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The one-strand degenerate action on the trivial module has scalar dot

Statement

Let $M = C$ be the trivial ${g l}_{2} (C)$ -module, let $V = C^{2}$ be the natural module, and restrict $M$ to ${s l}_{2} (C)$ . For tensor degree $d = 0$ the associated degenerate affine Hecke algebra is $H_{0} = C$ and acts on $M$ by scalars. For $d = 1$ , the Arakawa–Suzuki polynomial generator $y_{1}$ acts on $M \otimes V$ as $\frac{1}{2} {id}_{M \otimes V}$ . Thus the polynomial action factors through $C [y_{1}] / (y_{1} - \frac{1}{2}) ≅ C .$ This is a computation of the action on the indicated tensor object, not a claim about higher tensor degree or other modules.

Facts & Assumptions

Given: The trivial ${g l}_{2} (C)$ -module $M = C$ , its restriction to ${s l}_{2} (C)$ , the natural module $V = C^{2}$ , and tensor degrees $d = 0, 1$ .

[F1]

Category O, KLR, and Hecke route conventions: the Category-O route uses the degenerate affine Hecke action and keeps the source-normalized generator $y_{j} = x_{j} - \frac{1}{N} z_{M} + \frac{N - 1}{2}$ distinct from a nondegenerate Hecke generator.

[F2]

Arakawa-Suzuki Casimir operators on tensor functors: on $M \otimes V^{\otimes d}$ , the defined operators are $x_{j} = Ω_{0 j} + \sum_{k < j} P_{k j}$ and $s_{i} = P_{i, i + 1}$ .

[F3]

Arakawa-Suzuki Casimir operators on tensor functors: the source-normalized operators are $y_{j} = x_{j} - \frac{1}{N} z_{M} + \frac{N - 1}{2}$ .

[F4]

Arakawa-Suzuki Casimir operators on tensor functors: for trivial $M = C$ , $Ω_{0 j} = 0$ and $x_{1} = 0$ .

[F5]

The Casimir and place-permutation operators satisfy the degenerate affine Hecke relations: these operators commute with ${g l}_{N}$ and give the stated right degenerate affine Hecke action.

[F6]

Verma and finite-dimensional weight modules belong to O: every finite-dimensional $h$ -semisimple module for a finite-dimensional complex semisimple Lie algebra belongs to the BGG category $O$ .

[F7]

The classical BGG category O: $O$ consists of finitely generated, $h$ -semisimple modules on which each $U (n^{+})$ -orbit is finite-dimensional.

[F8]

The degenerate affine Hecke algebra: $H_{0} = C$ and $H_{1} = C [x_{1}]$ ; there are no $s_{i}$ generators in degree one.

[F9]

The special linear Lie algebra sl_2: ${s l}_{2} (C)$ has basis $e, f, h$ with $[h, e] = 2 e$ , $[h, f] = - 2 f$ , and $[e, f] = h$ .

[F10]

Simple, semisimple, and reductive Lie algebras: a finite-dimensional Lie algebra is simple when it is nonabelian with no nonzero proper ideals.

[F11]

Semisimple Lie algebras: a finite-dimensional Lie algebra is semisimple when its solvable radical is zero; the radical is the largest solvable ideal.

[F12]

Solvable radical: the solvable radical is a solvable ideal containing every solvable ideal of the Lie algebra.

[F13]

Derived series and solvable Lie algebras: a Lie algebra is solvable when its derived series eventually becomes zero.

Proof

1.1givenF6F7F9F10F11F12F13algebra

The adjoint action of $h$ has distinct eigenvalues $2, - 2, 0$ on the basis $e, f, h$ [F9]; for any nonzero ideal $I$ , Lagrange projection polynomials in $ad h$ preserve $I$ and isolate a nonzero basis component. If $e \in I$ , then $h = [e, f] \in I$ and $- 2 f = [h, f] \in I$ ; if $f \in I$ , then $h = [e, f] \in I$ and $2 e = [h, e] \in I$ ; if $h \in I$ , then $2 e = [h, e] \in I$ and $- 2 f = [h, f] \in I$ . Thus $I = {s l}_{2}$ , which is nonabelian since $[e, f] = h \neq 0$ , so it is simple [F10]. Its derived algebra equals itself by [F9], so it is not solvable [F13]; the solvable radical is an ideal [F11,F12], hence simplicity forces it to be zero, and [F11] makes ${s l}_{2}$ semisimple. The restricted module $M$ is one-dimensional and finite-dimensional, and $h = C h$ acts by zero, so it is $h$ -semisimple; [F6] puts it in $O$ as defined by [F7].

1.2givenF2F3F4F5F8algebra

For $d = 0$ , $M \otimes V^{\otimes 0} = M$ and $H_{0} = C$ , so the action is scalar [F8]. For $d = 1$ , [F2] and [F4] give $x_{1} = Ω_{01} = 0$ on the trivial factor, while $z_{M} = 0$ ; the normalization [F3] at $N = 2$ therefore gives $y_{1} = \frac{1}{2} {id}_{M \otimes V}$ . The action exists by [F5].

2.1F1F2F3F4F8algebra∎

By [F8], $H_{1} = C [x_{1}]$ ; the source-normalized route [F1] has $y_{1} = x_{1} + \frac{1}{2}$ by [F3]. Since [F2] and [F4] give $x_{1} = z_{M} = 0$ , evaluation at $y_{1} = \frac{1}{2}$ is the action, with kernel $(y_{1} - \frac{1}{2})$ by polynomial division. Thus the action factors through $C [y_{1}] / (y_{1} - \frac{1}{2}) ≅ C$ .

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Sources

Tomoyuki Arakawa and Takeshi Suzuki, Lie algebras and degenerate affine Hecke algebras of type A, J. Algebra 209 (1998), complete 16-page preprint (standard reference, not scraped)