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Equidistribution of Rational Matrices in their Conjugacy Classes


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.

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Correspondence to Yves Benoist.

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The second author partially supported by DMS 0333397.

Received: May 2005 Revision: March 2006 Accepted: June 2006

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Benoist, Y., Oh, H. Equidistribution of Rational Matrices in their Conjugacy Classes. GAFA, Geom. funct. anal. 17, 1–32 (2007).

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Keywords and phrases:

  • Equidistribution
  • conjugacy classes
  • Hecke operators
  • rational matrices
  • Hecke points

AMS Mathematics Subject Classification:

  • 11D45
  • 37A17
  • 37A25
  • 37A45