Why an extended group?
What had to be added to the finite Weyl group?
1 · Permutations and valuations
The finite Weyl group of GLₙ permutes diagonal entries; loops introduce integral valuations as additional data.
$$\Wf=S_n,\qquad\Xco=\ZZ^n,\qquad\Wt=\Xco\rtimes\Wf.$$Compute the two kinds of data in Definitions.
2 · 1965 · Iwahori and Matsumoto
Iwahori and Matsumoto developed a Bruhat decomposition and Hecke-ring description for p-adic Chevalley groups.
[IM, bibliographic record]For GLₙ, the motivating picture is a periodic lattice chain: preserving its individual lattices gives the Iwahori subgroup, while a rotation can shift the chain.
The two-dimensional matrices make that distinction visible.
3 · 2008 · Haines and Rapoport
Haines–Rapoport separate the normalizer quotient, its double-coset interpretation, and its two semidirect products.
[HR, Def. 7; Props. 8, 13; Lem. 14]Their framework also covers nonsplit groups; these pages calculate only the split GLₙ case.