Equidistribution of Rational Matrices in their Conjugacy Classes

Let G be a connected simply connected almost ℚ-simple algebraic group with G: = G(ℝ) non-compact and Γ ⊂ G_ℚ a cocompact congruence subgroup. For any homogeneous manifold x_0H ⊂ Γ∖G of finite volume, and a a ∈ G_ℚ, we show that the Hecke orbit Ta(x_0H) is equidistributed on Γ∖G as deg(a) → ∞, provided H is a non-compact commutative reductive subgroup of G. As a corollary, we generalize the equidistribution result of Hecke points ([COU], [EO_1]) to homogeneous spaces G/H. As a concrete application, we describe the equidistribution result in the rational matrices with a given characteristic polynomial.