Donkin-Koppinen filtration for GL(m|n) and generalized Schur superalgebras

Abstract

The paper contains results that characterize the Donkin-Koppinen filtration of the coordinate superalgebra $K[G]$ of the general linear supergroup $G=GL(m|n)$ by its subsupermodules $C_Γ=O_Γ(K[G])$. Here, the supermodule $C_Γ$ is the largest subsupermodule of $K[G]$ whose composition factors are irreducible supermodules of highest weight $λ$, where $λ$ belongs to a finitely-generated ideal $Γ$ of the poset $X(T)^+$ of dominant weights of $G$. A decomposition of $G$ as a product of subsuperschemes $U^-\times G_{ev}\times U^+$ induces a superalgebra isomorphism $ϕ^* : K[U^-]\otimes K[G_{ev}]\otimes K[U^+]\simeq K[G]$. We show that $C_Γ=ϕ^*(K[U^-]\otimes M_Γ\otimes K[U^+])$, where $M_Γ=O_Γ(K[G_{ev}])$. Using the basis of the module $M_Γ$, given by generalized bideterminants, we describe a basis of $C_Γ$. Since each $C_Γ$ is a subsupercoalgebra of $K[G]$, its dual $C_Γ^*=S_Γ$ is a (pseudocompact) superalgebra, called the generalized Schur superalgebra. There is a natural superalgebra morphism $π_Γ:Dist(G)\to S_Γ$ such that the image of the distribution algebra $Dist(G)$ is dense in $S_Γ$. For the ideal $X(T)^+_{l}$, of all weights of fixed length $l$, the generators of the kernel of $π_{X(T)^+_{l}}$ are described.