, Volume 15, Issue 4, pp 937-964
Date: 21 Aug 2010

Levi decompositions of a linear algebraic group

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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.