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The rational Cherednik commutator satisfies Jacobi compatibility

Statement

For the reflection data of Reflection data and parameters for a rational Cherednik algebra and the quotient presentation of The rational Cherednik algebra H_{t,c}(W,h), let $κ (u, v) = t ω (u, v) 1 - \sum_{s \in S} c_{s} ω_{s} (u, v) s (u, v \in V) .$ Then $κ$ is $W$ -equivariant, and its values satisfy $[u, κ (v, w)] + [v, κ (w, u)] + [w, κ (u, v)] = 0 (u, v, w \in V),$ where each bracket with a group-algebra element is computed using $[u, s] = (u - s (u)) s$ .

Facts & Assumptions

Given: The finite complex reflection group $W$ , its reflection representation $h$ , the reflection parameters $c$ , the forms $ω, ω_{s}$ , and $κ$ as defined in the two linked items.

[F1]

Reflection data and parameters for a rational Cherednik algebra: $W$ acts on $V = h \oplus h^{*}$ preserving $ω$ ; for each reflection $s$ , $V = \ker (1 - s) \oplus U_{s}$ with $\dim U_{s} = 2$ , and $ω_{s}$ has radical $\ker (1 - s)$ and is a nonzero alternating form on $U_{s}$ .

[F2]

The rational Cherednik algebra H_{t,c}(W,h): $c$ is constant on conjugacy classes, $ω_{s}$ is conjugated to $ω_{w s w^{- 1}}$ by $w$ , and the commutator parameter is $κ (u, v) = t ω (u, v) 1 - \sum_{s} c_{s} ω_{s} (u, v) s$ .

[F3]

The rational Cherednik algebra H_{t,c}(W,h): The smash-product relation gives $s u = s (u) s$ and hence $[u, s] = (u - s (u)) s$ .

Proof

technique · direct

Given: The data of [F1]–[F3].

1.1F1F2algebra

Under $w \in W$ , the summand indexed by $s$ in $κ (w u, w v)$ is $- c_{s} ω_{s} (w u, w v) s$ . Reindex by $r = w s w^{- 1}$ . Conjugacy invariance gives $c_{r} = c_{s}$ , and the reflected form satisfies $ω_{r} (w u, w v) = ω_{s} (u, v)$ by [F1]–[F2]. The identity term is invariant because $w$ preserves $ω$ . Thus $κ (w u, w v) = w κ (u, v) w^{- 1}$ .

1.2F2F3algebra

The scalar term $t ω (v, w) 1$ commutes with $u$ . For a fixed reflection $s$ , the contribution to the cyclic Jacobi sum is $- c_{s} (ω_{s} (v, w) (1 - s) u + ω_{s} (w, u) (1 - s) v + ω_{s} (u, v) (1 - s) w) s$ by [F3].

2.1F1step 1.2algebra∎

Project $u, v, w$ to $U_{s}$ along $\ker (1 - s)$ . On the two-dimensional space $U_{s}$ , for any alternating form $η$ and vectors $a, b, d$ , one has $η (b, d) a + η (d, a) b + η (a, b) d = 0$ : the left side is alternating trilinear in a two-dimensional space and therefore vanishes. Apply $1 - s$ to this identity with $η = ω_{s}$ ; because $ω_{s}$ vanishes on the radical, the result is exactly the parenthesized vector in step 1.2. Each reflection contribution is zero, so the cyclic sum vanishes.

Depends on

Used by

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Sources

Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism (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)