Induced Modules and Extensions of Representations, Ii
نویسندگان
چکیده
Grothendieck [7] showed that finite dimensional vector bundles over the projective line split into direct sums of line bundles (see also Andersen [1]). We begin this paper, after two preliminary sections, by proving this result (Theorem (3.5)) for induced bundles using representation theoretic techniques. The proof yields partial results in the infinite dimensional case and leads to the definition (3.0) of a unique decreasing filtration of any rational B-module V, where B is the Borel subgroup of a connected affine algebraic group P of semisimple rank one. This filtration is described in terms of the B-module structure of V (§§3, 5) and, in case V is finite dimensional, completely determines the homogeneous components of the associated vector bundles (on P/B) when the latter is written as a direct sum of line bundles. We call this the canonical filtration of V.
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