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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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