The Iwahori–Weyl group in Haines–Rapoport
Which hypotheses and conventions are being imported?
1 · Three located inputs
Definition 7 / Proposition 8: normalizer quotient and Iwahori double cosets. Proposition 13: splitting at a special vertex. Lemma 14: affine Weyl subgroup and alcove stabilizer.
[HR, pp. 4, 8–9]The source allows a general connected reductive group over a strictly henselian valued field; here \(\resfield((\unif))\) has algebraically closed residue field and \(G=\mathrm{GL}_n\) is split.
2 · Dictionary to the source
| Haines–Rapoport | These pages | Meaning |
|---|---|---|
| \(L\) | \(\FF\) | valued field |
| \(K_0\) | \(\Iwa\) | Iwahori subgroup |
| \(W_a\) | \(\Wa\) | affine Weyl group |
| \(X_*(T)_I\) | \(\ZZ^n\) | cocharacters; inertia acts trivially |
The source's letter \(I\) in a subscript means inertia, not our Iwahori subgroup.
3 · Derive the split quotient
A diagonal loop is a product of a diagonal unit and a unique valuation vector.
$$T(\FF)/T(\OO)\simeq\ZZ^n,\qquad N_G(T)(\FF)/T(\FF)\simeq S_n.$$Permutation matrices split the latter quotient; the multiplication rule proved in Definitions therefore yields
$$\Wt\simeq\ZZ^n\rtimes S_n.$$The determinant valuation kills precisely the zero-sum translations and all permutations, giving the second quotient used in Properties.
4 · Scope of the transfer
The integer lattice, determinant quotient, explicit \(\rot\), and infinite cyclic \(\Om\) are features of split GLₙ; they are not a restatement for every reductive group.
Our concrete matrix action fixes the sign of translation. Building conventions can reverse that sign; the formulas in these pages always mean multiplication by the displayed matrix.