In algebra, an Iwahori subgroup is a subgroup of a reductive algebraic group over a local field that is analogous to a Borel subgroup 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, and "parahoric" is a portmanteau of "parabolic" and "Iwahori". Iwahori & Matsumoto (1965) studied Iwahori subgroups for Chevalley groups over p-adic fields, and Bruhat & Tits (1972) 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.