Wide Regular Subalgebras of Symmetrizable Kac-Moody Algebras and an Extension of Schur's Lemma

Abstract

The behavior of representations under restriction is a central theme in Lie theory. We study wide regular subalgebras of symmetrizable Kac-Moody algebras, extending work of Douglas and Repka on semisimple Lie algebras. A subalgebra is wide if every irreducible integrable highest weight module remains indecomposable upon restriction. Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, root system $\Phi$, simple roots $\Pi$, and root space decomposition $\mathfrak{g}=\mathfrak{h}\oplus {⨁}_{\alpha \in \Phi }{\mathfrak{g}}_{\alpha }$. Denote by ${\Phi }_{\mathrm{re}}$ the set of real roots. To a regular subalgebra $\mathfrak{s}$ normalized by $\mathfrak{h}$, we associate a closed subset $T\subseteq \Phi$ by declaring $\alpha \in T$ if $\mathfrak{s}\cap {\mathfrak{g}}_{\alpha }\ne \{0\}$. Our main result is an extension of Schur's lemma: if $\mathfrak{h}\subseteq \mathfrak{s}$ and the real closure of $(T\cup (-T))\cap {\Phi }_{\mathrm{re}}$ contains $\Pi$, then $(\mathrm{End}V{)}^{\mathfrak{s}}=\mathbb{C}{\mathrm{Id}}_V$ for every irreducible integrable highest weight module $V$. As a consequence, this real-root closure condition yields a sufficient condition for wideness. In the affine case, we establish a converse: if $\mathfrak{s}$ is wide, then the closure of $T\cup (-T)$ in $\Phi$ is all of $\Phi$, and this implication holds without assuming that $\mathfrak{h}\subseteq \mathfrak{s}$. A key ingredient is a structural result showing that closed subsets of affine root systems are closed under arbitrary finite root sums that remain roots.