The PBW filtration conjecture for vacuum representations of affine Kac–Moody algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra, let $\widehat{\mathfrak{g}}$ be its corresponding affine algebra, and let $L_k$ be the vacuum level $k$ representation with highest weight vector $v_k$. Write

$$L_k^{\mathrm{gr}}\simeq \operatorname{U}(\mathfrak{g}^{\mathrm{ab}}\otimes t^{-1}\mathbb{C}[t^{-1}])/I_k,$$

where $I_k$ is the ideal of relations. Let $\theta$ be the longest root of $\mathfrak{g}$, choose $f_\theta\in\mathfrak{g}$ of weight $\theta$, and define

$$f_\theta(z)=\sum_{n\geq 0}z^n(f_\theta\otimes t^{-n-1}).$$

The PBW filtration conjecture. One has

$$I_k=\bigl(\operatorname{U}(\mathfrak{g})\oplus \operatorname{U}(\mathfrak{g}^{\mathrm{ab}}\otimes t^{-1}\mathbb{C}[t^{-1}])\bigr)\cdot \operatorname{span}\{\text{coefficients of }f_\theta(z)^{k+1}\},$$

that is, $I_k$ is the minimal $\operatorname{U}(\mathfrak{g})$-stable ideal containing the coefficients of $f_\theta(z)^{k+1}$. The coefficients act by zero on $L_k$. The conjectural description of the ideal $I_k$ gives the relations defining the PBW-graded vacuum representation; the paper proves it in the case $\mathfrak{g}=\mathfrak{sl}_t$, while the general simple Lie algebra case remains open.