Equality of Linear and Symplectic Orbits
  It is shown that the set of orbits of the action of the elementary symplectic transvection group on all unimodular elements of a symplectic module over a commutative ring of characteristic not 2 is identical with the set of orbits of the action of the corresponding elementary transvection group. This result is used to get improved injective stability estimates for $K_1$ of the symplectic transvection group over a non-singular affine algebras. 
Pratyusha Chattopadhyay and Ravi A. Rao
math.AC