Length-zero elements and the lattice chain

How does the rotation supply all the affine-space generators?

1 · Fix the subgroup convention

· The rotation normalizes the Iwahori Proved here

For the upper-triangular Iwahori and the rotation defined on the hub,

$$\rot\Iwa\rot^{-1}=\Iwa,\qquad\rot^n=\unif\,1_n.$$

Proof

Set $L_0=\OO^n$, $L_r=\OO^{n-r}\oplus(\unif\OO)^r$ for $0\le r<n$, and $L_{nm+r}=\unif^mL_r$.
An integral invertible matrix preserves these lattices precisely when its reduction preserves the flag spanned by the first $1,2,\ldots,n$ basis vectors; this is the upper-triangular Iwahori.
The rotation sends $L_j$ onto $L_{j+1}$ for every $j$. Conjugating a matrix which fixes every lattice by this rotation still fixes every lattice.

$$\rot\Iwa\rot^{-1}=\Iwa,\qquad\Iwa\rot\Iwa=\rot\Iwa,\qquad\ell(\rot)=0.$$

· Every index, including negative indices Proved here

Write $k=nm+r$, with $m\in\ZZ$, $0\le r<n$, using Euclidean division.

$$\boxed{\LLk=\rot^k\OO^n=\unif^m\big(\OO^{n-r}\oplus(\unif\OO)^r\big).}$$ $$L_{k+1}\subset L_k,\qquad\dim_{\resfield}(L_k/L_{k+1})=1.$$

Proof

For $0\le r<n$, rotate the first $r$ coordinates to the end, multiplying each by $\unif$.
Use $\rot^{nm}=\unif^m1_n$. This is valid for negative $m$, since the matrices are invertible over $\FF$.
Exactly one coordinate threshold rises at each step, so the quotient is a one-dimensional residue-field vector space.

· The length-zero complement Proved here

$$\kap|_{\Om}:\Om\xrightarrow{\sim}\ZZ,\qquad\Om=\{\rot^k:k\in\ZZ\},\qquad\Wt=\Wa\rtimes\Om.$$

Proof

The affine Weyl group acts simply transitively on the alcoves in the reduced apartment; consequently $\Wa\cap\Om=1$. This is the affine Coxeter input of [HR, Lem. 14].
Every extended element takes the base alcove to an alcove. An affine Weyl element carries that alcove back, giving $\Wt=\Wa\Om$.
The quotient by $\Wa$ is $\ZZ$ via $\kap$. Since $\kap(\rot)=1$, every class has exactly one representative $\rot^k$ in $\Om$.

$$g=(g\rot^{-\kap(g)})\rot^{\kap(g)},\qquad g\rot^{-\kap(g)}\in\Wa.$$

Normalization and extension to $LX$ are supplied by Lemma IIa. In the affine-space application,

$$S_k=\overline{\mathbb Q}_{\ell,L_k}[-k],\qquad G_{\rot^k}=\overline{\mathbb Q}_{\ell,L_k}=S_k[k].$$

Because a length-zero Schubert cell is a point, convolution translates the basic sheaf; IV gives the categorical proof.