Harish-Chandra modules for map and affine Lie superalgebras
Lucas Calixto, Vyacheslav Futorny, Henrique Rocha
TL;DR
The paper classifies simple Harish-Chandra modules over map superalgebras $\mathcal{G}=\mathfrak{g}\otimes A$ with $\mathfrak{g}$ basic classical, showing every simple HC module is either cuspidal bounded or parabolically induced from a cuspidal bounded module over a cuspidal Levi subalgebra; moreover, cuspidal bounded modules are evaluation modules. It introduces a tensor product framework for infinite-dimensional settings and a shadow-based parabolic induction mechanism to reduce classification to more tractable subcategories. Extending to affine Kac-Moody Lie superalgebras, it proves a bijection between simple finite weight modules for loop algebras and their affine extensions with trivial central action, and for type I, a Kac induction argument reduces the bounded HC-module classification to the even part, yielding a practical reduction to well-understood components. Overall, the results unify and extend known classifications by showing that, in these super settings, simple bounded and cuspidal modules can be effectively described via evaluation modules and even-part reductions, enabling concrete descriptions in key families such as loop, affine, and type I algebras.
Abstract
We obtain a classification of simple modules with finite weight multiplicities over basic classical map superalgebras. Any such module is parabolic induced from a simple cuspidal bounded module over a cuspidal map superalgebra. Further on, any simple cuspidal bounded module is isomorphic to an evaluation module. As an application, we obtain a classification of all simple Harish-Chandra modules for basic classical loop superalgebras. Extending these results to affine Kac-Moody Lie superalgebras obtained by adding the degree derivation, we construct a family of bounded simple modules of level zero, and conjecture that all bounded simple cuspidal modules belong to this family. Finally, we show that for affine Kac-Moody Lie superalgebras of type I the Kac induction functor reduces the classification of all simple bounded modules to the classification of the same class of modules over the even part, whose classification is claimed by Dimitrov and Grantcharov.