L The actual enlarged rank-two inducing carrier
On this page · Visual guide: Low-rank repair enlarges an actual compact group · Permalink · Source line 2
Status: ordinary proof. The construction below supplies an actual arithmetic finite Heisenberg pair and exact recovery of an already fixed representation. It does not change the previously audited zero-dimensional BK quotient at rank two. No Lean verification or H5 comparison is claimed.
L.1 1. Exact setting and result
On this page · Permalink · Source line 11
Let F=F_2((t)) and G=GL2(F). Let [A,q,0,beta] be a minimal simple stratum with K=F[beta] totally ramified of degree2 and A its complete period-two hereditary order. Assume
q=2m+1>=3, m>=1.Write R=rad(A), \(U_A^a=1+R^a\) for a>=1, and \(U_K^a=1+\mathfrak p_K^a\). Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
J=J^1=H^1=U_K^1 U_A^(m+1).Fix the full actual extension Lambda of theta to
N=K×J,and the already fixed irreducible supercuspidal candidate
Pi=c-Ind_N^G Lambda.Lambda is one-dimensional here. Its value on a uniformizer of K, including the noncompact scalar normalization, is part of the fixed data.
Define the larger compact open arithmetic subgroup
P=U_K^1 U_A^m.Then the following statements hold.
J is normal of index2 in P, and P∩N=J.
E=Ind_J^P theta is an actual irreducible two-dimensional P-representation.
There is an explicitly defined normal subgroup H_new⊂J of index2 such that P/H_new≅F2² and theta_new=theta| H_new has nondegenerate commutator pairing on that quotient. E is its Heisenberg representation.
E embeds into the restriction of the same Pi, as the subspace of inducing functions supported on NP.
With X=c-Ind_J^G theta, a uniformizer varpi_K and s=Lambda(varpi_K), there is an exact quotient
Pi ≅ X/(T−s)X, (Tf)(g)=f(varpi_K g).Induction in stages identifies X with c-Ind_P^G E. This gives an exact same-candidate recovery using the retained noncompact scalar, not merely a nonzero cyclic-generation assertion.
The new Heisenberg pair is an actual pair of compact subgroups of GL2(F), but is not asserted to be the BK \(H^1/J^1\) pair of a maximal simple type. Whether the original H2 realization relation allows this enlargement is a separate obligation.
L.2 2. The compact enlargement is a group, with normal subgroup of index two
On this page · Permalink · Source line 90
K× normalizes A and every power of its radical, so U_K^1 normalizes U_A^m. Therefore P is a subgroup and is compact open. It is contained in U_A^1 because m>=1 and \(U_K^1\subset U_A^1\).
The standard radical calculation gives
[U_A^a,U_A^b]⊂U_A^(a+b), a,b>=1.Indeed, for x∈R^a and y∈R^b, the difference (1+x)(1+y)−(1+y)(1+x) is in R^(a+b), and multiplication by unit inverses preserves that ideal.
Consequently
[U_A^m,U_K^1]⊂U_A^(m+1),
[U_A^m,U_A^(m+1)]⊂U_A^(2m+1)⊂U_A^(m+1).These facts, and normalization by U_K^1, show that J is normal in P. They also show
[P,J]⊂U_A^(m+1). (1)To calculate the index, use
U_A^m∩J=U_K^m U_A^(m+1).For if ku∈U_A^m with k∈U_K^1 and u∈U_A^(m+1), then \(k\in K^\times\cap U_A^m=U_K^m\). The reverse inclusion is immediate. The complete period-two A-grade \(R^m/R^{m+1}\) has F2-dimension2, and its field grade \(\mathfrak p_K^m/\mathfrak p_K^{m+1}\) has dimension1. Thus
[P:J]=[U_A^m:U_K^m U_A^(m+1)]=2. (2)Finally, if kj∈P with k∈K× and j∈J, then \(k\in Pj^{-1}\subset U_A^1\). Hence k∈U_K^1 and kj∈J. Therefore
P∩N=J. (3)These arguments use the actual group intersections. They do not replace P or J by a vector-space surrogate.
L.3 3. Irreducibility and the conjugate-character ratio
On this page · Permalink · Source line 150
For the minimal simple character, the primary BK intertwining theorem gives
I_G(theta)=J K×J=N.Here K× normalizes J and theta. In particular, for g∈P\J, (3) says g does not intertwine theta. Because J is normal in P, this means that the two characters
theta,
theta^g(j)=theta(gjg^−1)are distinct. The index-two induction E=Ind_J^P theta is therefore irreducible. This also follows directly from the two distinct J-weight lines, which are interchanged by g.
Define the actual character of J
lambda(j)=theta^g(j)/theta(j).It is nontrivial. By (1),
lambda(j)=theta(gjg^−1j^−1),and the argument of theta lies in U_A^(m+1). The full simple character has exact restriction
theta(1+x)=psi_F(tr_F(beta x)), x∈R^(m+1).Since F has characteristic2, its additive characters take values in {1,−1}. Consequently
lambda(J)={1,−1}. (4)No assertion about shallow values of theta is used here. Theta may have a larger finite image on J. Only its ratio with its actual conjugate is proved to have order2.
As g²∈J, conjugation by g² fixes the character theta. Thus \(\lambda^g=\lambda^{-1}=\lambda\). It follows that
H_new=ker(lambda)is normal in P. It has index2 in J and index4 in P.
L.4 4. The actual finite Heisenberg pair
On this page · Permalink · Source line 213
Use the function model of E with f(jp)=theta(j)f(p) and right translation by P. Coordinates are f(1),f(g). Then for j∈J,
E(j)=diag(theta(j),theta^g(j))
=theta(j) diag(1,lambda(j)),while
E(g)=[[0,1],[theta(g²),0]]. (5)Therefore E(g)²=theta(g²)I. Choose j_0∈J with lambda(j_0)=−1, and put
Z=theta(j_0)^−1 E(j_0)=diag(1,−1),
X=E(g).The exact matrix identity is
XZX^−1=−Z. (6)The projective image of J has order2, generated by Z, while X is off-diagonal and contributes an independent projective involution. Thus the projective image of P is exactly C2×C2, with nontrivial commutator (6). Its projective kernel is precisely H_new. In particular
P/H_new≅F2².On H_new, E is the scalar theta_new=theta| H_new. This character is P-invariant: conjugation by J fixes theta and conjugation by g changes theta by lambda, which is trivial on H_new. The resulting alternating form
(pH_new,p'H_new)↦theta_new([p,p'])is nondegenerate, since its two displayed generators have commutator −1. Therefore (H_new,P,theta_new) is an actual arithmetic finite Heisenberg pair, and its actual irreducible carrier E has dimension2.
All representations here have finite image on the compact group P: theta has open kernel, the induction has finite dimension, and an open normal core of that kernel acts trivially. Thus no infinite projective-image issue is being suppressed.
This proof does not require rescaling the actual matrix g inside GL2(F). Its represented square theta(g²) can be an arbitrary retained scalar. Rescaling X is only allowed later in a C× pushout if that is the attachment contract.
L.5 5. Explicit embedding into the same fixed candidate
On this page · Permalink · Source line 276
Realize Pi as functions F:G→C with
F(nx)=Lambda(n)F(x),compact support modulo N, and right translation by G. For f∈E define i(f) by
i(f)(np)=Lambda(n)f(p), n∈N, p∈P,and zero off NP. This is well-defined because N∩P=J and Lambda| J=theta. The set NP is a finite union of N-cosets, by [P:J]=2, so the support is compact modulo N. The map i is P-equivariant and injective.
Its image is exactly the two-dimensional P-stable subspace of Pi consisting of inducing functions supported on NP. Thus the enlarged carrier is already inside the given Pi. No new candidate is chosen, and all noncompact values of Lambda remain fixed.
Since Pi is irreducible and i(E) is nonzero, its G-span is Pi. More precisely, there is the canonical G-equivariant quotient
Q_E:c-Ind_P^G E→Pi,
Q_E(F)=sum_(Px∈P\G) Pi(x^−1)i(F(x)). (7)The sum is finite and independent of representatives. If x is replaced by px, then F(px)=E(p)F(x) and P-equivariance of i cancels the additional Pi(p^−1). Formula (7) is nonzero and hence onto. The next section identifies its kernel exactly.
L.6 6. Exact recovery by the noncompact Hecke eigenvalue
On this page · Permalink · Source line 316
Let X=c-Ind_J^G theta, in the same left-covariant/right-translation convention. Induction in stages gives an explicit isomorphism
c-Ind_P^G E → X,
F ↦ [x↦F(x)(1)]. (8)The inverse sends f∈X to the E-valued function F(x)(p)=f(px). Both covariance statements follow directly from the definitions, and the finite index [P:J] ensures the support conditions agree.
Fix a uniformizer varpi=varpi_K. Because K× normalizes theta, the operator
(Tf)(x)=f(varpi x) (9)is a well-defined invertible G-equivariant operator on X. Put s=Lambda(varpi), which is nonzero. Since k_K=F2 and J contains U_K^1,
N/J≅K×/U_K^1≅Z,with generator varpi. Define
Q:X→Pi,
(Qf)(x)=sum_(ell∈Z) s^(−ell) f(varpi^ell x). (10)For each x the sum is finite, since f has compact support modulo the open subgroup J. Equation (10) satisfies
(Qf)(varpi x)=s(Qf)(x),
(Qf)(jx)=theta(j)(Qf)(x), j∈J,and therefore the required N-covariance. Its support is compact modulo N. It commutes with G-right translation and satisfies
QT=sQ. (11)It is onto: a function in Pi has finite support modulo N, and on each occupied N-coset its value can be lifted to a function supported on one J-coset. Extend those finitely many choices using theta.
To compute the kernel, decompose X as the algebraic direct sum of its subspaces supported on the left N-cosets in G. Each such summand is the space of finitely supported sequences on N/J≅Z, with the fixed theta covariance. The operator T is a bilateral shift. On that summand, (10) is evaluation of its Laurent polynomial at the nonzero scalar s, up to the harmless convention of replacing the Laurent variable by its inverse. The kernel is therefore the principal ideal generated by T−s. The elementary identity
ker(C[z,z^−1]→C, z↦s)=(z−s)proves, summand by summand, that
ker Q=(T−s)X.All vectors occupy finitely many such summands, so no completion or closure is needed. Thus
Pi ≅ X/(T−s)X ≅ (c-Ind_P^G E)/(T−s)(c-Ind_P^G E), (12)where T is transported through (8).
Under (8), formula (10) agrees with (7): on the standard copy of E supported on P, it gives exactly i, and G-equivariance then gives equality everywhere. Thus (12) identifies the kernel of the canonical recovery map, not merely the kernel of an unrelated surjection.
The same description can be phrased as evaluation of the type Hecke algebra at the character determined by Lambda. Its intertwining support is N, each supported intertwining space is one-dimensional, and the retained extension Lambda normalizes the Laurent generator. The explicit formulas above avoid any ambiguity about convolution orientation.
The scalar s is essential. E alone, being compact data, does not determine it. Changing s can change the unramified normalization of the candidate. The construction retains the original Lambda and s, so it recovers the exact original Pi rather than an unspecified member of its inertial class.
L.7 7. Full scalar-pushout attachment and remaining marking obligations
On this page · Permalink · Source line 418
Form the C× pushout of the actual pair (H_new,P,theta_new). It is a Heisenberg extension of the actual symplectic plane P/H_new. Any supplied source plane with the same chosen alternating form has a center-preserving pushout isomorphism into it. Every source full-character value is then preserved in the center, as in the retained full-character attachment theorem. Stone–von Neumann identifies its source carrier with the actual two-dimensional E.
Together with the embedding i and exact quotient (12), this provides an actual arithmetic carrier and recovery inside the same fixed candidate. It closes the carrier-capacity failure of the old rank-two eta by changing the compact arithmetic pair, not by increasing the old eta or discarding its dimension calculation.
At p=2, a C× pushout does not automatically identify a separately marked native mu_2 extension or quadratic refinement. The source finite central extension can be retained through its chosen embedding, while a required comparison with the target's preselected square function remains an additional question. No arbitrary native Arf invariant is asserted here.
Nontrivial stabilizer operators, transport graphs, and original H2's requirement that a target be a particular BK type-side quotient are likewise separate. In particular, the theorem proves that the new pair is an actual arithmetic finite Heisenberg pair, but does not rename it as the unchanged maximal-simple-type pair. No new hypothesis is silently added to original Theorem16.2.
L.8 8. Scope, smallest boundary, and sources
On this page · Permalink · Source line 450
The inequality q>=3 matters. If q=1, then m=0 and the displayed enlargement is not the group used above. Over residue field F2 the complete Iwahori unit group equals U_A^1, and a naive level-zero replacement does not give the asserted index-two enlargement. No q=1 claim is made. The source levels q=3h with positive odd h satisfy the required q>=3.
The proof uses the actual one-dimensional Lambda because the maximal parameter field has full degree and residue field F2. It does not apply the same scalar argument automatically to a higher-rank inducing representation.
Primary BK passages: 3.1.7–8 (odd minimal H/J equality); 3.2.1–2 (deep restriction and K× normalization); 3.3.2 and 5.1.1/5.1.8 (intertwining); and 6.2.1–5 (the fixed maximal-type extension and compact induction). The subgroup, matrix, Mackey-support, and Laurent-evaluation calculations are proved explicitly above. No external correspondence theorem is used in this enlargement.
Retained primary text:
closure_recovery/extracted/Paper_XI_Continuation158_General_Sectors_and_Arit
hmetic_Constraints/scratch158/information158/input_cache/BK_source.txt
SHA256 d69db0827256707a34cd2562af68b9de11fc889ab3f43535ec47b620d58ca9c8The previously reviewed negative statement concerned eta of the original BK \(J^1/H^1\). That quotient remains zero. The new positive statement concerns a different, explicitly constructed arithmetic pair inside the same ambient representation, with exact noncompact-normalization-aware recovery.