Reductification of parahoric group schemes

TL;DR

This paper demonstrates that parahoric group schemes over henselian discretely valued fields become reductive after a finite Galois extension, and establishes a parahoric analogue of the Grothendieck--Serre conjecture in good residue characteristics.

Contribution

It extends the reductification results of parahoric group schemes to wildly ramified extensions and proves a new case of the Grothendieck--Serre conjecture for parahoric torsors.

Findings

Existence of a reductive integral model after Galois extension

Reduction of triviality of torsors to stacky reductive groups

Extension of previous tamely ramified results to wildly ramified cases

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.