Relative symplectic subquadrangle hemisystems of the Hermitian surface

We introduce the notion of relative subquadrangle regular system of a generalized quadrangle. A relative subquadrangle regular system of order m on a generalized quadrangle S of order (s, t) is a set \({\mathcal R}\) of embedded subquadrangles with a prescribed intersection property with respect to a given subquadrangle T such that every point of S T lies on exactly m subquadrangles of \({\mathcal R}\) . If m is one half of the total number of such subquadrangles on a point we call \({\mathcal R}\) a relative subquadrangle hemisystem with respect to T. We construct two infinite families of symplectic relative subquadrangle hemisystems of the Hermitian surface \({{\mathcal H}(3,q^2)}\) , q even.