as in (3) in R-d, and fixed interval J(p) in J(p), for each p is an element of P, we consider the counting function N(Omega(t); (J(p))(p is an element of P)) := (pi, lambda pi is an element of Omega t) (:) Sigma(lambda omega,p is an element of Jp) , (for all p is an element of P) vertical bar Cr(pi)vertical bar(2). In the main result in this paper, Theorem 1.1, we give, under some mild conditions on the Omega(t), the asymptotic distribution of the function N(Omega(t); (J(p))(p is an element of P)), as t -> infinity. We show that at the finite places outside I the eigenvalues of the Hecke operator T(p2) are equidistributed compatibly with the Sato-Tate measure, whereas at the archimedean places the eigenvalues lambda(pi) are equidistributed with respect to the Plancherel measure. As a consequence, if we pick an infinite place 1 and we prescribe lambda(pi,j) is an element of Omega(j) for all infinite places j not equal 1 and lambda(pi,p) is an element of J(p) for all finite places p in P for fixed sets and fixed intervals J(p) subset of J(p), with positive measure and then allow lambda(pi,l) to run over larger and larger regions, then there are infinitely many representations pi in such a set, and their positive density is as described in Theorem 1.1.