Iwahori subgroup

In algebra, an Iwahori subgroup is a subgroup of a reductive algebraic group over a nonarchimedean local field that is analogous to a Borel subgroup of an algebraic group. A parahoric subgroup is a proper 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.

Definition

More precisely, Iwahori and parahoric subgroups can be described using the theory of affine Tits buildings. The (reduced) building B(G) of G admits a decomposition into facets. When G is quasisimple the facets are simplices and the facet decomposition gives B(G) the structure of a simplicial complex; in general, the facets are polysimplices, that is, products of simplices. The facets of maximal dimension are called the alcoves of the building.

When G is semisimple and simply connected, the parahoric subgroups are by definition the stabilizers in G of a facet, and the Iwahori subgroups are by definition the stabilizers of an alcove. If G does not satisfy these hypotheses then similar definitions can be made, but with technical complications.

When G is semisimple but not necessarily simply connected, the stabilizer of a facet is too large and one defines a parahoric as a certain finite index subgroup of the stabilizer. The stabilizer can be endowed with a canonical structure of an O-group, and the finite index subgroup, that is, the parahoric, is by definition the O-points of the algebraic connected component of this O-group. It is important here to work with the algebraic connected component instead of the topological connected component because a nonarchimedean local field is totally disconnected.

When G is an arbitrary reductive group, one uses the previous construction but instead takes the stabilizer in the subgroup of G consisting of elements whose image under any character of G is integral.

Examples

• The maximal parahoric subgroups of GL_n(K) are the stabilizers of O-lattices in Kⁿ. In particular, GL_n(O) is a maximal parahoric. Every maximal parahoric of GL_n(K) is conjugate to GL_n(O). The Iwahori subgroups are conjugated to the subgroup I of matrices in GL_n(O) which reduce to an upper triangular matrix in GL_n(k) where k is the residue field of O; parahoric subgroups are all groups between I and GL_n(O), which map one-to-one to parabolic subgroups of GL_n(k) containing the upper triangular matrices.
• Similarly, the maximal parahoric subgroups of SL_n(K) are the stabilizers of O-lattices in Kⁿ, and SL_n(O) is a maximal parahoric. Unlike for GL_n(K), however, SL_n(K) has n conjugacy classes of maximal parahorics.
• When G is commutative, it has a unique maximal compact subgroup and a unique Iwahori subgroup, which is contained in the former. These groups do not always agree. For example, let L be a finite separable extension of K of ramification degree e. The torus L^×/K^× is compact. However, its Iwahori subgroup is O_L^×/O_K^×, a subgroup of index e whose cokernel is generated by a uniformizer of L.

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, S2CID 125864274
• Bruhat, F.; Tits, Jacques (1984), "Groupes réductifs sur un corps local II. Schémas en groupes. Existence d'une donnée radicielle valuée", Publications Mathématiques de l'IHÉS, 60: 5–184, doi:10.1007/BF02700560, ISSN 1618-1913, MR 0756316, S2CID 122178785
• Bruhat, F.; Tits, Jacques (1984), "Schémas en groupes et immeubles des groupes classiques sur un corps local", Bulletin de la Société Mathématique de France, 112: 259–301, doi:10.24033/bsmf.2006, MR 0788969
• 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 (25): 5–48, doi:10.1007/BF02684396, ISSN 1618-1913, MR 0185016, S2CID 4591855
• 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., vol. XXXIII, Providence, R.I.: American Mathematical Society, pp. 29–69, MR 0546588