Equidistribution of rational matrices in their conjugacy classes
Let \({\mathbf G}\) be a connected simply connected almost \({\mathbb Q}\)-simple algebraic \({\mathbb Q}\)-group with \(G={\mathbf G}_{\mathbb R}\) non-compact and \(\Gamma\) a cocompact congruence subgroup in \({\mathbf G}_{\mathbb Q}\). Let \(\mu_X\) be the \(G\)-invariant probability measure on \(X:=\Gamma \setminus G\) and \(\pi:G\rightarrow X\) the standard projection. For any \(a\in {\mathbf G}_{\mathbb Q}\) the set \(\Gamma a \Gamma\) is a finite union of distinct \(\Gamma\)-orbits: \[ \Gamma a \Gamma=\bigcup_{1\leq i\leq \deg(a)}\Gamma a_i, \] where \(\deg(a)\) is the degreee of \(a\). For \(x=\pi(g)\in X\), the finite set \[ T_a(x):=\pi(a\pi^{-1}(x))=\{\pi(a_ig)\mid i=1,\ldots,\deg(a)\} \] is the Hecke orbit of \(x\). Let \(H\) be a closed commutative reductive non-compact subgroup of \(G\) and \(\lambda _H\) a left Haar measure on \(H\). For any \(H\)-orbit \(Y:=x_0H\subset X\) with finite volume the Hecke orbit of \(Y\) is \[ T_a(Y):=\pi(a\pi^{-1}(Y)). \] Let \(\lambda_a\) be the unique \(H\)-invariant measure on \(X\) supported on \(T_a(Y)\) which is locally equal to \(\lambda_H\) on each \(H\)-orbit in \(T_a(Y)\). The authors prove that for any \(H\)-orbit \(x_0H\subset X\) with finite volume, the Hecke orbit \(T_a(x_0H)\) becomes equidistributed with respect to \(\mu_X\) as \(\deg(a)\to\infty.\) This means that for any \(f\in C_c(X)\) one has \[ \lim_{\deg(a)\to\infty}\frac{1}{\lambda_a(X)}\int_Xf \,d\lambda_a=\int_Xf \,d\mu_X. \] As a concrete application they describe the equidistribution result in the rational matrices with a given characteristic polynomial.
scientific article; zbMATH DE number 5152555