Levi decompositions of a linear algebraic group

TL;DR

This paper investigates the existence and conjugacy of Levi factors in connected linear algebraic groups over fields of positive characteristic, providing conditions under which Levi factors exist and are conjugate, especially in the context of parahoric group schemes.

Contribution

It offers new sufficient conditions for the existence and conjugacy of Levi factors in linear algebraic groups over fields of positive characteristic, with applications to parahoric group schemes.

Findings

Levi factors exist under certain conditions in positive characteristic.

Any two Levi factors of a parahoric group scheme are geometrically conjugate.

Conditions for Levi factor existence depend on the splitting over unramified extensions.

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. We give in this paper some sufficient conditions for the existence and the conjugacy of Levi factors of G. Let 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 A-group schemes P whose generic fibers P/K coincide with G; these are known as *parahoric group schemes*. The special fiber P/k of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that P/k has a Levi factor, and that any two Levi factors of P/k are geometrically conjugate.