In algebra, an Iwahori subgroup is a subgroup of a

reductive algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

over a nonarchimedean

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...

that is analogous to a

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 general linear group ''GLn'' (''n x n'' invertible matrices), the 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 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 general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...

of an algebraic group. Iwahori subgroups are named after

Nagayoshi Iwahori was a Japanese mathematician who worked on algebraic groups over local fields who introduced Iwahori–Hecke algebras and Iwahori subgroups. Publications * See also *Chevalley–Iwahori–Nagata theorem In mathematics, the Chevalley–Iwahori� ...

, and "parahoric" is a

portmanteau A portmanteau word, or portmanteau (, ) is a blend of wordsTits 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 mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...

; 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 In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...

. 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 ''K''^''n''. 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 sugroups 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 ''K''^''n'', and SL_''n''(''O'') is a maximal parahoric. Unlike for GL_''n''(''K''), however, SL_''n''(''K'') has 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

* * * * *{{Citation , last1=Tits , first1=Jacques , title=Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1 , url=https://www.ams.org/publications/online-books/pspum331-index , publisher=

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, location=Providence, R.I. , series=Proc. Sympos. Pure Math. , mr=546588 , year=1979 , volume=XXXIII , chapter=Reductive groups over local fields , chapter-url=https://web.archive.org/web/20061011070408/https://www.ams.org/online_bks/pspum331/pspum331-ptI-2.pdf , pages=29–69 Linear algebraic groups Representation theory