A sufficient criterion

· The data to check Definition

Let \(\Grp\) be a split connected reductive group over a non-Archimedean local field \(\Loc\), and \(\Iw\) an Iwahori subgroup. Let \(X(\Loc)\) be a locally compact left \(\Grp(\Loc)\)-space, \(\Module=C_c^\infty(X(\Loc))^{\Iw}\), and \(\basic\in\Module\) a chosen basic function. Let \(\Gamma\subseteq N_{\Grp(\Loc)}(\Iw)/\Iw\) be a subgroup. Suppose

$$\Module=\varinjlim_{\nu}\LevelModule{\nu},\qquad \dim\LevelModule{\nu}<\infty.$$

Each \(\LevelModule{\nu}\) has a basis \(e_{\nu,1},\ldots,e_{\nu,d_{\nu}}\) of indicators of finite-level strata. The ordered labels include every terminal stratum.

· Translate coverage plus triangularity Proved here

For each level \(\nu\), assume there are \(\gamma_{\nu,1},\ldots,\gamma_{\nu,d_{\nu}}\in\Gamma\) whose translates lie in \(\LevelModule{\nu}\) and satisfy

$$T_{\gamma_{\nu,j}}*\basic=\sum_{i=1}^{d_{\nu}}A_{\nu,ij}e_{\nu,i}.$$ $$A_{\nu,ij}=0\quad(i<j),\qquad A_{\nu,jj}\ne0.$$

Then

$$\boxed{\mathbb C[\Gamma]*\basic=\Module,\qquad \Hecke*\basic=\Module.}$$

Proof

1. Normalizer representatives make the Hecke operator an ordinary translation: $$T_{\gamma}*\basic(x)=\basic(\gamma^{-1}x).$$
2. The finite matrix is invertible: $$\det A_{\nu}=\prod_{j=1}^{d_{\nu}}A_{\nu,jj}\ne0.$$
3. Every level, hence every function, lies in the generated module: $$\LevelModule{\nu}=\operatorname{span}_j\{T_{\gamma_{\nu,j}}*\basic\}\subseteq\mathbb C[\Gamma]*\basic.$$

· Translation reaches every simple Proved here

Let \(\Category\) be a triangulated category with an action of the geometric Iwahori–Hecke category and a chosen object \(\BasicSheaf\). Assume:

1. Every object comes through an exact fully faithful functor from a bounded finite-level category whose perverse heart has finite length; the level functors preserve the stated objects and triangles.
2. For every simple object \(\Simple\) at every level, there are \(\gamma\in\Gamma\) and \(d\in\mathbb Z\) such that

$$\Simple\cong\gamma_!\BasicSheaf[d],\qquad \delta_\gamma*_I\BasicSheaf\cong\gamma_!\BasicSheaf.$$

Here \(\delta_\gamma\) is the point kernel on the normalizer coset, and \(\gamma_!:\Category\to\Category\) is translation. Then

$$\boxed{\Category=\operatorname{Thick}\{\delta_\gamma*_I\BasicSheaf:\gamma\in\Gamma\}.}$$

Proof

1. Translate coverage places every simple in the proposed thick subcategory: $$\Simple[-d]\cong\delta_\gamma*_I\BasicSheaf.$$
2. A Jordan–Hölder filtration builds each finite-length perverse object from these simples by extensions, hence triangles.
3. Bounded perverse truncation triangles build each finite-level complex from its finitely many perverse cohomology objects: $$ {}^p\tau_{\le j-1}K\longrightarrow{}^p\tau_{\le j}K\longrightarrow{}^pH^j(K)[-j]\xrightarrow{+1}.$$
4. Every object belongs to some level, so the finite-level conclusion gives the asserted equality.

· A smooth affine symmetric counterexample Proved here

Put \(\BaseGroup=\mathrm{GL}_2\), take its upper-triangular Iwahori \(\BaseIwahori\), and let \(\MaxCompact=\mathrm{GL}_2(\Ring)\). The smooth affine variety \(X=\BaseGroup\) is spherical and symmetric for \(\Grp=\BaseGroup\times\BaseGroup\), acting by \((g_1,g_2)x=g_1xg_2^{-1}\); its stabilizer at the identity is the diagonal subgroup. Set

$$\GroupHecke=C_c(\BaseIwahori\backslash\BaseGroup(\Loc)/\BaseIwahori).$$

For the ordinary finite simple reflection \(s\) of GL₂, the basic function on this integral model is

$$\basic=\mathbf1_{\MaxCompact}=T_e+T_s.$$

Both factors have the length-zero shifts of the model, but the generated module is the proper two-sided ideal

$$\Hecke*\basic=\GroupHecke(T_e+T_s)\GroupHecke.$$ $$\boxed{\GroupHecke(T_e+T_s)\GroupHecke\subsetneq\GroupHecke.}$$ $$\GroupHecke=C_c^\infty(X(\Loc))^{\BaseIwahori\times\BaseIwahori}.$$

Proof

1. The finite Bruhat decomposition of GL₂ over the residue field has two cells, so

$$\MaxCompact=\BaseIwahori\sqcup\BaseIwahori s\BaseIwahori.$$

2. The Iwahori–Hecke presentation gives a unital algebra character \(\signchar:\GroupHecke\to\mathbb C\): send every affine simple generator to \(-1\), and every length-zero generator to \(1\).

$$\signchar(T_s)=-1,\qquad\signchar(T_\omega)=1\quad(\omega\in\Omega_{\BaseGroup}).$$

The quadratic relations vanish at −1, the braid relations hold for identical scalars, and conjugation by Ω preserves these values. [Sht, Corollary 2.33]

3. Left and right Hecke operators produce exactly the displayed two-sided ideal: inversion in the second factor changes the right-convolution kernel and permutes the full Hecke algebra. This ideal is killed by the character, while the unit is not:

$$\signchar(T_e+T_s)=0,\qquad\signchar(T_e)=1.$$

Therefore \(T_e=\mathbf1_{\BaseIwahori}\) cannot lie in the generated module.