Transfers for non-stable K₁-functors of classical type

Let k be a field. Let G be an absolutely almost simple simply connected k-group of type A_l, l>=2, or D_l, l>=4, containing a 2-dimensional split torus. If G is of type D_l, assume moreover that char k is different from 2. We show that the Nisnevich sheafification of the non-stable K_1-functor K_1^G, also called the Whitehead group of G, on the category of smooth k-schemes is A^1-invariant, and has oriented weak transfers for affine varieties in the sense of Panin-Yagunov-Ross. If k has characteristic 0, this implies that the Nisnevich sheafification of K_1^G is birationally invariant. We also prove a rigidity theorem for \A1-invariant torsion presheaves with oriented weak transfers over infinite fields. As a corollary, we conclude that K_1^G(R)=K_1^G(k) whenever R is a Henselian regular local ring with a coefficient field k.