Iwahori generation · Proposition IV.2

The group

Where the conditions of Proposition III fail for \(X=H\): at I3, on the closed pieces.

1 · The case

\(X=H\) under \(H\times H\), \((h_1,h_2)\cdot x=h_1xh_2^{-1}\), with \(W\ne1\) (S2). As in Proposition II, the Hecke action is two-sided:

$$\mathcal C_I=\mathcal D_c(I\backslash LH/I),\qquad c:=\mathrm{IC}_0=\underline k_{K/I},\qquad\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathcal A\ast c\ast\mathcal B:\ \mathcal A,\mathcal B\in\mathcal H_{c,I}\}.$$

Proposition II proves \(\mathcal C_I^{\mathrm{Iwa}}\ne\mathcal C_I\). This page locates the failure among the conditions of Proposition III: it is I3, at the closed pieces.

$$\text{closed piece }I\omega I\ \longrightarrow\ \text{only }Y=c\text{ allowed}\ \longrightarrow\ A\simeq\textstyle\bigoplus_s\Delta_\omega[d_s]\in\mathcal C_I^{\mathrm{Iwa}}\ \longrightarrow\ \Phi(A)=0\ne\Phi(\Delta_\omega)\ \longrightarrow\ \text{contradiction}$$

2 · Definitions
  • Orbits, closures and closed pieces $$LH=\bigsqcup_{w\in\widetilde W}IwI,\qquad\overline{IwI}=\bigsqcup_{v\le w}IvI,\qquad\widetilde W=W_{\mathrm{aff}}\rtimes\Omega .$$

    \(\le\) is the Bruhat order, so every orbit closure is a finite union of orbits and, for \(M\gg0\), the pieces of a level are the double cosets \(IwI\) it contains. The closed ones are \(I\omega I=\omega I\), \(\omega\in\Omega\) of length zero, one in each connected component of \(LH\).

  • The functor \(\Phi\) of Proposition II

    \(\Phi=\mathrm{Av}^r_\psi:\mathcal C_I\to\mathcal W\simeq\mathrm{Vect}^{\Omega}\), right Whittaker averaging (II, definitions). It is exact, commutes with left convolution, \(\Phi(\mathcal A\ast\mathcal F)=\mathcal A\ast\Phi(\mathcal F)\), and

    $$\mathcal C_I^{\mathrm{Iwa}}\subseteq\ker\Phi,\qquad\Phi(\Delta_e)\ne0$$

    by Proposition II.

3 · Proposition IV.2

Proposition IV.2 · The groupProved

Assume S2. For every \(\omega\in\Omega\), no object of \(\mathcal C_I^{\mathrm{Iwa}}\) is a nonzero sum of shifts of \(\Delta_\omega\):

$$\boxed{\ A\in\mathcal C_I^{\mathrm{Iwa}},\ \ \operatorname{supp}A\subseteq I\omega I,\ \ A|_{I\omega I}\simeq\bigoplus_{s=1}^r\underline k[d_s]\ \Longrightarrow\ r=0 .\ }$$

In particular I3 fails at every closed piece \(I\omega I\).

Proof

Step 1 reduces I3 at a closed piece to the boxed claim; step 2 proves it with \(\Phi\); step 3 is the function shadow.

  1. Method At a closed piece, I3 asks for such an \(A\).

    $$P=I\omega I\ \text{closed}\ \Longrightarrow\ Y=c,\quad A=\mathcal G\ast c\in\mathcal C_I^{\mathrm{Iwa}},\quad A\simeq\bigoplus_{s=1}^r\Delta_\omega[d_s].$$

    Proof
    • No piece lies below a closed piece, so the only admissible \(Y\) in I3 is \(\mathrm{IC}_0=c\), and \(A=\mathcal G\ast c\) lies in \(\mathcal C_I^{\mathrm{Iwa}}\).
    • \(\operatorname{supp}A\subseteq I\omega I\), which is closed, so \(A=i_*(A|_{I\omega I})\simeq\bigoplus_s\Delta_\omega[d_s]\).
  2. Result \(\Phi\) separates \(\Delta_\omega\) from \(\mathcal C_I^{\mathrm{Iwa}}\).

    $$0=\Phi(A)\simeq\bigoplus_{s=1}^r\Phi(\Delta_\omega)[d_s],\qquad\Phi(\Delta_\omega)\ne0\ \Longrightarrow\ r=0 .$$

    Proof
    • \(A\in\mathcal C_I^{\mathrm{Iwa}}\subseteq\ker\Phi\), and \(\Phi\) is exact.
    • \(\Delta_\omega\) is invertible, \(\Delta_{\omega^{-1}}\ast\Delta_\omega=\Delta_e\), and \(\Phi\) is left \(\mathcal H_{c,I}\)-linear: \(\Phi(\Delta_e)=\Delta_{\omega^{-1}}\ast\Phi(\Delta_\omega)\). Since \(\Phi(\Delta_e)\ne0\), \(\Phi(\Delta_\omega)\ne0\).
  3. Result Functions: the sign character.

    $$\varepsilon(T_\omega)=1,\qquad\varepsilon\big(\mathcal H_Ie_K\mathcal H_I\big)=0\ \Longrightarrow\ a\,T_\omega\in\mathcal H_Ie_K\mathcal H_I\ \text{only for }a=0 .$$

    Proof
    • \(\varepsilon(T_w)=(-1)^{\ell(w)}\) (II, step 1) and \(\ell(\omega)=0\).
    • The class of \(A\) is \(\big(\sum_s(-1)^{d_s}\big)T_\omega\), which can vanish with \(r>0\) (take \(\Delta_\omega\oplus\Delta_\omega[1]\)); so the sheaf statement needs \(\Phi\), not \(\varepsilon\).
Remarksonly I3 fails · two-sided action · tori · contrast with \(\mathbb A^n\)