In algebra, an Iwahori subgroup is a subgroup of a reductive algebraic group over a local field

Local field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

 that is analogous to a Borel subgroup

Borel subgroup

In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...

 of an algebraic group. A parahoric subgroup is a subgroup that is a finite union of double cosets of an Iwahori subgroup, so is analogous to a parabolic subgroup of an algebraic group. Iwahori subgroups are named after Nagayoshi Iwahori

Nagayoshi Iwahori

Nagayoshi Iwahori is a Japanese mathematician working on algebraic groups over local fields who introduced Iwahori–Hecke algebras and Iwahori subgroups....

, and "parahoric" is a portmanteau of "parabolic" and "Iwahori". studied Iwahori subgroups for Chevalley groups over p-adic fields, and extended their work to more general groups.

Roughly speaking, an Iwahori subgroup of an algebraic group G(K), for a local field K with integers O and residue field k, is the inverse image in G(O) of a Borel subgroup of G(k).

A reductive group over a local field has a Tits system (B,N), where B is a parahoric group, and the Weyl group of the Tits system is an affine Coxeter group.