Parabolic induction for Takiff algebras and character formulas in Chari's category $\tilde{\mathcal{O}}$

Abstract

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, and let ${\mathcal{C}}_N\mathfrak{g}=\mathfrak{g}\otimes (\mathbb{C}[t]/t^{N+1}\mathbb{C}[t])$ denote the $N$-th Takiff algebra of $\mathfrak{g}$. We introduce a notion of parabolic subalgebras of ${\mathcal{C}}_N\mathfrak{g}$ and classify them in terms of $(N+2)$-partitions of a simple root system of $\mathfrak{g}$. We then study the corresponding parabolically induced ${\mathcal{C}}_N\mathfrak{g}$-modules and establish a necessary and sufficient criterion for their irreducibility, extending Jantzen's criterion for parabolically induced $\mathfrak{g}$-modules. As an application, we derive character formulas for almost all irreducible Harish-Chandra modules in Chari's category $\tilde{\mathcal{O}}$ over the (centerless) affine Lie algebra associated with $\mathfrak{g}$. In particular, when $\mathfrak{g}={\mathfrak{sl}}_3(\mathbb{C})$, we obtain explicit character formulas for all irreducible Harish-Chandra modules in this category.