Please reference this paper for notation in this question.

I'm trying to understand two claims made in the above paper (they may be related). First, in the construction of $\mathcal{H}_\lambda$ on page 4, we start with the usual irreducible representation $V_\lambda$ of the semisimple Lie algebra $\mathfrak{g}$, declare the positive powers of $z$ to act trivially and the central element $c$ to act by multiplication by $l$, and then induce an action of the rest of the affine Lie algebra $\hat{\mathfrak{g}}$ on $V_\lambda$. The result is called $\mathcal{V}_\lambda$ and it is isomorphic to $U(\hat{\mathfrak{g}}_-) \otimes_{\mathbb{C}} V_\lambda$ as a $\hat{\mathfrak{g}}_-$-module. As a consequence, it is a highest weight module, and by theory analogous to that of semisimple Lie algebras, it has a maximal submodule.

I understand all of that. What I do not understand is the claim that this maximal submodule $\mathcal{Z}_\lambda$ is generated by $(X_\theta \otimes z^{-1})^{l - \lambda(H_\theta) + 1} v_\lambda$. Following Remark 3.6 on page 8, I managed to prove that this element is annihilated by $\hat{\mathfrak{g}}_+$, and indeed that no lower nonzero power of $X_\theta \otimes z^{-1}$ acting on $v_\lambda$ is annihilated by all of $\hat{\mathfrak{g}}_+$. Thus I'm lead to believe that this should somehow imply that $\mathcal{Z}_\lambda$ is a highest weight module with $(X_\theta \otimes z^{-1})^{l - \lambda(H_\theta) + 1} v_\lambda$ its highest weight vector, and hence the claim follows, but I can't see why this is the case.

The second claim is labeled (*) on page 7: The endomorphism $X_{-\theta} \otimes f$ of $\mathcal{H}$ is locally nilpotent for all $f \in \mathcal{O}(U)$. From what I can tell, this is equivalent to the claim that $X_{-\theta} \otimes f$ is a locally nilpotent endomorphism of $\mathcal{H}_\lambda$ for any $f \in \mathcal{C}((z))$. Following the given hint, I can see why this is true for $f \in \mathcal{C}[[z]]$, but I don't know about negative powers of $z$. I think this might be related to the first claim: Perhaps once we have pushed any element of $\mathcal{V}_\lambda$ into a suitably low weight space by acting on it by $X_{-\theta} \otimes z^{-1}$ repeatedly, it actually falls into some proper submodule of $\mathcal{V}_\lambda$, and is therefore quotiented out when we pass to $\mathcal{H}_\lambda$. Again, I can't see why this is actually true though.

Any ideas on either claim are greatly appreciated.


I answer the first question only.

The space $\mathcal{V}_\lambda$ is highest weight for the affine Lie algebra, hence a quotient of a Verma $\Delta(\mu)$ of the affine Lie algebra. The maximal submodule $\mathcal{Z}_\lambda$ is thus the image of the maximal submodule of the Verma.

The maximal submodule of a Verma is known to be generated by the vectors $f_i^{\langle \mu,\alpha_i^\vee\rangle+1}v_\mu$ as $\alpha_i$ runs over the simple roots for the affine root system. In your case, the vectors in this set corresponding to finite simple roots all map to zero in the quotient $\mathcal{V}_\lambda$ by its construcion from a finite dimensional irreducible. Therefore the maximal submodule of $\mathcal{V}_\lambda$ is generated by the image of the vector for the affine root, which is the generator you have written down.


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