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A Hecke action alone does not imply a double centralizer

Statement refuted

Refuted claim: if a Hecke algebra action on $M \otimes V^{\otimes d}$ is natural in the $g$ -module $M$ and commutes with $g$ , then the two double centralizer equalities $φ (U (g)) = {End}_{H_{d}} (M \otimes V^{\otimes d})$ and $ψ (H_{d}) = {End}_{g} (M \otimes V^{\otimes d})$ hold.

Facts & Assumptions

Given: The trivial module $M = C^{2}$ , the natural module $V = C^{2}$ , the tensor product $M \otimes V$ with basis $u_{a} \otimes v_{b}$ (The elementary tensors of two bases form the product basis of the tensor product, The tensor product $M \otimes_{R} N$ from the additive group underlying the free $Z$ -module on $M \times N$ , elementary tensors, and finite tensor sums), and the Casimir operator $x_{1} = Ω^{M, V}$ .

[F1]

On $M \otimes V$ the Casimir acts as $Ω^{M, V} = \sum_{i, j} ρ_{M} (E_{i j}) \otimes ρ_{V} (E_{j i})$ ; since $ρ_{M} = 0$ , $x_{1} = 0$ (Arakawa-Suzuki Casimir operators on tensor functors, Classical complex matrix Lie algebras).

[F2]

The natural action of ${g l}_{2}$ on $M \otimes V$ has image $1 \otimes M_{2} (C) \subseteq M_{4} (C)$ , of dimension $4$ ; the four matrix units act as $1 \otimes E_{i j}$ and span this algebra, and products of these operators remain in it, proving the enveloping-image assertion (Classical complex matrix Lie algebras, Representations of Lie algebras).

Counterexample

Let $g = {g l}_{2} (C)$ (Classical complex matrix Lie algebras), let $M = C^{2}$ be the trivial $g$ -module (all matrix units act by $0$ ), let $V = C^{2}$ be the natural module (Representations of Lie algebras), and let $d = 1$ . The Arakawa-Suzuki construction of Arakawa-Suzuki Casimir operators on tensor functors provides an action of $H_{1} = C [x_{1}]$ (The degenerate affine Hecke algebra) on $M \otimes V ≅ C^{4}$ commuting with ${g l}_{2}$ (The Casimir and place-permutation operators satisfy the degenerate affine Hecke relations).

Proof technique: direct.

Given: The operators and modules of the Counterexample.

1.1F1algebra

The Hecke action. By [F1] the generator $x_{1}$ acts as $0$ on $M \otimes V$ , so the $H_{1}$ -action is the evaluation action $x_{1} \mapsto 0$ (the quotient $H_{1} / (x_{1}) = C$ of The degenerate affine Hecke algebra specialized to the scalar $0$ ).

2.1step 1.1algebra

The first centralizer. Since $H_{1}$ acts by the scalars $C$ on the four-dimensional space, ${End}_{H_{1}} (M \otimes V) = {End}_{C} (C^{4}) = M_{4} (C)$ , of dimension $16$ .

3.1F2step 2.1algebra

The image of the enveloping algebra. By [F2] the image of $U ({g l}_{2})$ in $End (M \otimes V)$ is the subalgebra $1 \otimes M_{2} (C)$ , of dimension $4$ . Hence the first double centralizer equality fails by a factor of $4$ in dimension.

4.1F2step 3.1given∎

The second centralizer. Writing an endomorphism as a $2 \times 2$ block matrix indexed by the basis of $M$ , commutation with every $1 \otimes E_{i j}$ forces each block to be a scalar matrix. Thus the ${g l}_{2}$ -endomorphisms are $M_{2} (C) \otimes 1$ , of dimension $4$ , while $ψ (H_{1}) = C \cdot i d$ has dimension $1$ ; hence the second equality also fails. The missing hypothesis is the truncated/deformed setup: the double centralizer theorem applies to the finite-W-algebra tensor space $V_{c}^{\otimes d}$ over a pyramid where the Hecke action is the cyclotomic quotient action and the algebra on the opposite side is the finite W-algebra, not $U (g)$ itself. Its associated graded image algebras are the corresponding images of $U (g^{e})$ and $C_{l} [x_{1}, \dots, x_{d}] ⋊ C [S_{d}]$ ; this does not identify the deformed actions with the graded actions (The finite W and degenerate cyclotomic Hecke actions satisfy a double centralizer, The type-A finite W-algebra and the Whittaker category).

Remarks

The example is not a counterexample to the higher-level Schur-Weyl theorem; it shows exactly that a natural commuting Hecke action on an arbitrary tensor module is not sufficient, and that the truncation (the deformed tensor space over a pyramid) is a needed hypothesis.
The trivial module is legitimate as an input to the Arakawa-Suzuki construction: the construction assigns an $H_{d}$ -action to every $g$ -module $M$ .

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Sources

Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article) (standard reference, not scraped)
Jonathan Brundan and Alexander Kleshchev, Representations of shifted Yangians and finite W-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918 (complete 109-page arXiv memoir) (standard reference, not scraped)