Une preuve courte du principe de Selberg pour un groupe $p$-adique

In 1992, Blanc and Brylinski showed the following property for a $p$-adic group $G$, called the ``abstract Selberg principle'': the orbital integrals on conjugacy classes of non-compact elements of the Hattori rank of a finitely generated projective smooth representation of $G$ vanish. The proof is by explicit computations of ``low'' level ($0$ and $1)$ cyclic and Hochschild cohomologies. Here we intend to show that this property is actually a direct consequence of two facts: Clozel's integration formula (which leads us to assume the defining characteristic to be zero) and the triviality of the action of unramified characters on the $K_0$ of $G$ (which is also proven here, using a standard $K$-theoretic argument due to Grothendieck).