Tame cuspidal representations in non-defining characteristics
Let k be a non-archimedean local field of odd residual characteristic p. Let G be a (connected) reductive group that splits over a tamely ramified field extension of k. We revisit Yu's construction of smooth complex representations of G(k) from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. 
Moreover, we show that an analogous construction yields smooth, irreducible, cuspidal representations over an arbitrary algebraically closed field R of characteristic different from p and that this construction provides all smooth, irreducible, cuspidal R-representations if p does not divide the order of the Weyl group of G.