On the Dynkin index of a principal ${\mathfrak{sl}}_2$-subalgebra

Abstract

Let $g$ be a simple Lie algebra over an algebraically closed field of characteristic zero. The goal of this note is to prove a closed formula for the Dynkin index of a principal $s{l}_2$-subalgebra of $g$. The key step in the proof uses the "strange formula" of Freudenthal--de Vries. As an application, we (1) compute the Dynkin index any simple $g$-module regarded as $s{l}_2$-module and (2) obtain an identity connecting the exponents of $g$ and the dual Coxeter numbers of both $g$ and the Langlands dual $g^{\vee }$.

6 pages, to appear in "Advances in Math"