### Manin’s and Peyre’s conjectures on rational points and adelic mixing

Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$. We prove Manin’s conjecture on the asymptotic (as $T\to \infty $) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X\left(\mathbb{A}\right)$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $\mathbf{G}\left(K\right)$ on $X\left(\mathbb{A}\right)$. Our approach is based on the mixing property of ${L}^{2}(\mathbf{G}\left(K\right)\setminus \mathbf{G}\left(\mathbb{A}\right))$ which we obtain with a rate of convergence.