  We verify in an elementary way a result of Peterson for the maximal
orthogonal and Lagrangian Grassmannians, and then find Vafa-Intriligator type
formulas which compute their 3-point, genus zero Gromov-Witten invariants.
Finally we study total positivity of the related Peterson varieties and show
that Rietsch's conjecture about the total positivity holds for these cases.
