Non-symmetric Cauchy kernels for the classical groups

We give non-symmetric versions of the Cauchy kernel and Littlewood’s kernels, corresponding to the types An, Bn, Cn and Dn, of the classical groups. We show that these new kernels are diagonal in the basis of two families of key polynomials (one of them being Demazure characters) obtained as images of dominant monomials under isobaric divided differences. We define scalar products such that the two families of key polynomials are adjoint to each other.