TOTALLY REAL AND COMPLEX SUBSPACES OF A RIGHT QUATERNIONIC VECTOR SPACE WITH A HERMITIAN FORM OF SIGNATURE ( n, 1)

We study totally real and complex subsets of a right quaternionic vector space of dimension n+ 1 with a Hermitian form of signature ( n, 1) and extend these notions to right quaternionic projective space. Then we give a necessary and sufficient condition for a subset of a right quaternionic projective space to be totally real or complex in terms of the quaternionic Hermitian triple product. As an application, we show that the limit set of a non -elementary quaternionic Kleinian group I' is totally real (resp. commutative) with respect to the quaternionic Hermitian triple product if and only if I' leaves a real (resp. complex) hyperbolic subspace invariant.