Semistandard filtrations in highest weight categories.
The author defines the new concept of a semistandard filtration in a highest weight category. Basic properties of semistandard filtrations are presented. For any semistandard filtration of a module \(M\), a semistandard multiplicity associated to a standard module in \(M\) is defined, and proved that it is independent on the choice of the semistandard filtration. The special case of semistandard filtrations of maximal submodules of standard modules is considered, and a characterization of the semistandard multiplicities in terms of the dimensions of \(\text{Ext}^1\) groups between irreducible modules is given. Applications to some general inequalities on the behavior of \(\text{Ext}^1\) in the presence of a suitable exact functor are given. Specific results about \(\text{Ext}^1_G(L,L')\) are proved for irreducible modules \(L,L'\) of a semisimple algebraic group \(G\). These are related to known issues for \(\text{Ext}^n\) and parity conditions involving standard and irreducible modules with singular high weights in the presence of the Lusztig conjecture for representations of \(G\) in positive characteristics.
scientific article; zbMATH DE number 5566087
scientific article