, Volume 15, Issue 4, pp 937-964

Levi decompositions of a linear algebraic group

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Abstract

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G.

Let $ \mathcal{A} $ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth $ \mathcal{A} $ -group schemes $ \mathcal{P} $ whose generic fibers $ {{\mathcal{P}} \left/ {K} \right.} $ coincide with G; these are known as parahoric group schemes. The special fiber $ {{\mathcal{P}} \left/ {K} \right.} $ of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that $ {{\mathcal{P}} \left/ {K} \right.} $ has a Levi factor, and that any two Levi factors of $ {{\mathcal{P}} \left/ {K} \right.} $ are geometrically conjugate.

Dedicated to the memory of Vladimir Morozov and to his contributions to mathematics
Research of the author was supported in part by the US NSA award H98230-08-1-0110.