Q1 · Shorten the picture key

Replace the legend paragraph above the proof pictures; this is an editorial request.

Q2 · State the bijection

Replace “indexes” in Definition 1.2 with the displayed map; the existing definition of \(I\) is retained.

Q3 · Finite, affine, and extended Weyl groups

Translation lattices

Zero-sum translations form a proper subgroup of all translations.

1. 1. Separate the three groups

   $$ (t^\lambda w)(t^\mu v)=t^{\lambda+w\mu}wv,\qquad\kappa(t^\lambda w)=\sum_i\lambda_i.$$

   Table 1. Which translations are allowed?

   Group	Translation vector
   \(S_n\)	\(\lambda=0\)
   \(W_{\mathrm{aff}}\)	\(\sum_i\lambda_i=0\)
   \(\widetilde W\)	\(\lambda\in\mathbb Z^n\)

   Proof

   1. Conjugating a diagonal matrix by \(w\) permutes its entries, yielding the multiplication rule.
   2. The coroots of \(\mathrm{GL}_n\) are \(\varepsilon_i-\varepsilon_j\); their integer span is exactly \(Q^\vee\). Permutations preserve coordinate sum, so \(\kappa\) is a surjective homomorphism with the displayed kernel.
   3. For \(n\ge2\), \(t^{\varepsilon_1-\varepsilon_2}\in W_{\mathrm{aff}}\setminus S_n\), while \(t^{\varepsilon_n}\in\widetilde W\setminus W_{\mathrm{aff}}\). Thus neither inclusion is equality.
   Sources

2. 2. Split the determinant quotient

   $$x=\underbrace{x\tau^{-r}}_{u\in W_{\mathrm{aff}}}\tau^r,\qquad r=\kappa(x),\qquad\tau^n=t\,1_n.$$
   Proof

   1. Each full cycle multiplies every basis vector by \(t\), so \(\tau^n=t1_n\). Its determinant has valuation one, giving \(\kappa(\tau^r)=r\).
   2. Hence \(\kappa(x\tau^{-r})=0\). This gives existence; applying \(\kappa\) to two decompositions fixes \(r\), and then fixes \(u\).
   3. The kernel is normal, and \(\langle\tau\rangle\cap\ker\kappa=1\), proving the semidirect product. This complement contains permutation parts when \(n>1\); it is different from the diagonal translation subgroup \(\mathbb Z^n\).
   Sources

Length and rotation

Length counts affine reflections after the length-zero factor has been removed.

1. 3. Define the length

   $$\ell:\widetilde W\to\mathbb Z_{\ge0},\qquad\ell(x)=\dim_k(IxI/I).$$

   For \(n\ge2\), the affine simple reflections for this upper-triangular convention are

   $$s_i=(i\ i+1)\quad(1\le i<n),\qquad s_0=t^{\varepsilon_n-\varepsilon_1}(1\ n).$$ $$\ell_{\mathrm{Cox}}(u)=\min\{m:u=s_{i_1}\cdots s_{i_m},\ i_j\in\{0,\ldots,n-1\}\},\quad u\in W_{\mathrm{aff}}.$$
   Proof

   1. The affine Bruhat-cell theorem gives \(IxI/I\cong\mathbb A^{\ell(x)}\), with the extended Coxeter length. On \(W_{\mathrm{aff}}\), this is the minimum number of the displayed affine simple reflections.
   2. For translations, root-coordinate valuation differences give \(\ell(t^\lambda)=\sum_{i<j}|\lambda_i-\lambda_j|\). For each pair, conjugation moves the two opposite root-subgroup thresholds by opposite integers; the total quotient dimension contributed is their absolute difference.
   3. Thus \(\ell(t^{\varepsilon_n})=n-1\), but \(\ell(t^{(a,\ldots,a)})=0\). A nonzero translation can have either positive or zero length.
   Sources

2. 4. Identify the zero-length elements

   $$L_{nm+a}=t^m(\mathcal O^{n-a}\oplus(t\mathcal O)^a),\qquad m\in\mathbb Z,\quad0\le a<n.$$ $$\tau L_j=L_{j+1},\qquad\tau I\tau^{-1}=I,\qquad\ell(u\tau^r)=\ell_{\mathrm{Cox}}(u).$$

   Example · \(\mathrm{GL}_2\).

   $$s_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad s_0=\begin{pmatrix}0&t^{-1}\\t&0\end{pmatrix},\quad\tau=\begin{pmatrix}0&1\\t&0\end{pmatrix}.$$

   Table 2. Matrices separating the three groups and length zero.

   Element	Membership	Length
   \(s_1\)	\(S_2\)	1
   \(s_0\)	\(W_{\mathrm{aff}}\setminus S_2\)	1
   \(\tau\)	\(\widetilde W\setminus W_{\mathrm{aff}}\)	0
   \(\tau s_1=\operatorname{diag}(1,t)\)	\(\widetilde W\setminus W_{\mathrm{aff}}\)	1
   \(\tau^2=t1_2\)	\(\Omega\)	0

   Proof

   1. The subgroup fixing every lattice of this periodic chain is exactly \(I\): fixing \(L_0\) gives integral invertibility, and fixing the intermediate lattices gives the upper-triangular flag condition modulo \(t\).
   2. The matrix \(\tau\) shifts the chain, so conjugation by it preserves its common stabilizer. Every \(\tau^r\) therefore gives a point cell.
   3. Right translation by \(\tau^r\) induces an isomorphism \(IuI/I\to Iu\tau^rI/I\), because \(\tau^r\) normalizes \(I\). Thus length is unchanged.
   4. For \(D=\tau s_1=\operatorname{diag}(1,t)\) in \(\mathrm{GL}_2\), the stabilizer \(I\cap DID^{-1}\) requires the lower-left entry to lie in \(t^2\mathcal O\), rather than \(t\mathcal O\). Its missing coefficient gives the cell $$ID I/I=\{(1+ctE_{21})DI:c\in k\}\cong\mathbb A^1,$$ where \(E_{21}\) is the matrix unit. Thus \(\ell(D)=1\), while \(I\tau I/I\) is a point.
   5. Use the unique decomposition from step 2. Coxeter length vanishes only at \(u=1\), so \(\ell(x)=0\) holds exactly for \(x\in\langle\tau\rangle\). For \(n=1\), there are no affine reflections and every cell is a point.
   Sources

The convolution argument

The proof uses point cells in every determinant component.

1. 5. Translate the basic object

   $$I\tau^rI=\tau^rI,\qquad\mathrm{IC}_{\tau^r}=\delta_{\tau^rI},\qquad\delta_{\tau^rI}*_IM=(\tau^r)_!M.$$
   Proof

   1. The convolution correspondence over the point \(\tau^rI\) is \(\tau^rI\times^I F^n\cong F^n\), via \(v\mapsto[\tau^r,v]\). The action map becomes \(v\mapsto\tau^rv\), giving the formula.
   2. For the page’s basic object \(\mathrm{IC}_0=\underline{\overline{\mathbb Q}}_{\ell,\mathcal O^n}\), this gives the constant object on \(\tau^r\mathcal O^n\). Under its stated normalization \(S_r=\underline{\overline{\mathbb Q}}_{\ell,\tau^r\mathcal O^n}[-r]\), this is \(S_r[r]\).
   3. When \(n\ge2\), \(S_n\) and \(W_{\mathrm{aff}}\) lie entirely in \(\kappa=0\). They omit the elements \(\tau^r\), \(r\ne0\), used for these point-cell translations. This explains the group distinction; it does not by itself rule out other generation arguments using affine reflections.
   Sources

Proposed edit. Add a “Weyl groups and length” subtab under the \(\mathrm{GL}_n\) page, with the three-group comparison, the two splittings, the geometric and Coxeter definitions of length, and the \(\mathrm{GL}_2\) example above. Reuse the existing explanation where appropriate; the mathematical page and its navigation await approval.

Solve run: gl-n-weyl-groups.

Related request · A self-contained subtab

The answer above includes the requested definition of length; the subtab is a proposed owner edit.