On local descent for unitary groups

We study the local descent from irreducible, supercuspidal, self-conjugate representations of GL2n(E)GL2n(E) to irreducible, supercuspidal and generic representations of the quasi-split unitary group U2n(F)U2n(F) in 2n   variables, corresponding to a quadratic extension E/FE/F of p  -adic fields. We construct the descent and prove that it is nontrivial, supercuspidal, generic and irreducible. We write the relations with poles of local gamma factors and functoriality. We also consider representations as above of GL2n+1(E)GL2n+1(E).