O Exact enlarged-carrier recovery and original-contract comparison
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Evidence: a bounded contract audit and elementary conditional recovery lemmas. This note does not prove that the proposed compact enlargement is a group, that its induced representation is Heisenberg, or that the source markings attach to it. Those are the arithmetic inputs to be verified by the separate rank-two and rank-four constructions.
Verdict: an enlarged compact carrier is not, merely by being Heisenberg or an equivalent type, a new instance of the original H2 quotient. It can serve as a common assembly carrier for the already supplied H2 targets if its actual occurrence, marked attachment, and same-candidate recovery are proved and retained. The recovery mechanism differs from the finite sector idempotent used when the packet lies inside the original inducing carrier.
O.1 1. What original H2 literally requires
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Original Section10 fixes an actual maximal simple character theta, its actual groups H_theta¹ subset J_theta¹ subset J_theta, and
V_theta=J_theta¹/H_theta¹.Its Heisenberg representation is the particular eta_theta containing theta. Definition11.1 supplies an equivariant symplectic isometry from a source envelope to that type-side quotient, together with the central character, and at p=2 the compatible quadratic refinements and central-extension lift. Standard Fact10.2 separately supplies the canonical beta extension of this simple-character datum.
Consequently the following implications are invalid without more proof:
Xi is a finite Heisenberg representation
therefore Xi=eta_theta' for a maximal simple character theta';
(P,Xi) is equivalent to a known arithmetic type
therefore it has that type's J¹/H¹ or canonical beta extension.An equivalence of generated representation categories does not identify the particular finite group quotient in Definition11.1. If a fixed conductor and torsion profile forces every maximal simple-character carrier in the candidate to have dimension one or two, constructing a larger Xi in that candidate does not refute the dimension theorem. Xi is another arithmetic carrier.
There is nevertheless an important distinction between replacing an H2 target and assembling already supplied H2 targets. The accepted sector construction itself uses a finite sector that need not be a whole maximal simple-character carrier. Its old actual break targets remain unchanged and are attached to that sector. An enlarged P-carrier can play the analogous assembly role, provided the complete dictionary and recovery below are proved. Calling it merely an arbitrary vector space would understate that result; calling it a new canonical maximal simple type would overstate it.
O.2 2. Precise conditional setup
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Let G=GL_n(F), Z=F×I_n, and let the already constructed candidate be
pi=c-Ind_N^G Lambda,with its fixed actual extended type and fixed central character omega. Let J be a compact open normal subgroup of N such that
eta=Lambda|J is irreducible, N/J is infinite cyclic.In the proposed F2-residue, full-degree totally ramified setting, J is the actual compact type group J¹=J, not a new group called J merely to fit this notation. Let P be an actual compact open subgroup of G satisfying
J subset P, N intersect P=J.Define the genuine finite-dimensional compact induction
Xi=Ind_J^P eta. (1)Neither irreducibility nor Heisenberg structure of Xi is assumed to follow from its dimension. They must be proved separately when needed.
If the known eta intertwining satisfies I_G(eta)=N, then Xi is irreducible: the finite Mackey endomorphism formula gives
End_P(Ind_J^P eta)=End_J(eta)=C,because no nontrivial double coset in P lies in I_G(eta). Finite-dimensional smooth representations of compact P factor through a finite group and are semisimple, so scalar endomorphisms imply irreducibility. This criterion requires its actual group and intertwining hypotheses; dimension alone is not a substitute.
O.3 3. Xi occurs in the same pi, and can be an equivalent compact type
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The subspace of the actual compact-induction model of pi consisting of functions supported on NP is stable under P. Mackey restriction identifies that subspace with
Ind_(N intersect P)^P(Lambda|_(N intersect P))=Xi. (2)Thus Xi is an actual compact P-subrepresentation of the same pi, with no change of arithmetic candidate or noncompact scalar.
Induction in stages also gives the exact G-isomorphism
c-Ind_P^G Xi isomorphic to c-Ind_J^G eta. (3)If (J,eta) is already a type for a stated Bernstein block and Xi is irreducible, then (P,Xi) detects the same irreducible representations:
Hom_P(Xi,V) isomorphic to Hom_J(eta,V).This is the precise equivalent-type statement that follows from (1). It does not assert that Xi is the eta of another maximal simple character, or attach a canonical beta extension to Xi. The known canonical extension in Lambda remains the one originally supplied.
O.4 4. Exact recovery before fixing the center: a quotient
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Choose u in N mapping to a generator of N/J, and retain the actual matrix Lambda(u). Use the model
X=c-Ind_J^G eta,
f(jg)=eta(j)f(g),with G acting by right translation. Define the invertible G-equivariant shift operator
(Tf)(g)=Lambda(u)^(-1)f(ug). (4)Normality of J and the actual intertwining relation for Lambda(u) ensure that Tf has the same J-covariance. Its normalization explicitly retains the fixed noncompact extension; changing Lambda(u) changes the relation below.
Define
(Qf)(g)=sum_(k in Z) Lambda(u)^(-k)f(u^k g). (5)The sum is finite on each argument and has the required compact support modulo N. It is N-covariant and therefore lies in pi. It is surjective and satisfies Q(Tf)=Q(f). Decomposing support into N/J-orbits reduces its kernel to the elementary augmentation kernel on finitely supported sequences on Z. That kernel consists exactly of consecutive differences. Hence
X/(T-1)X isomorphic to pi. (6)Through (3), this is a recovery of pi from the enlarged compact carrier plus the original noncompact relation. It does not select the correct relation from the finite Heisenberg packet alone.
In general this quotient cannot be replaced by a G-equivariant split inclusion before central normalization. A nonzero vector of X has finite support modulo compact J. The valuation of its determinant support is a finite subset of Z. A scalar uniformizer z=tI_n shifts these valuations by n; consequently X has no nonzero eigenvector for z. Every vector of pi is such an eigenvector, with eigenvalue omega(t). Therefore no nonzero G-map pi -> X exists. In particular (6) does not split.
This is why the Block159 sector summand projector cannot simply be quoted: there the finite sector was inside Lambda and its stabilizer contained the noncompact center. Here Xi is larger than Lambda and lies in pi| P, with P compact. These are different induction problems.
O.5 5. After retaining the center: a finite Hecke idempotent
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There is also an exact summand version once the already fixed central character is imposed. Put J_Z=ZJ and P_Z=ZP. Assume, as in the proposed totally ramified full-degree setting, that
N/J_Z is finite cyclic of order ell(here ell=n). Let Lambda_0=Lambda| J_Z. Because Xi occurs in the same pi, its action on Z intersect P agrees with omega. It therefore extends to P_Z by letting Z act through omega; call this Xi_omega. Induction in stages gives
Y=c-Ind_(P_Z)^G Xi_omega
isomorphic to c-Ind_(J_Z)^G Lambda_0. (7)Using the same normalized shift (4), now on Y, one has T^ell=1. Indeed u^ell belongs to J_Z and the J_Z covariance cancels Lambda(u)^ell. The finite operator
e_Lambda=(1/ell) sum_(j=0)^(ell-1) T^j (8)is a G-equivariant idempotent whose image is exactly the fixed pi. To see this directly, N-covariant functions embed in J_Z-covariant functions because N/J_Z is finite. On that embedded pi the finite version of (5) equals ell times the identity. Its average is precisely (8), and its image consists exactly of the N-covariant functions.
Equivalently, Ind_(J_Z)^N Lambda_0 is Lambda tensored with the regular representation of N/J_Z. Formula(8) selects the Lambda summand. It uses the already fixed Lambda(u), including its root choice, so it preserves the same candidate. Other root choices select the other extensions in this finite induction. They are not interchangeable input data.
This is a finite Hecke/induction projector after central-character normalization. It is not the original finite image-algebra projector onto a sector of eta. A final artifact must retain (7), (8), and the normalization Lambda(u), rather than describing recovery merely as “the carrier is an equivalent type.”
O.6 6. The exact remaining admissibility checks
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For either proposed low-rank enlargement, the new proof must supply all of the following concrete inputs:
Actual compact group: prove that the proposed P is a subgroup, is compact open, contains the actual J, and satisfies P intersect N=J. Compute [P:J] and the resulting Xi dimension.
Actual carrier: verify eta=Lambda| J, its actual intertwining if the irreducibility criterion is used, and the Mackey realization (2).
Actual finite Heisenberg structure: exhibit the scalar preimage in P and the resulting elementary abelian projective quotient. Compute its represented commutator and prove nondegeneracy. An irreducible carrier of dimension two or four need not automatically have this structure.
Original attachments: attach the unchanged supplied actual H2 break targets, retaining their full source character values and dyadic finite extension/lift/square markings. The precise transported-versus-native distinction remains the one proved in
DYADIC_MARKINGS167.md.Recovery data: retain the actual Lambda, central character, and noncompact generator normalization, and prove the isomorphisms and quotient/idempotent above for those exact groups.
When these are supplied, the result is stronger than occurrence of an unrelated finite vector space: it is a marked arithmetic carrier in the same candidate, equivalent through explicit induction and recovering that candidate exactly. For C={1} and one vertex, it can repair the failed low-rank assembly-carrier realization under the retained scalar-pushout contract. The original H2 targets remain their original simple-character quotients; they are not renamed Xi.
If the intended target instead requires the source packet itself to equal J_theta'¹/H_theta'¹ for a maximal simple character in this same candidate, the enlargement does not solve it. The retained low-rank dimension obstruction still applies. A separate theorem identifying Xi with such an eta would be needed, and under the fixed obstructed profile it cannot be inferred from type equivalence.
No H5 condition is added or discharged. If H5 is subsequently supplied, it must concern this fixed recovered candidate. Nontrivial stabilizer actions, canonical native quadratic comparison, and arbitrary transport graphs remain outside this bounded argument.
O.7 Evidence and self-review
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Read directly: original Section10 and Definition11.1 in the raw manuscript; the Block159 NONINVARIANT_SECTOR_RECONSTRUCTION159.md; current PROOF167.md and TARGET167.md; the retained dyadic marking theorem. The Mackey, induction, shift-quotient, and central-normalized projector arguments above are proved conditionally for the displayed groups, rather than assumed for the proposed congruence formula. No actual low-rank P computation or new Lean execution is claimed in this contract note.