Length zero and Iwahori generation
How an alcove symmetry becomes a generator of functions and sheaves.
1 · The setup
Let \(\Loc\) be a non-Archimedean local field with valuation ring \(\Ring\), uniformizer \(\unif\), and residue cardinality \(\resq\); fix an integer \(\nDim\ge2\).
$$\Grp=\mathrm{GL}_{\nDim},\qquad X(\Loc)=\Loc^{\nDim},\qquad \Iw=\{g\in\mathrm{GL}_{\nDim}(\Ring):\bar g\text{ upper triangular}\}.$$ $$\Hecke=C_c(\Iw\backslash\Grp(\Loc)/\Iw),\qquad\operatorname{vol}(\Iw)=1.$$ $$\Module=C_c^\infty(X(\Loc))^{\Iw},\qquad \basic=\mathbf1_{\Ring^{\nDim}}.$$HKP identifies a group-algebra part of the Hecke algebra coming from the length-zero subgroup Ω.
[HKP, §7.1, p. 19]The question is when its action on the chosen basic function already spans the whole module.
2 · The phenomenon to explain
· Affine space is generated by its lattice shift Proved here
Define \(\shift:\Loc^{\nDim}\to\Loc^{\nDim}\) by \(\shift e_1=\unif e_{\nDim}\) and \(\shift e_i=e_{i-1}\) for \(2\le i\le\nDim\). Put \(\Lat_{\idx}=\shift^{\idx}\Ring^{\nDim}\) for every \(\idx\in\mathbb Z\), and let \(\Om=\langle\shift\rangle\) be the length-zero subgroup.
$$T_{\shift^{\idx}}*\basic=\mathbf1_{\Lat_{\idx}}.$$ $$\boxed{\mathbb C[\Om]*\basic=\Module=\Hecke*\basic.}$$There is one module generator, \(\basic\). The two finite-level bases are the shell indicators with the terminal stratum, and the nested lattice indicators.
Proof
The chain-stabilizer proof gives length-zero translation; the finite-level basis proof gives spanning and independence.
3 · A path through the question
The lattice model is the smallest moving example. The history page explains the sources; the criterion separates the proved sufficient condition from the research question.
4 · What to be able to do
- Recover τ from the periodic lattice chain and distinguish one step from one period.
- Derive the length-zero convolution operator from its single double coset.
- Convert shell values into a finite sum of lattice indicators.
- State the support and local-system hypotheses needed to generalize generation.
- Compare the HKP, Chen–Yi and Shtotland results without extending their conclusions beyond their hypotheses.
Proved here The model and the sufficient criteria. Proved (cited) The named literature results. Question Recognition of generation from weighted orbit data in a specified broader class.
5 · Cost of these pages
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