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 vertical bar 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- x G(ev) x U+ induces a superalgebra isomorphism phi* : K[U-]circle times K[G(ev)]circle times K[U+]similar or equal to K[G]. We show that C-Gamma = phi* (K[U-]circle times M-Gamma circle times 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) -> 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.