Levi Factors of the Special Fiber of a Parahoric Group Scheme and Tame Ramification

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, and let P be a parahoric group scheme over A with generic fiber P/K = G. The special fiber P/k is a linear algebraic group over k. If G splits over an unramified extension of K, we proved in some previous work that the special fiber P/k has a Levi factor, and that any two Levi factors of P/k are geometrically conjugate. In the present paper, we extend a portion of this result. Following a suggestion of Gopal Prasad, we prove that if G splits over a tamely ramified extension of K, then the geometric special fiber P/kalg has a Levi factor, where kalg is an algebraic closure of k.