TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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KZ monodromy on category O factors through the reflection Hecke algebra

Statement

For every $M \in O_{c} (W, h)$ , the KZ braid-group representation $KZ (M)$ factors through the parameter-specific finite reflection Hecke algebra $H_{q (c)} (W)$ defined in The KZ monodromy functor and the reflection Hecke algebra. No semisimplicity or genericity assumption on $c$ is imposed.

Facts & Assumptions

Given: The data and hypotheses in the Statement.

[F1]

Cherednik category O has a standard-filtered projective generator: for fixed $c$ , a direct summand of a generalized regular standard is a finite projective generator of $O_{c}$ ; every object is a quotient of a finite sum of that generalized standard.

[F2]

Hecke relations specialize in the generalized-standard family proves the exact meridian polynomial identity on all generalized regular standards, by convergent nonresonant gauges and holomorphic continuation of the polynomial-free parameter family.

[F3]

Dunkl localization gives flat connections on the regular locus: KZ is exact.

[F4]

The KZ monodromy functor and the reflection Hecke algebra specifies the target algebra and every defining meridian relation.

Proof

technique · direct

Given: The stated hypotheses and Facts.

1.1F1F2choosealgebra

Choose a generalized regular standard $G_{n, c}$ whose finite sum surjects onto $M$ , using [F1]. By [F2], every specified meridian polynomial annihilates $KZ (G_{n, c})$ , and hence also a finite direct sum of these modules.

2.1F3F4step 1.1algebra∎

Exactness [F3] gives a surjection from that direct sum to $KZ (M)$ . An annihilating relation passes to a quotient, so every defining relation in [F4] annihilates this fiber. The braid-group action therefore factors through $H_{q (c)} (W)$ , for the original arbitrary parameter and object. This uses a generating quotient, rather than the invalid inference from exactness and relations on composition factors.

Depends on

Used by

KZ identifies the torsion quotient with Hecke modules under the Hecke-rank hypothesis Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof and Xiaoguang Ma, Lecture Notes on Cherednik Algebras (standard reference, not scraped)
Victor Ginzburg, Nicolas Guay, Eric Opdam, and Raphael Rouquier, On the category O for rational Cherednik algebras (standard reference, not scraped)