Local transfer for quasi-split classical groups and congruences mod $\ell$ (with an appendix by Guy Henniart)

Let G be the group of rational points of a quasi-split p -adic special orthogonal, symplectic or unitary group for some odd prime number p . Following Arthur and Mok, there are an integer N\geqslant 1 , a p -adic field E and a local functorial transfer from isomorphism classes of irreducible smooth complex representations of G to those of \mathrm{GL}_{N}(E) . By fixing a prime number \ell different from p and an isomorphism between the field of complex numbers and an algebraic closure of the field of \ell -adic numbers, we obtain a transfer map between representations with \ell -adic coefficients. Now consider a cuspidal irreducible \ell -adic representation \pi of G : we can define its reduction mod \ell , which is a semisimple smooth representation of G of finite length, with coefficients in a field of characteristic \ell . Let \pi' be a cuspidal irreducible \ell -adic representation of G whose reduction mod \ell is isomorphic to that of \pi . We prove that the transfers of \pi and \pi' have reductions mod \ell which may not be isomorphic, but which have isomorphic supercuspidal supports. When G is not the split special orthogonal group \mathrm{SO}_{2} , we further prove that the reductions mod \ell of the transfers of \pi and \pi' share a unique common generic component.