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DefinitionDefinition: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-04 (gpt-6.1-sol)
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Nondegenerate principal admissible affine level

Definition

For the root data of def-cft-root-data-for-principal-w-levels, write h=hg, h∨=hg∨, r∨ for the lacing number, and hg~∨ for the normalized dual Coxeter datum of a supplied Langlands-dual simple Lie algebra. A nondegenerate principal admissible level is k=−h∨+pq, where p,q∈Z>0, gcd⁡(p,q)=1, and one of the following mutually source-specified tests holds:

gcd⁡(q,r∨)=1,p≥h∨,q≥h;

or

gcd⁡(q,r∨)=r∨,p≥h,q≥r∨hg~∨.

The second test is conditional on the dual-root realization and normalization in the preceding definition. Since p/q>0, every such level is noncritical: k≠−h∨.

Facts & Assumptions

Given: A finite-dimensional complex simple g, its normalized root data, and positive coprime integers p,q.

[F1]

Root normalization and dual data for principal W levels: h, h∨, and r∨ have the stated root and form conventions; the supplied dual realization and its optional Choice-based construction are precise.

[F2]

Dual Coxeter number for the normalized invariant form: hg∨ is the dual Coxeter number for the invariant form normalized by long roots of squared length 2.

[C1]

This is the nondegenerate principal-W threshold predicate. It is stronger than the general admissible-number predicate; no converse equivalence is asserted. The actual forward implication is proved in the linked justifier.

Proof

1.1givenF1F2C1algebra

Positive coprime integers and the supplied normalized root data make every gcd test and inequality a well-defined arithmetic condition. The second branch uses the supplied dual realization from F1; no unsupported existence claim is inserted. The two tests agree in the simply-laced case where their gcd conditions coincide.

2.1givenF1F2C1step 1.1algebra∎

Positivity gives p/q>0, hence k+hvee is nonzero and the level is noncritical. This proves the Definition's stated consequence. The stronger denominator thresholds distinguish nondegeneracy; they are not an equivalence with all admissible affine numbers.

Proof status

The arithmetic predicate and noncriticality assertion are proved. Its exact forward admissibility implication is proved by the attached justifier; the false stronger converse is not claimed.

Depends on

Dependency tree · 2 levels

Sources