SIMPLE BOUNDED HIGHEST WEIGHT MODULES OF BASIC CLASSICAL LIE SUPERALGEBRAS
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Springer Science+Business Media New York (2020)
SIMPLE BOUNDED HIGHEST WEIGHT MODULES OF BASIC CLASSICAL LIE SUPERALGEBRAS MARIA GORELIK∗
DIMITAR GRANTCHAROV∗∗
Department of Mathematics The Weizmann Institute of Science Rehovot 7610001, Israel
Department of Mathematics University of Texas, Arlington Arlington, TX 76019, USA
[email protected]
[email protected]
Abstract. We classify all simple bounded highest weight modules of a basic classical Lie superalgebra g. In particular, our result leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of g.
Introduction The representation theory of Lie superalgebras has been extensively studied in the last several decades. Remarkable progress has been made on the study of the (super)category O, see for example [S1] and the references therein. On the other hand, the theory of general weight modules of Lie superalgebras is still at its beginning stage. An important advancement in this direction was made in 2000 in [DMP] where the classification of the simple weight modules with finite weight multiplicities over classical Lie superalgebras was reduced to the classification of the so-called simple cuspidal modules. This result is the superanalog of the Fernando-Futorny parabolic induction theorem for Lie algebras. The classification of the simple cuspidal modules over reductive finite-dimensional simple Lie algebras was completed by Mathieu, [M], following works of Benkart, Britten, Fernando, Futorny, Lemire, Joseph, and others, [BBL], [BL], [F], [Fu], [Jo]. One important result in [M] is that every simple cuspidal module is a twisted localization of a simple bounded highest weight module, where, a bounded module by definition is a module whose set of weight multiplicities is bounded. The maximum weight multiplicity of a bounded module is called the degree of the module. The presentation of the simple cuspidal modules via twisted localization of highest weight modules was extended to the case of classical Lie superalgebras in DOI: 10.1007/S00031-020-09616-x Supported in part by the Minerva foundation with funding from the Federal German Ministry for Education and Research. ∗∗ Supported in part by Simons Collaboration Grant 358245. Received January 4, 2019. Accepted July 4, 2020. Corresponding Author: Dimitar Grantcharov, e-mail: [email protected] 2010 Mathematics Subject Classification. Primary: 17B10. Keywords and phrases: Lie superalgebras, odd reflections, weight modules, character formulas. ∗
MARIA GORELIK, DIMITAR GRANTCHAROV
[Gr]. In this way, the classification of simple weight modules with finite weight multiplicities of a classical Lie superlagebra k is reduced to the classification of the simple bounded highest weight modules of k. The latter modules are easily classified for Lie superalgebras of type I. For Lie superalgebras of type II a classification is obtained for Lie supealgebras o
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