Reductification of parahoric group schemes

Abstract

Parahoric group schemes are certain possibly non-reductive, smooth, affine integral models of reductive group schemes defined over a henselian discretely valued field $K$ whose residue field is perfect. We show that any such group scheme $\mathscr{P}$ becomes reductive, in a particular regard, after a (possibly wildly ramified) finite Galois extension $L/K$. More precisely, we prove that there exists a reductive integral model $\mathscr{G}$ of the base change $\mathscr{P}_L$ such that $\mathscr{P}$ can be recovered as the smoothening of the subgroup of Galois invariants of the Weil restriction of $\mathscr{G}$. Our work extends results of Balaji--Seshadri and Pappas--Rapoport from the tamely ramified and simply-connected semisimple setting. As an application, we establish a parahoric analogue of the Grothendieck--Serre conjecture in sufficiently good residue characteristics. Specifically, we confirm that generically trivial parahoric torsors are trivial whenever the generic reductive group is simply-connected. The proof proceeds by reducing the problem to a statement about a stacky reductive group over a stacky discrete valuation ring.