Isometric Actions of Quaternionic Symplectic Groups

Denote by Sp(k, l) the quaternionic symplectic group of signature (k, l).We study the deformation rigidity of the embedding Sp(k, l) Sp(1) H , where H is either Sp(k + 1, l) or Sp(k, l + 1) , this is done by studying a natural non-associative algebra m coming from the affine structure of Sp(1) .We compute the automorphism group of m and as a consecuence of this, we are able to compute the isometry group of Sp(1) at least up to connected components.Using these results, we obtain a uniqueness result on the structure of Sp( 1) together with an isometric left Sp(k, l)-action and classify its finite volume quotients up to finite coverings.Finally, we classify arbitrary isometric actions of Sp(k, l) into connected, complete, analytic, pseudo-Riemannian manifolds of dimension bounded by dim(Sp(1)) that admit a dense orbit.