A canonical filtration for certain rational modules
Let G be a simple, simply connected algebraic group and let \(B\subset G\) be a Borel subgroup. A filtration of a rational G-module is called a ''good filtration'' if its successive quotients are each obtained by induction from one-dimensional B-modules. The author gives a cohomological characterization of rational G-modules of countable dimension which admit a good filtration; this extends a result of \textit{S. Donkin} [Math. Z. 177, 1-8 (1981; Zbl 0455.20029)] for finite dimensional G-modules. Moreover, the author demonstrates that a rational G-module of arbitrary dimension satisfying Donkin's cohomological condition admits a ''canonical filtration'' associated to a natural partial ordering of induced modules. The usefulness of the author's criterion is demonstrated by a theorem providing a calculation in all degrees of the rational cohomology of a rational G-module obtained by a single Frobenius twist from one with a canonical filtration.
simple, simply connected algebraic group
scientific article; zbMATH DE number 3849485