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

Abstract

Let \(\cal{A}\) 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 \(\cal{P}\) be a parahoric group scheme over \(\cal{A}\) with generic fiber \({\cal{P}}_{/K} = G\). The special fiber \({\cal{P}}_{/k}\) 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 \({\cal{P}}_{/k}\) has a Levi factor, and that any two Levi factors of \({\cal{P}}_{/k}\) 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 \({\cal{P}}_{/k_{\rm{alg}}}\) has a Levi factor, where k alg is an algebraic closure of k.