Relative hemisystems on the Hermitian surface

Let S be a generalized quadrangle of order (q ²,q) containing a subquadrangle S′ of order (q,q). Then any line of S either meets S′ in q+1 points or is disjoint from S′. After Penttila and Williford (J. Comb. Theory, Ser. A 118:502–509, 2011), we call a subset H of the lines disjoint from S′ a relative hemisystem of S with respect to S′, provided that for each point x of S?S′, exactly half of the lines through x disjoint from S′ lie in H. A new infinite family of relative hemisystems on the generalized quadrangle $\mathcal{H}(3,q^{2})$ admitting the linear group PSL(2,q) as an automorphism group is constructed. The association schemes arising from our construction are not equivalent to those arising from the Penttila–Williford relative hemisystems.