[Paper Review] Rationality of the exceptional W-algebras $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$ associated with subregular nilpotent elements of $\mathfrak{sp}_4$

Justine Fasquel|arXiv (Cornell University)|Sep 20, 2020

Algebraic structures and combinatorial models30 references4 citations

TL;DR

This paper proves the rationality of the exceptional $χ$-algebra $χ_k(\mathfrak{sp}_4, f_{\text{subreg}})$ at admissible levels $k = -3 + p/3$ and $k = -3 + p/4$, establishing that its simple modules are completely reducible and computing their characters. The work confirms a conjecture of Kac-Wakimoto for this case and explicitly describes the nontrivial action of the component group on the module category.

ABSTRACT

We prove the rationality of the exceptional W-algebras associated with the simple Lie algebra $\mathfrak{sp}_4$ and subregular nilpotent elements, proving a new particular case of a conjecture of Kac-Wakimoto. Moreover, we describe the simple $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$-modules and compute their characters. We also explicit the nontrivial action of the component group on the set of these simple modules.

Motivation & Objective

• To prove the rationality of the exceptional $χ$-algebra $χ_k(\mathfrak{sp}_4, f_{\text{subreg}})$ at admissible levels $k = -3 + p/3$ and $k = -3 + p/4$, confirming a conjecture of Kac-Wakimoto.
• To classify and compute the characters of all simple $χ_k(\mathfrak{sp}_4, f_{\text{subreg}})$-modules.
• To explicitly describe the nontrivial action of the component group on the set of simple modules.

Proposed method

• Utilizes the Drinfeld-Sokolov reduction functor $H_f$ applied to integrable highest weight modules $χ_k(\lambda^{(s)}_{i,j})$ of the affine Kac-Moody algebra $>hat{\mathfrak{sp}}_4$.
• Employs the character formula from [26, Section 3] to compute the characters of the reduced modules $H_f(\widehat{L}_k(\lambda^{(s)}_{i,j}))$.
• Applies the affine Weyl group action and twisted orbit sums via the extended affine Weyl group $\widetilde{W}$, particularly using the element $y = r_{\alpha_1} r_{\alpha_2} t_{-\varpi_1^\vee}$.
• Relies on the fact that the associated variety of $L_k(\mathfrak{sp}_4)$ is the subregular nilpotent orbit $\mathbb{O}_{\text{subreg}}$ when $q = 3$ or $q = 4$, linking admissible levels to the exceptional pair condition.
• Uses the isomorphism $H_f(\widehat{L}_k(\lambda^{(s)}_{i,j})) \cong L(\xi^{(s)}_{i,j}, \chi^{(s)}_{i,j})$ to identify the simple modules and their characters.
• Establishes that the reduction is concentrated in degree zero, i.e., $H^l_f(\widehat{L}_k(\lambda^{(s)}_{i,j})) = 0$ for $l \neq 0$, confirming the module structure.

Experimental results

Research questions

• RQ1 Is the exceptional $χ$-algebra $χ_k(\mathfrak{sp}_4, f_{\text{subreg}})$ rational at admissible levels $k = -3 + p/3$ and $k = -3 + p/4$?
• RQ2 What are the characters of the simple $χ_k(\mathfrak{sp}_4, f_{\text{subreg}})$-modules, and how are they computed via Drinfeld-Sokolov reduction?
• RQ3 How does the component group of the nilpotent orbit $\mathbb{O}_{\text{subreg}}$ act nontrivially on the set of simple modules?

Key findings

• The $χ$-algebra $χ_k(\mathfrak{sp}_4, f_{\text{subreg}})$ is rational at admissible levels $k = -3 + p/3$ with $(p,3) = 1$, $p \geq 3$, and $k = -3 + p/4$ with $(p,2) = 1$, $p \geq 4$, confirming the Kac-Wakimoto conjecture in this case.
• All simple $χ_k(\mathfrak{sp}_4, f_{\text{subreg}})$-modules arise as the zeroth cohomology of the Drinfeld-Sokolov reduction of integrable highest weight $>hat{\mathfrak{sp}}_4$-modules.
• The character of each simple module $L(\xi^{(s)}_{i,j}, \chi^{(s)}_{i,j})$ is given by the formula $\operatorname{ch}_{L(\xi^{(s)}_{i,j}, \chi^{(s)}_{i,j})}(q,z) = \operatorname{ch}_{H_f(\widehat{L}_k(\lambda^{(s)}_{i,j}))}(q,z)$, explicitly computed via affine Weyl group orbit sums.
• The component group of the subregular nilpotent orbit $\mathbb{O}_{\text{subreg}}$ acts nontrivially on the set of simple modules, as shown by the structure of the character formulas and the action of the Weyl group elements.
• For $k = -3 + p/3$, the simple modules are labeled by $L(\xi^{(3)}_{i,j}, \chi^{(3)}_{i,j})$ with $i,j$ satisfying $1 \leq i \leq j \leq p-1$, $i+j \leq p$, and the character formula is $\operatorname{ch}_{L(\xi^{(3)}_{i,j}, \chi^{(3)}_{i,j})}(q,z) = \frac{q^{\chi^{(3)}_{i,j}+2p-2i-\frac{j}{2}}z^{-\frac{2p}{3}+\frac{1}{2}}}{\prod_{n>0}(1-q^n)^2} \prod_{n>0}(1-q^{n-1}z)^{-1}(1-q^n z^{-1})^{-1} \sum_{w,\eta} \epsilon(wt_{3\eta}) q^{\cdots} z^{\cdots}$.
• For $k = -3 + p/4$, the simple modules are labeled by $L(\xi^{(2')}_{i,j}, \chi^{(2')}_{i,j})$, and the character formula is analogous, with the same structure of sum over $W \times Q^\vee$ and twisted Weyl group action.