Associated Graded Structure of the Singular Hecke Order

Write the singular order as

O_sing = Z_3 × R_loc,   R_loc = {(a,b) in Z_3 × Z_3 : a ≡ b mod 3}.

Then

O_sing / J ≅ F_3 × F_3,

with simple characters:

• chi_pair: E=0, N=0
• chi_v7: E=1, N=0

Graded pieces

The singular order has four natural graded module patterns:

• O/J ≅ chi_v7 ⊕ chi_pair
• J/3O ≅ chi_pair
• for every r >= 1, J^r / J^{r+1} ≅ chi_v7 ⊕ chi_pair ⊕ chi_pair
• for every r >= 0, 3^r O / 3^{r+1} O ≅ chi_v7 ⊕ chi_pair ⊕ chi_pair

So the infinite depth does not produce new simple types. It repeats one graded representation pattern forever:

chi_v7 ⊕ 2 chi_pair.

Read

This sharpens the residue-node picture.

• The semisimple head has one split branch chi_v7 and one glued branch chi_pair.
• The first nilpotent shadow J/3O is purely chi_pair.
• Every deeper graded layer is semisimple of the same type chi_v7 ⊕ 2 chi_pair.

So the local singularity has one exceptional first shadow, followed by an eventually periodic graded representation pattern.