archives@tulane.edu / Let G be a connected reductive complex algebraic group. We study the inclusion posets of diagonal G-orbit closures in a product of two partial flag varieties. In this dissertation, we show some results for G=SL_n and G=SO_{2n}. If the diagonal action is of complexity zero, then the poset is a graded lattice. If the diagonal action is of complexity one, then the poset is isomorphic to one of a finite number of posets that we determine explicitly. / 1 / Tien Minh Le