History

From double cosets to a presentation

The Iwahori–Matsumoto basis answers a question that was first asked for finite groups: if a group is cut into double cosets of a subgroup, what is the multiplication table of the algebra those double cosets span? Each event below changed the answer, and the last one is the use made of it in this repository.

1964 · Hecke rings of finite Chevalley groups Context

Obstacle: For a finite Chevalley group $G(\Fq)$ with Borel subgroup $B(\Fq)$, the endomorphism ring of the permutation representation on $G(\Fq)/B(\Fq)$ is spanned by the double cosets $B(\Fq)wB(\Fq)$, $w\in W$. Its multiplication table was not known in general.
Advance: Iwahori gave generators and defining relations: one generator $S_i$ per simple reflection, the braid relations of $W$, and the quadratic relation $S_i^2=q\cdot1+(q-1)S_i$ [Iwa64, Thm 4.1]. The ring is a $q$-deformation of the group ring of $W$.
What stayed open: The same question for the infinite group $G(F)$ over a local field $F$, and for which subgroup in place of $B(\Fq)$.
Key example and picture: $\SL_2(\Fq)$ acting on the $q+1$ points of the projective line: the nontrivial double coset is the adjacency operator $A$ of the complete graph, and $A^2=(q-1)A+q$ is the quadratic relation. The same count reappears on the tree.
Developed on: Hecke algebras §5, SL₂ II §4

1965 · Iwahori–Matsumoto: the $p$-adic Chevalley group and its Iwahori subgroup Proved (cited)

Obstacle: Over a local field, $G(F)$ is not finite and $G(\cO)$ is too large a subgroup for a Bruhat-type decomposition indexed by a Coxeter group.
Advance: Iwahori and Matsumoto introduce the subgroup $I$ (their $B$), the inverse image of a Borel subgroup under reduction modulo the uniformizer, and prove a Bruhat decomposition $G(F)=\bigsqcup_{w\in\Wt}IwI$ indexed by the extended affine Weyl group [IM65, Thm 2.16]. They compute the index $|IwI/I|=q^{\len(w)}$ [IM65, Prop 3.2], give the relations of the Hecke ring [IM65, Prop 3.4] and its presentation [IM65, Thm 3.5], and describe the whole ring as a twisted tensor product with the group ring of the length-zero subgroup $\Om$ [IM65, Prop 3.8]. The sign character appears on their last page [IM65, §3.2, p. 279].
What stayed open: Non-split groups; the centre of the algebra and its relation to the spherical Hecke algebra of $G(\cO)$; the representation theory of $G(F)$ that the algebra controls.
Key example and picture: The affine Weyl group acting on the Euclidean space $\Xs\otimes\R$ cut into alcoves by the affine root hyperplanes [IM65, §1.3]; the length of $w$ is the number of hyperplanes between the fundamental alcove and its image [IM65, Prop 1.10].
Developed on: the source page, the Concept, Groups §4

1972 · Bruhat–Tits: buildings and Iwahori subgroups of all reductive groups Context

Obstacle: Iwahori–Matsumoto treat Chevalley groups. For a general reductive group over a local field one needs an intrinsic definition of $I$ and of the affine Weyl group.
Advance: Bruhat and Tits build a polysimplicial complex, the building, on which $G(F)$ acts; Iwahori subgroups are stabilizers of chambers, and the affine Weyl group is the group of the apartment [BT72].
What stayed open: Explicit presentations for non-split groups (the parameters $q$ then vary with the reflection).
Key example and picture: For $\SL_2$ the building is a tree in which every vertex has $q+1$ neighbours; $K$ fixes a vertex and $I$ fixes an edge [Ser80, Ch. II, §1.1]. This site draws everything on that tree.
Developed on: SL₂ I: the tree

1976 · Borel: what the algebra controls Context

Obstacle: Why study the Hecke algebra of $I$ rather than of $K$?
Advance: Borel shows that $V\mapsto V^I$ is an equivalence between the smooth representations of $G(F)$ generated by their $I$-fixed vectors and the modules over the Iwahori–Hecke algebra [Bor76, Thm 4.10, Cor 4.11]. Representations with $K$-fixed vectors are among them, but so are others, such as the Steinberg representation, whose $I$-fixed line carries the sign character.
What stayed open: The classification of the simple modules (Kazhdan–Lusztig, 1987) and the structure of the centre.
Key example and picture: The two one-dimensional modules of the algebra: the index character $T_w\mapsto q^{\len(w)}$ and the sign character $T_w\mapsto(-1)^{\len(w)}$ (the Concept §6).
Developed on: the Concept §6, the Iwahori level, the group case

1980s–1989 · Bernstein's presentation and the centre Context

Obstacle: The Iwahori–Matsumoto basis hides the commutative algebra of translations and makes the centre invisible.
Advance: Bernstein (in lectures) and Lusztig in print [Lus89] give a second presentation: $\cH=\C[\Xs]\otimes\Hzero$ as a vector space, with elements $\Th_\lambda$ spanning a copy of the group algebra of $\Xs$ and one relation between $\Th_\lambda$ and $T_s$. The centre is $\C[\Xs]^W$, and it is the spherical Hecke algebra of $K$ through the Satake isomorphism.
What stayed open: Nothing for the split case; the graded version and the geometric realization are later chapters.
Key example and picture: For $\SL_2$: $\Th_{\alpha^\vee}=q^{-1}T_{s_0}T_{s_1}$ up to the normalization recalled on the source page, and the centre is spanned by $\Th_{n\alpha^\vee}+\Th_{-n\alpha^\vee}$.
Developed on: Source: Haines–Kottwitz–Prasad 2010, the Concept §5

2010 · Haines–Kottwitz–Prasad: a self-contained account Proved (cited)

Obstacle: The proofs of Bernstein's presentation, of Macdonald's formula and of the Satake isomorphism were scattered.
Advance: A short exposition proving them from the module $M=\Cc(T(\cO)U(F)\backslash G(F)/I)$, which is free of rank one over $\cH$ [HKP10, Lemma 1.6.1]; it states the Iwahori–Matsumoto relations in the form used here [HKP10, §7.2, p. 20] and identifies the spherical Hecke algebra with the corner $\eK\cH\eK$ [HKP10, §4.1].
What stayed open: Not applicable: an exposition.
Key example and picture: The rank-one computation of the intertwiner for $\SL_2$ [HKP10, §1.12].
Developed on: Source: Haines–Kottwitz–Prasad 2010

2026 · Where this site is used Context

Obstacle: The proof workspace on Iwahori generation compares, for a spherical variety $X$, the part of $\Cc(I\backslash X(F))$ generated by the basic function $\one_{X(\cO)}$ with the whole module. For the group case it needs the facts $\one_K=\sum_{w\in W}T_w$, $[K:I]=\sum_{w\in W}q^{\len(w)}$, the idempotent $\eK$, and the sign character killing $\one_K$.
Advance: These are consequences of the Concept (§5–6); the workspace's classical pages use them [Draft].
What stayed open: See the workspace's to-do list.
Key example and picture: The group $X=H$ with $G=H\times H$: the Iwahori-generated part is the two-sided ideal of $\eK$, which the sign character shows to be proper.
Developed on: Iwahori level, the criterion, the group case