Homomorphisms of unitary groups

In Problem XVII from [4], Weisfeiler indicates a general formulation for homomorphisms between classical groups in terms of a congruence subgroup coming from an induced integral structure in the group. As he observes, certainly not all homomorphisms arise in this manner, but this formulation provides a good foundation for extending the fairly well understood situation for isomorphisms. We establish here Weisfeiler's conjecture for unitary groups, and homomorphisms which do not kill involutions. For monomorphisms this problem is usually solved via group theoretical arguments using derived groups and centralizers, but for homomorphisms these methods are hard to apply and it seems necessary to use a more geometrical approach. In the proof here the central new idea involves moving subspaces in a controlled manner, intersecting them and counting dimensions. It should also be possible to use this technique in other situations. Let V and W be right vector spaces over the division rings k and K with involutions (both denoted by *) where