Levi decompositions of a linear algebraic group

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 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.