Questions and transfer
Test the distinctions: one step or one period, two bases or two generators, algebra or action.
1 · Predict and explain
Compute a negative index
For dimension three, compute \(\Lat_{-1}\) and the result of applying τ.
Expose a misconception
Do the standard and closure bases imply two Hecke-module generators? What happens to the zero stratum?
No. A vector-space basis lists independent vectors; a module generator produces vectors using operators. The vector \(\basic\) generates all lattice indicators under τ. At a finite level the zero stratum lifts to the terminal lattice; on the full field its singleton indicator is not locally constant.
2 · Transfer the criterion
A model with several chains
Suppose a normalizer subgroup permutes a basis of finite-support functions, and the basis labels split into three orbits. How many orbit-representative basis vectors suffice? Can a single basis vector generate all three orbits under this subgroup?
Three representative basis vectors suffice. One basis vector reaches only its own orbit; its subgroup-algebra span misses the other two. This is a statement about the chosen normalizer subgroup, not about what extra reflection operators could do.
Constructive exit task
Choose a G-space and a basic function. Write a five-line proposed test: (1) finite-level strata; (2) normalizer action; (3) translates of the basic support; (4) a finite triangular matrix or a missing basis vector; (5) the additional local-system check needed for sheaves.
Model answer for affine space: use the lattice quotients; τ shifts their indices; the translates are all nested lattice indicators; the matrix against shells plus the terminal lattice has diagonal one; connected stabilizers leave only the constant local systems. The chain then reaches every simple up to shift.