Reduction theory for a rational function field

Details Reduction theory for a rational function field Reduction theory for a rational function field

2003-10-18

arxiv.org/abs/math/0310289v1

Proc. Indian Acad. Sci. (Math. Sci.), Vol. 113, No. 2, May 2003, pp. 153-163

Mathematics

Representation Theory

10 pages

Scientific paper

10.1007/BF02829764

Let $G$ be a split reductive group over a finite field $\Fq$. Let $F=\Fq(t)$ and let $\A$ denote the ad\`eles of $F$. We show that every double coset in $G(F)\bsl G(\A)/ K$ has a representative in a maximal split torus of $G$. Here $K$ is the set of integral ad\`elic points of $G$. When $G$ ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.

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