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KZ monodromy of standard modules satisfies the reflection Hecke relations

Statement

For every finite-dimensional $W$ -module $E$ , each positive reflection meridian acts on $KZ (Δ_{c} (E))$ with eigenvalues among $μ_{H, 0}, \dots, μ_{H, e_{H} - 1}$ from The KZ monodromy functor and the reflection Hecke algebra. Consequently $KZ (Δ_{c} (E))$ is a module over the defined finite reflection Hecke algebra $H_{q (c)} (W)$ . No semisimplicity assumption on $O_{c}$ or on the Hecke algebra is made.

Facts & Assumptions

Given: The data and hypotheses in the Statement.

[F1]

Hecke relations specialize in the generalized-standard family proves the exact meridian polynomial identities on every generalized standard at every parameter, with the current endpoint and residue conventions.

[F2]

Standard modules for rational Cherednik category O: $Δ_{c} (E)$ is induction of the finite $W$ -module $E$ with $h$ acting by zero.

[F3]

The KZ monodromy functor and the reflection Hecke algebra: the specified reflection Hecke algebra is the braid-group algebra modulo these polynomial identities.

Proof

technique · direct

Given: The stated hypotheses and Facts.

1.1F1F2algebra

A standard is an induced nilpotent module with augmentation ideal acting by zero; equivalently it is a direct summand of a finite sum of the degree-zero generalized regular standard, by finite-group averaging. [F1] therefore gives $\prod_{j = 0}^{e_{H} - 1} (T_{H} - μ_{H, j}) = 0$ on its KZ fiber for every $H$ and every $c$ . This proves an annihilating polynomial, including resonant parameters; no diagonalizability at such parameters is inferred from a characteristic polynomial.

2.1F3step 1.1algebra∎

Every eigenvalue of an operator annihilated by that polynomial is one of its roots. Since every defining relation of [F3] acts by zero, the braid representation factors through the stated Hecke quotient. This proves all assertions without a semisimplicity hypothesis.

Depends on

Used by

KZ monodromy on category O factors through the reflection Hecke algebra Theorem

Dependency tree · two levels

13 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)