A combinatorial approach to Donkin-Koppinen filtrations of general linear supergroups

For a general linear supergroup (Formula presented.) we consider a natural isomorphism (Formula presented.) where G_ev is the even subsupergroup of G, and U ^–, U ⁺ are appropriate odd unipotent subsupergroups of G. We compute the action of odd superderivations on the images (Formula presented.) of the generators of (Formula presented.) extending results established in [8] and [7]. We describe a specific ordering of the dominant weights (Formula presented.) of (Formula presented.) for which there exists a Donkin-Koppinen filtration of the coordinate algebra (Formula presented.) Let (Formula presented.) be a finitely generated ideal (Formula presented.) of (Formula presented.) and (Formula presented.) be the largest (Formula presented.) -subsupermodule of (Formula presented.) having simple composition factors of highest weights (Formula presented.) We apply combinatorial techniques, using generalized bideterminants, to determine a basis of G-superbimodules appearing in Donkin-Koppinen filtration of (Formula presented.) considered initially in [9].