Longest weight vectors and excellent filtrations
Let B be a Borel subgroup in a connected simply connected semisimple algebraic group over an algebraically closed field. In the hope of some day explaining why some features of the representation theory of B are independent of the characteristic, we now develop some homological algebra related with the existence of certain filtrations of B modules. Compare with the ``good filtrations'' of \textit{S. Donkin} [Rational Representations of Algebraic Groups (Lect. Notes Math. 1140, 1985; Zbl 0586.20017)], or the ``excellent filtrations'' of \textit{P. Polo} [Variétés de Schubert et excellentes filtrations. (Preprint 1987)]. See also \textit{A. Joseph} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 389-419 (1985; Zbl 0589.22014)]. In particular, we find a special filtration of the algebra of regular functions on B, viewed as a B bimodule.
connected simply connected semisimple algebraic group
scientific article; zbMATH DE number 4046020