A characterization of groups of parahoric type

TL;DR

This paper generalizes the known relationship between parahoric subgroups of reductive groups over local fields and reductive groups over residue fields by characterizing a broader class of groups called groups of parahoric type.

Contribution

It introduces and characterizes groups of parahoric type, extending the classical correspondence to a wider class of linear algebraic groups under certain conditions.

Findings

Any group of parahoric type is k-isomorphic to a quotient of a parahoric subgroup by its h-th congruence subgroup.

The classical isomorphism between parahoric quotients and reductive groups over k is generalized.

The paper establishes conditions under which these isomorphisms hold.

Abstract

Let F be a local henselian nonarchimedean field of residual field k, and let G be the group of F-points of a connected reductive group defined over F. It is well-known that the quotient of any parahoric subgroup of G by its first congruence subgroup is isomorphic to the group of k-points of a reductive group defined over k, and conversely; in this paper, we generalize this result by studying a class of linear algebraic groups named groups of parahoric type; we prove that, under certain conditions, any such group is k-isomorphic to the quotient of a parahoric subgroup of some reductive group over F by its h-th congruence subgroup for a suitable h.