Weyl groups and length

Why distinguish permutations, affine reflections, and length-zero translations?

Setup and action

Let \(n\ge1\), \(k=\overline{\mathbb F}_q\), \(\mathcal O=k[[t]]\), \(F=k((t))\), \(G=\mathrm{GL}_n\), \(T\) diagonal and \(B\) upper triangular. The action is \(G\times\mathbb A^n\to\mathbb A^n\), \((g,v)\mapsto gv\), on column vectors; \(e_i\in F^n\) is the standard column basis and \(\varepsilon_i\in\mathbb Z^n=X_*(T)\) the standard cocharacter basis.

$$I(R)=\{g\in\mathrm{GL}_n(R[[t]]):g\bmod t\in B(R)\},\qquad LX(R)=R((t))^n,$$ $$t^\lambda=\operatorname{diag}(t^{\lambda_1},\ldots,t^{\lambda_n}),\quad \tau e_1=te_n,\quad \tau e_i=e_{i-1}\ (i>1),\quad L_a(R)=\tau^aR[[t]]^n.$$

Here \(R\) is any \(\mathbb F_q\)-algebra, \(LX(k)=F^n\), and \(L_a(k)=\tau^a\mathcal O^n\). Geometric objects are base-changed to \(k\) for sheaves; all formulas have models over \(\mathbb F_q\).

Three groups. For \(n\ge2\), the three groups are distinct: permutations, zero-sum translations with permutations, and all translations with permutations.

$$\widetilde W=N_G(T)(F)/T(\mathcal O),\qquad\Omega=\langle\tau\rangle,\qquad\kappa(t^\lambda w)=\operatorname{val}\det(t^\lambda w)=\sum_i\lambda_i.$$$$S_n\ \subsetneq\ W_{\mathrm{aff}}=Q^\vee\rtimes S_n\ \subsetneq\ \widetilde W=\mathbb Z^n\rtimes S_n,\qquad Q^\vee=\{\lambda:\textstyle\sum_i\lambda_i=0\}.$$

Proposition · Two splittings and length. Determinant valuation \(\kappa:\widetilde W\to\mathbb Z\) has kernel \(W_{\mathrm{aff}}\), and

$$\widetilde W=W_{\mathrm{aff}}\rtimes\Omega,\qquad\kappa(\tau)=1,\qquad\ell(u\tau^r)=\ell_{\mathrm{Cox}}(u),\qquad\Omega=\{x:\ell(x)=0\}.$$

For \(n=1\), \(S_1=W_{\mathrm{aff}}=1\), while \(\widetilde W=\Omega\cong\mathbb Z\), with length zero throughout.

Proof plan. Steps 1–2 · Translation latticesSteps 3–4 · Length and rotationStep 5 · Convolution

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 2. Which translations are allowed?
    GroupTranslation 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

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

    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 3. Matrices separating the three groups and length zero.
    ElementMembershipLength
    \(s_1\)\(S_2\)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

Write \(LG(R)=G(R((t)))\); the geometric action map is \(m:LG\times^I LX\to LX\), \([g,v]\mapsto gv\).

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 \(\tau^rI\times^I LX\cong LX\), via \(v\mapsto[\tau^r,v]\). The action map becomes \(v\mapsto\tau^rv\), giving the formula.
    2. For the 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