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 $C_{\Gamma}=O_{\Gamma}(K[G])$. Here, the supermodule $C_{\Gamma}$ is the largest subsupermodule of $K[G]$ whose composition factors are irreducible supermodules of highest weight $\lambda$, where $\lambda$ belongs to a finitely-generated ideal $\Gamma$ 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 $\phi^* : K[U^-]\otimes K[G_{ev}]\otimes K[U^+]\simeq K[G]$. We show that $C_{\Gamma}=\phi^*(K[U^-]\otimes M_{\Gamma}\otimes K[U^+])$, where $M_{\Gamma}=O_{\Gamma}(K[G_{ev}])$. Using the basis of the module $M_{\Gamma}$, given by generalized bideterminants, we describe a basis of $C_{\Gamma}$. Since each $C_{\Gamma}$ is a subsupercoalgebra of $K[G]$, its dual $C_{\Gamma}^*=S_{\Gamma}$ is a (pseudocompact) superalgebra, called the generalized Schur superalgebra. There is a natural superalgebra morphism $\pi_{\Gamma}:Dist(G)\to S_{\Gamma}$ such that the image of the distribution algebra $Dist(G)$ is dense in $S_{\Gamma}$. For the ideal $X(T)^+_{l}$, of all weights of fixed length $l$, the generators of the kernel of $\pi_{X(T)^+_{l}}$ are described.