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Filtrations of $$G$$-modules. (English) Zbl 0748.20026
Let $$G$$ be a semi-simple, simply connected algebraic group over an algebraically closed field $$K$$. Let $$B$$ be a Borel subgroup of $$G$$. For a character $$\lambda: B\to\mathbb{G}_ m$$, let $$L_ \lambda$$ denote the associated line bundle on $$G/B$$. Let $$F(\lambda)=H^ \circ(G/B,L_ \lambda)$$. If characteristic of $$k$$ is zero, then for a dominant character $$\lambda$$, $$F(\lambda)$$ is an irreducible $$G$$-module, and in fact any finite dimensional $$G$$-module is of the form $$F(\lambda)$$, for some dominant $$\lambda$$; further any finite dimensional $$G$$-module $$M$$ is completely reducible, i.e., $$M$$ is a direct sum of irreducible $$G$$- modules. This is no longer the case if characteristic of $$k\neq 0$$. The author proves that (when $$\text{char }k\neq 0$$) the $$G$$-module $$F(\lambda)\otimes F(\mu)$$, for $$\lambda$$, $$\mu$$ dominant, has a good filtration (here, a filtration of a $$G$$-module $$M$$ is said to be good if each subquotient is isomorphic to some $$F(\lambda)$$). This result was first proved by Donkin for $$G$$ not containing any component of type $$E_ 7$$, $$E_ 8$$. Donkin’s proof is based on a case by case analysis. Also Donkin’s proof is long. In this paper the author gives a uniform proof for $$G$$ of any type. This paper makes an important contribution to the Theory of Algebraic Groups and Representation theory.

MSC:
 20G05 Representation theory for linear algebraic groups 20G15 Linear algebraic groups over arbitrary fields 14M17 Homogeneous spaces and generalizations
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