Iwahori subgroup

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.

References

• Bruhat, F.; Tits, Jacques (1972), "Groupes réductifs sur un corps local", Publications Mathématiques de l'IHÉS 41: 5–251, doi:10.1007/bf02715544, ISSN 1618-1913, MR 0327923 
• Iwahori, N.; Matsumoto, H. (1965), "On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups", Publications Mathématiques de l'IHÉS (25): 5–48, ISSN 1618-1913, MR 0185016 
• Tits, Jacques (1979), "Reductive groups over local fields" (PDF), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 29–69, MR 546588