Equidistribution of Rational Matrices in their Conjugacy Classes

Abstract.

Let G be a connected simply connected almost$\mathbb{Q}$-simple algebraic group with $G \,{\text{: = }}{\mathbf{G}}{\text{(}}\mathbb{R}{\text{)}}$ non-compact and $\Gamma \, \subset \,{\mathbf{G}}_{\mathbb{Q}}$ a cocompact congruence subgroup. For any homogeneous manifold $x_{0} H\, \subset \,\Gamma \backslash G$ of finite volume, and a $a\, \in \,{\mathbf{G}}_{\mathbb{Q}}$, we show that the Hecke orbit T _a (x ₀ H) is equidistributed on $\Gamma \backslash G$ as ${\text{deg}}(a)\, \to \,\infty$, provided H is a non-compact commutative reductive subgroup of G. As a corollary, we generalize the equidistribution result of Hecke points ([COU], [EO1]) to homogeneous spaces G/H. As a concrete application, we describe the equidistribution result in the rational matrices with a given characteristic polynomial.