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

The paper contains results that characterize the Donkin-Koppinen filtration of the coordinate superalgebra $K[G]$ of the general linear supergroup $G=GL(m\mathrm{\mid }n)$ by its subsupermodules $C_{\mathrm{\Gamma }}=O_{\mathrm{\Gamma }}(K[G])$. Here, the supermodule $C_{\mathrm{\Gamma }}$ 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 $\mathrm{\Gamma }$ 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 ${\varphi }^{\ast }:K[U^{-}]\otimes K[G_{ev}]\otimes K[U^{+}]\simeq K[G]$. We show that $C_{\mathrm{\Gamma }}={\varphi }^{\ast }(K[U^{-}]\otimes M_{\mathrm{\Gamma }}\otimes K[U^{+}])$, where $M_{\mathrm{\Gamma }}=O_{\mathrm{\Gamma }}(K[G_{ev}])$. Using the basis of the module $M_{\mathrm{\Gamma }}$, given by generalized bideterminants, we describe a basis of $C_{\mathrm{\Gamma }}$. Since each $C_{\mathrm{\Gamma }}$ is a subsupercoalgebra of $K[G]$, its dual $C_{\mathrm{\Gamma }}^{\ast }=S_{\mathrm{\Gamma }}$ is a (pseudocompact) superalgebra, called the generalized Schur superalgebra. There is a natural superalgebra morphism ${\pi }_{\mathrm{\Gamma }}:Dist(G)\to S_{\mathrm{\Gamma }}$ such that the image of the distribution algebra $Dist(G)$ is dense in $S_{\mathrm{\Gamma }}$. 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.